Worked Example Library

T1 · Linear FunctionsLesson 1 · What is a function?

Let's take something boring and turn it into math. Your cell plan charges $25 a month for the base fee, plus $0.10 for every minute you go over your data. Total cost depends on how many overage minutes you rack up.

"Find a function that gives the total monthly bill based on overage minutes, then use it to predict the bill for 0, 50, and 200 minutes."

Step 1: Identify the parts.

Two ingredients: a fixed piece (the $25 base fee that you pay no matter what) and a variable piece ($0.10 times however many minutes). The minutes are what change, that's our input.
→ x = overage minutes
Step 2: Translate the words into a rule.

"Total = base fee + (rate × minutes)" becomes:
$f(x) = \htmlClass{we-k1}{25} + \htmlClass{we-k2}{0.10}\htmlClass{we-k3}{x}$
Step 3: Plug in different inputs.

To "evaluate" a function at a value, we substitute the value for x and simplify.

Input Substitute Output

x = 0 25 + 0.10(0) $25.00
x = 50 25 + 0.10(50) $30.00
x = 200 25 + 0.10(200) $45.00
Step 4: Read it back in English.

"With 50 overage minutes, the bill is $30." Notice that the function never gives two answers for the same input, 50 minutes always produces $30. That's the "exactly one output" rule in action.
→ that's what makes it a function

Input	Substitute	Output
x = 0	25 + 0.10(0)	$25.00
x = 50	25 + 0.10(50)	$30.00
x = 200	25 + 0.10(200)	$45.00

Answerf(x) = 25 + 0.10x · $25, $30 and $45

Watch: What, Exactly, Is a Function? · 3:27 Practise: Lesson 1 quick check Try it: Interactive Line Graph Explorer The full lesson

T1 · Linear FunctionsLesson 2 · Two ways to reason

Your dorm Wi-Fi keeps dropping. Watch how you naturally use both kinds of reasoning to solve it, induction first, then deduction once you've found the rule.

"The Wi-Fi cuts out at random. Or does it? You're going to figure out why, and predict when it'll happen next."

1
Step 1: Gather observations.

You jot down every time the Wi-Fi has dropped this week. No theories yet, just the facts.
- Monday, dropped at 7:02 PM
- Tuesday, dropped at 7:00 PM
- Wednesday, dropped at 7:05 PM
→ this is your data
Step 2: Spot a pattern (induction).

Three nights, same time. You generalize: "the Wi-Fi drops around 7 PM." That's a conjecture, a pattern-based guess. It might be wrong, but it's a strong starting point.
→ specific cases → general pattern
Step 3: Find the actual rule.

You ask the RA. Turns out: at 7 PM, 200 students start streaming Netflix simultaneously, overloading the dorm router. Now you have a real rule, not just a pattern: "router overloads when too many devices stream at once."
Step 4: Predict the future (deduction).

It's Thursday at 6:55 PM. You apply the rule: "if 200 students stream at 7 PM, the router will overload." You don't have to wait and see, you can deduce it. At 7:01 PM, sure enough, Wi-Fi dies.
→ rule + case → certain conclusion
Step 5: The takeaway.

Induction got you close, you spotted the pattern and made a useful conjecture. Deduction got you certain, once you knew the actual rule, the answer was guaranteed. Most real problem-solving uses both, in this exact order.

AnswerInduction spots the pattern; deduction makes it certain

Watch: The Two Ways Humans Reason · 3:09 Practise: Lesson 2 quick check The full lesson

T1 · Linear FunctionsLesson 3 · Close enough

You've got eight items in your cart. You'd like to know roughly what they'll add up to before you reach the register, for budgeting, for sanity, for not being surprised. Watch how rounding turns a hopeless mental-math problem into a 10-second one.

"You have 8 items in your cart with the prices listed below. Estimate the total before checkout, then compare against the actual receipt."

Step 1: Pick a place value to round to.

For dollars and cents, the easiest target is the nearest dollar. That gets rid of the cents entirely and leaves you with whole numbers. easy to add in your head.
→ round to the nearest $1
Step 2: Round each item.

Look at the cents. If 50¢ or more, round up; otherwise, round down.

Actual Rounded

$3.49 $3
$7.89 $8
$2.19 $2
$5.99 $6
$1.29 $1
$8.49 $8
$4.79 $5
$6.99 $7
Step 3: Add the rounded numbers in your head.

Group them however feels easiest. 3 + 8 = 11, 2 + 6 = 8, 1 + 8 = 9, 5 + 7 = 12. Now add the four sums: 11 + 8 + 9 + 12 = 40.
$Estimate \approx $40$
Step 4: Compare to reality.

Actual receipt total: $41.12. Your $40 estimate was off by about a dollar, on a $40 bill, after roughly 10 seconds of mental math. That's the magic of rounding to compatible numbers: small individual errors tend to cancel out when you sum them.
→ if the receipt had said $89, you'd know to look closer

Actual	Rounded
$3.49	$3
$7.89	$8
$2.19	$2
$5.99	$6
$1.29	$1
$8.49	$8
$4.79	$5
$6.99	$7

AnswerEstimate ≈ $40 (receipt: $41.12)

Watch: Close Enough Is a Superpower · 3:13 Practise: Lesson 3 quick check The full lesson

T1 · Linear FunctionsLesson 4 · Slope as a rate of change

You've been working a part-time job and stashing what you can. You checked your balance twice. Find the slope, your savings rate.

"On day 10, your account had $180. On day 30, it had $480. What's the slope of your savings, your rate of saving in dollars per day?"

Step 1: Write the two points as ordered pairs.

Time goes on the x-axis (it's the input), money on the y-axis (it depends on time). Translate the sentences into points.
$(\htmlClass{we-k3}{10},\ \htmlClass{we-k4}{180}) \quad \text{ and } \quad (\htmlClass{we-k3}{30},\ \htmlClass{we-k4}{480})$ → x = days, y = dollars
Step 2: Label them so the formula doesn't trip you up.

Pick either point as (x₁, y₁). The other becomes (x₂, y₂). It doesn't matter which you pick, as long as you're consistent on both top and bottom.

Label Value

x₁ 10
y₁ 180
x₂ 30
y₂ 480
Step 3: Plug into the formula.

Slope is rise over run, or $\htmlClass{we-k2}{m} = \frac{\htmlClass{we-k4}{y_2 - y_1}}{\htmlClass{we-k3}{x_2 - x_1}}$.
$\htmlClass{we-k2}{m} = \frac{\htmlClass{we-k4}{480} - \htmlClass{we-k4}{180}}{\htmlClass{we-k3}{30} - \htmlClass{we-k3}{10}} = \frac{300}{20} = \htmlClass{we-k2}{15}$
Step 4: Translate it back to English.

The slope is 15, but 15 what? Look at your axes: y was dollars, x was days. So the slope is 15 dollars per day. You're saving about $15 each day. That's what slope means, never just the number, always the units.
→ slope = rate, with units attached

Label	Value
x₁	10
y₁	180
x₂	30
y₂	480

Answerm = $15 per day

Watch: Slope Is Just "How Fast" · 3:01 Practise: Lesson 4 quick check Try it: Interactive Line Graph Explorer The full lesson

T1 · Linear FunctionsLesson 5 · Linear modeling and prediction

Back to that study-abroad fund from the hook. You have $200 saved today, and you put away $75 each week. Will you hit your $1,800 goal by week 24? Let's build the model and find out.

"You currently have $200 saved. You save $75 per week. Build a model for your savings, then predict how much you'll have at week 24, and decide whether you'll hit $1,800 in time."

Step 1: Identify the slope and the y-intercept.

Read the situation slowly. The per word tells you the slope. The starting amount is the y-intercept, what's true when zero weeks have passed.

Phrase	What it means
"$75 per week"	m = 75 (slope, dollars per week)
"$200 today"	b = 200 (y-intercept, starting savings)

→ x = weeks, y = total savings

Step 2: Plug them into y = mx + b.

Now it's just substitution. The slope replaces m; the y-intercept replaces b.
$y = \htmlClass{we-k2}{75}\htmlClass{we-k3}{x} + \htmlClass{we-k1}{200}$ → that's your linear model
Step 3: Sanity-check before predicting.

Try a value you can verify. At x = 0 (today), the model predicts y = 75(0) + 200 = $200. ✓ matches your current savings. At x = 1 (next week), y = 75(1) + 200 = $275, that's $200 plus one week of $75. ✓ Both check out, so the model is set up correctly.
Step 4: Predict at week 24.

Plug x = 24 into the model.
You'll have $2,000 by week 24, comfortably past your $1,800 goal. In fact, you can flip the question: solving 1,800 = 75x + 200 gives x ≈ 21.3 weeks, so you'll cross the finish line in about week 22.
$y = \htmlClass{we-k2}{75}(\htmlClass{we-k3}{24}) + \htmlClass{we-k1}{200} = 1{,}800 + 200 = \$2{,}000$ → that's a prediction, not a guess

Answery = 75x + 200 · $2,000 by week 24, so yes

Watch: One Equation. Predict the Future. · 3:02 Practise: Lesson 5 quick check Try it: Interactive Line Graph Explorer The full lesson

T1 · Linear FunctionsLesson 6 · Spreadsheets, the finale

A driver-ed instructor records a car's stopping distance at six different speeds. We'll use Excel to discover the linear model behind the data. using =SLOPE() and =INTERCEPT(), then chart it, predict at new speeds, and decide whether the model holds up.

"Given six (speed, distance) data points, find the linear model d = ms + b. Predict the stopping distance at a new speed, plot the data with a trendline, and interpret what the slope and intercept mean in real life."

Step 1: Enter the data.

Set up two columns: speed in column B (the input, what you control) and distance in column C (the output, what you measure). The headers live in row 16, the six data points in rows 17 through 22.

A

B

C

D

16

speed (m/s)

distance (m)

17

10

8

18

15

22

19

20

30

20

25

46

21

30

55

22

35

70

→ s is the input, d is the output
Step 2: Use =SLOPE() to find m.

Excel has a built-in function that computes the slope of the best-fit line straight from your data, no formula derivation needed. In an empty cell, type:
$=SLOPE(C17:C22, B17:B22)$ → Excel returns m ≈ 2.43
Step 3: Use =INTERCEPT() to find b.

Same idea, same argument order: y-values first, x-values second.
$=INTERCEPT(C17:C22, B17:B22)$
Excel returns b ≈ -16.14. Format both cells to two decimal places, the rubric will check.

D26 f_x =SLOPE(C17:C22, B17:B22)

B

C

D

E

25

Intercept (b)

-16.14

26

Slope (m)

2.43
Step 4: Write the equation. Predict at new speeds.

Plugging the values into y = mx + b. but using context-appropriate variables, since this isn't a generic graph but a real situation:
$d = 2.43 s - 16.14$
Now use the equation to predict stopping distance at speeds the data didn't include. At 40 m/s (about 90 mph):
$d = 2.43 (40) - 16.14 = 97.20 - 16.14 \approx 81 m$
Roughly the length of an Olympic swimming pool. In Excel, you'd put predicted speeds in column E and the prediction formula =2.43*E17-16.14 in column F, then drag down.
Step 5: Make a scatter plot. Add a trendline.

Select your data range (B16:C22). Click Insert → Scatter, pick the X-Y points version, not the connected line. Then right-click any data point in the chart, choose Add Trendline, select Linear, and check Display Equation on chart. Excel draws the best-fit line and prints the equation right next to it.

The trendline runs through the cloud of data points, exactly the same equation Excel computed in steps 2 and 3. Two paths to the same answer, which is a useful sanity check.
Step 6: Interpret the slope and intercept.

The numbers don't matter unless you can translate them back to the real world.
Slope (m = 2.43): for every 1 m/s increase in speed, stopping distance increases by 2.43 m. That makes sense, go faster, need more distance to stop. ✓
Y-intercept (b = −16.14): at 0 m/s, the model predicts −16.14 m of stopping distance. That does not make physical sense. A car that isn't moving stops in 0 meters, not negative ones.
So is the model wrong? Not quite, it's a linear approximation that fits the data well between 10 and 35 m/s but breaks down outside that range. This is the extrapolation warning from Lesson 5: trust your model inside the data, be skeptical outside it.
→ slope: makes sense. intercept: does not.

Answerd = 2.43s − 16.14 · about 81 m at 40 m/s

Watch: A Spreadsheet Is a Grid of Functions · 3:18 Practise: Lesson 6 quick check Try it: Excel Fundamentals The full lesson

T2 · Conversions & BudgetingLesson 1 · What is a percent, really?

Three steps. Same three steps every percent problem uses.

"You're at the register with a $40 shirt. Sales tax is 7%. What will you actually pay at checkout?"

Step 1: Convert the percent to a decimal.

The % symbol means "divide by 100." To strip the symbol, shift the decimal point two places to the left.
$\htmlClass{we-k1}{7\%} = \frac{7}{100} = \htmlClass{we-k1}{0.07}$
Step 2: Multiply the rate by the base.

The base is the price you're being taxed on, $40. Multiply the decimal rate by the base to get the tax amount.
$\htmlClass{we-k1}{0.07} \times \htmlClass{we-k2}{40} = \$2.80$
Step 3: Add the tax to the base.

Sales tax sits on top of the price, so the total at the register is base + tax.
$\htmlClass{we-k2}{\$40.00} + \$2.80 = \$42.80$ → that's what you pay
Step 4: Sanity check.

7% is a small slice. The total should be a little more than $40, and it is. If you'd computed $68 or $4, the decimal slipped somewhere. Always pause and ask: "does this answer feel right?"

Answer$42.80 at the register

Watch: Percent Just Means "Out of 100" · 2:25 Practise: Lesson 1 quick check Try it: Percent Sandbox The full lesson

T2 · Conversions & BudgetingLesson 2 · Three masks, one number.

Three forms, three steps. We'll move from the form on the left of the triad to the form on the right.

"You scored 3 out of 4 on a quiz. Express that score as a decimal and as a percent."

Step 1: Fraction to decimal: divide top by bottom.

The fraction $\tfrac{3}{4}$ literally means "3 divided by 4." Just do the division.
$\htmlClass{we-k4}{3} \div \htmlClass{we-k2}{4} = \htmlClass{we-k1}{0.75}$
Step 2: Decimal to percent: multiply by 100.

To turn a decimal into a percent, multiply by 100, or shift the decimal point two places to the right.
$\htmlClass{we-k1}{0.75} \times 100 = \htmlClass{we-k1}{75\%}$
Step 3: All three forms together.

$\tfrac{3}{4}$, 0.75, and 75\% are three notations for the same value. Use whichever one fits the context.
$\frac{\htmlClass{we-k4}{3}}{\htmlClass{we-k2}{4}} = \htmlClass{we-k1}{0.75} = \htmlClass{we-k1}{75\%}$ → same value, three masks
Step 4: Sanity check.

$\tfrac{3}{4}$ should be more than half (0.5 / 50%) and less than the whole (1.0 / 100%). 0.75 and 75% both fall in that range. Good.

Answer3/4 = 0.75 = 75%

Watch: One Number, Three Disguises · 2:59 Practise: Lesson 2 quick check Try it: Percent Sandbox The full lesson

T2 · Conversions & BudgetingLesson 3 · Percent in the wild.

Two percent moves stacked on one purchase. Take the discount first, then tax the discounted price. Order matters.

"You grab an $80 jacket on sale at 25% off. At the register, sales tax is 7%. What's the final total?"

Step 1: Compute the discount amount.

The discount is 25% of the original price. Convert 25% to a decimal (0.25), then multiply by $80.
$\htmlClass{we-k1}{0.25} \times \htmlClass{we-k2}{80} = \$20$
Step 2: Subtract the discount from the base.

Sale price = original − discount.
$\htmlClass{we-k2}{\$80} - \$20 = \$60$
Step 3: Compute the sales tax (on the sale price).

Sales tax applies to the sale price, not the original. 7% of $60.
$\htmlClass{we-k4}{0.07} \times 60 = \$4.20$
Step 4: Add the tax to get the total.

Final = sale price + tax.
$\$60 + \$4.20 = \$64.20$ → what you actually pay
Step 5: Sanity check.

Original was $80. Discount takes it down to $60. Tax adds a few dollars back. The final answer of $64.20 is between $60 and $80, exactly where it should be.

Answer$64.20 total

Watch: Tip, Tax, Discount: One Move · 3:41 Practise: Lesson 3 quick check Try it: Receipt Stacker The full lesson

T2 · Conversions & BudgetingLesson 4 · Multiply by 1, in disguise.

Same five steps for every single-factor conversion.

"You ran 4 miles this morning. Your fitness app wants the distance in feet. How many feet did you run?"

Step 1: Identify the conversion factor.

You need a fraction that relates miles and feet. The known equivalence: 1 mile = 5280 feet. That gives you a conversion factor.
$\frac{\htmlClass{we-k3}{5280\ \text{ft}}}{1\ \text{mi}} = 1$
Step 2: Put the unwanted unit on the bottom.

You're starting with miles and you want to end with feet. So miles is the unwanted unit. Put it on the bottom of the fraction so it cancels with the miles in your starting quantity.
Step 3: Multiply, and watch the units cancel.

Set up the multiplication. Notice the mi on top of "4 mi" and the mi on the bottom of the factor cancel each other out (top × bottom = 1). The ft on top is what survives.
$4 mi \times 5280 ft 1 mi = 4 \times 5280 ft$
Step 4: Compute the number.

4 × 5280 = 21,120. The surviving unit is feet.
$\htmlClass{we-k2}{4} \times \htmlClass{we-k3}{5280} = 21{,}120\ \text{ft}$ → that's how far you ran
Step 5: Sanity check.

4 miles is more than 1 mile, so the answer should be more than 5280 ft. It is (21,120 > 5280). And it should be a fairly big number, since feet are tiny compared to miles. 21,120 feels right.

Answer21,120 ft

Watch: Multiply by 1, Convert Anything · 3:14 Practise: Lesson 4 quick check Try it: Cancellation Sandbox The full lesson

T2 · Conversions & BudgetingLesson 5 · Chain the move.

No single conversion factor takes you from miles to inches. Chain two.

"You ran 5 miles. How many inches is that? Use 1 mile = 5280 feet and 1 foot = 12 inches."

Step 1: Plan the chain.

Start unit is mi, target unit is in. There's no direct mi → in factor, but you know mi → ft (5280) and ft → in (12). Stack them.
$mi \to ft \to in$
Step 2: Set up the first factor.

Cancel the mi from "5 mi" by putting mi on the bottom of the first factor, ft on top.
$5 mi \times 5280 ft 1 mi$
Step 3: Set up the second factor.

Cancel the ft that survived the first factor by putting ft on the bottom of the next factor, in on top.
$5 mi \times 5280 ft 1 mi \times 12 in 1 ft$
Step 4: Cancel the units along the chain.

Two pairs cancel: mi in the start meets mi in the first factor's denominator, ft in the first factor's numerator meets ft in the second factor's denominator. Only in is left.
$5 mi \times 5280 ft 1 mi \times 12 in 1 ft$
Step 5: Multiply the numbers.

5 × 5280 × 12. Run the arithmetic.
$\htmlClass{we-k2}{5} \times \htmlClass{we-k3}{5280} \times \htmlClass{we-k4}{12} = 316{,}800\ \text{in}$ → 5 miles in inches

Answer316,800 in

Watch: 60 mph Is How Many Feet per Second? · 3:22 Practise: Lesson 5 quick check Try it: Cancellation Sandbox The full lesson

T2 · Conversions & BudgetingLesson 6 · Tell your money what to do.

Five steps. Each one a small piece of the math you learned this week.

"Your take-home pay is $3,000 a month. Rent is $900, groceries average $450, transport (gas + transit) runs $300, and other expenses (utilities, phone, fun, subscriptions) total $900. What's your savings rate?"

Step 1: Identify the income.

Take-home pay (after tax). For this budget, that's the $3,000 we have to allocate.
$\text{income} = \htmlClass{we-k2}{\$3{,}000}$
Step 2: List the expense categories.

Each category is a line item. Mix of fixed (rent) and variable (groceries, transport, other).

Category Amount % of income

Rent $900 30%
Food $450 15%
Transport $300 10%
Other $900 30%
Step 3: Sum the expenses.

Add the four categories. This is the total monthly spend.
$\htmlClass{we-k3}{\$900} + \htmlClass{we-k3}{\$450} + \htmlClass{we-k3}{\$300} + \htmlClass{we-k3}{\$900} = \$2{,}550$
Step 4: Compute savings.

Savings is what's left after expenses.
$\htmlClass{we-k2}{\$3{,}000} - \$2{,}550 = \$450$ → left over each month
Step 5: Compute the savings rate.

Savings rate is savings as a percent of income. Same percent shape from Lesson 1: $\frac{\text{part}}{\text{base}}$, then $\times 100$.
$\frac{450}{\htmlClass{we-k2}{3{,}000}} \times 100 = \htmlClass{we-k1}{15\%}$

Category	Amount	% of income
Rent	$900	30%
Food	$450	15%
Transport	$300	10%
Other	$900	30%

Answer$450 saved · 15% savings rate

Watch: Tell Your Money Where to Go · 2:52 Practise: Lesson 6 quick check The full lesson

T3 · SavingsLesson 1 · What interest actually is.

Four steps. Same four steps every interest problem uses, no matter which formula comes next.

"Your friend asks to borrow $500 to fix their transmission. You agree on 6% annual interest, paid back in two years. How much will they owe you at the end?"

Step 1: Name the four pieces.

P = $500 (the principal you lent). r = 6% (the rate). t = 2 years (the term). A = ? (what you'll get back, principal plus interest).
Step 2: Convert the percent to a decimal.

The % symbol means "divide by 100." Shift the decimal two places to the left.
$\htmlClass{we-k1}{6\%} = \frac{6}{100} = \htmlClass{we-k1}{0.06}$
Step 3: Compute the interest.

Interest equals principal times rate times time. Multiply across.
$I = \htmlClass{we-k2}{P} \times \htmlClass{we-k1}{r} \times \htmlClass{we-k3}{t} = \htmlClass{we-k2}{500} \times \htmlClass{we-k1}{0.06} \times \htmlClass{we-k3}{2} = \$60$
Step 4: Add interest back to principal.

The future value is what your friend hands back: the original $500 plus the $60 of interest that accrued.
$A = \htmlClass{we-k2}{P} + I = \htmlClass{we-k2}{500} + 60 = \$560$ → that's the full payback
Step 5: Sanity check.

6% of $500 is about $30 per year. Over two years, around $60 of interest. The total ($560) is a little more than the principal ($500). If you'd computed $830 or $20, the decimal slipped somewhere. Always pause and ask: does this answer feel right?

Answer$560 owed ($60 interest)

Watch: What Is Interest? It's Rent on Money · 3:14 Practise: Lesson 1 quick check Try it: Financial Formulas Calculator The full lesson

T3 · SavingsLesson 2 · Simple interest, the warmup formula.

Three steps. Same recipe as Lesson 1, with the fractional time spelled out.

"A 26-week Treasury bill has a face value of $10,000 and pays 4.5% simple interest. How much interest does it earn at maturity, and what's the total payout?"

Step 1: Convert the time to years.

The formula expects t in years. 26 weeks is six months — half a year.
$\htmlClass{we-k3}{t} = \frac{6}{12} = \htmlClass{we-k3}{0.5}\ \text{years}$
Step 2: Convert the percent to a decimal.

Same step as Lesson 1. Drop the %, shift the decimal two places left.
$\htmlClass{we-k1}{4.5\%} = \frac{4.5}{100} = \htmlClass{we-k1}{0.045}$
Step 3: Apply I = Prt.

Multiply principal by rate by time. All three pieces are now in compatible units: dollars, decimal rate, years.
$I = \htmlClass{we-k2}{10{,}000} \times \htmlClass{we-k1}{0.045} \times \htmlClass{we-k3}{0.5} = \$225$
Step 4: Find the accumulated amount.

Either add interest to principal (A = P + I), or use the factored form A = P(1 + rt). Both give the same answer.
$A = \htmlClass{we-k2}{P} + I = \htmlClass{we-k2}{10{,}000} + 225 = \$10{,}225$ → payout at maturity
Step 5: Sanity check.

4.5% per year on $10,000 is about $450 annually. Over half a year, around $225. The math agrees. If you'd computed $4,500 or $22.50, the decimal slipped on either the rate or the time.

AnswerI = $225 · A = $10,225

Watch: Simple Interest Explained: I = Prt in One Multiply · 3:07 Practise: Lesson 2 quick check Try it: Financial Formulas Calculator The full lesson

T3 · SavingsLesson 3 · Compound interest, and where it leads.

Same principal, same rate, same time. Four compounding frequencies. Watch the answers converge.

"Deposit $1,000 at 5% APR for 10 years. What's the future value when compounded annually, monthly, daily, or continuously?"

Step 1: Annual compounding (n = 1).

Match each number to its letter, then plug into $A = \htmlClass{we-k2}{P}\left(1 + \frac{\htmlClass{we-k1}{r}}{\htmlClass{we-k4}{n}}\right)^{\htmlClass{we-k4}{n}\htmlClass{we-k3}{t}}$ with n = 1 (once a year).
$\begin{aligned} A &= \htmlClass{we-k2}{1{,}000}\left(1 + \frac{\htmlClass{we-k1}{0.05}}{\htmlClass{we-k4}{1}}\right)^{\htmlClass{we-k4}{1} \times \htmlClass{we-k3}{10}} \\ &= \htmlClass{we-k2}{1{,}000}\,(1.05)^{10} \\ &= \$1{,}628.89 \end{aligned}$
Step 2: Monthly compounding (n = 12).

Same formula, n=12. The exponent is n × t = 12 × 10 = 120 months.
$\begin{aligned} A &= \htmlClass{we-k2}{1{,}000}\left(1 + \frac{\htmlClass{we-k1}{0.05}}{\htmlClass{we-k4}{12}}\right)^{\htmlClass{we-k4}{12} \times \htmlClass{we-k3}{10}} \\ &= \htmlClass{we-k2}{1{,}000}\,(1.004167)^{120} \\ &= \$1{,}647.01 \end{aligned}$
Step 3: Daily compounding (n = 365).

n=365 means interest is added once per day. The exponent is 365 × 10 = 3,650 days.
$A = \htmlClass{we-k2}{1{,}000}\left(1 + \frac{\htmlClass{we-k1}{0.05}}{\htmlClass{we-k4}{365}}\right)^{\htmlClass{we-k4}{365} \times \htmlClass{we-k3}{10}} = \$1{,}648.66$
Step 4: Continuous compounding.

Use $A = Pe^{rt}$. No n; the formula is its own limit case.
$A = \htmlClass{we-k2}{1{,}000}\,e^{\htmlClass{we-k1}{0.05} \times \htmlClass{we-k3}{10}} = \htmlClass{we-k2}{1{,}000}\,e^{0.5} \approx \$1{,}648.72$ → the ceiling
Step 5: Compare and sanity-check.

Four answers: $1,628.89, $1,647.01, $1,648.66, $1,648.72. The continuous case is the largest, but only by about $20 over the annual case. Bigger n always wins, but by less than you'd think. The big lift on this $1,000 came from compound vs. simple interest ($1,648.72 vs. $1,500), not from picking daily over monthly.

Answer$1,628.89 · $1,647.01 · $1,648.66 · $1,648.72

Watch: Compound Interest Explained: Interest That Earns Interest · 3:47 Practise: Lesson 3 quick check Try it: Compound Interest Lab The full lesson

T3 · SavingsLesson 4 · The rate you really earn.

Two accounts with the same headline rate but different compounding schedules. APY makes the comparison honest.

"Bank A offers 5% APR compounded annually. Bank B offers 5% APR compounded daily. Compute each account's APY and decide which one actually pays more in a year."

Step 1: Identify r and n for each.

Both have r = 0.05. Bank A: n = 1 (annual). Bank B: n = 365 (daily).
Step 2: Compute Bank A's APY.

With n = 1, the formula collapses: APY equals the APR exactly.
$\begin{aligned} Y_A &= \left(1 + \frac{\htmlClass{we-k1}{0.05}}{\htmlClass{we-k4}{1}}\right)^{\htmlClass{we-k4}{1}} - 1 \\ &= 1.05 - 1 = 0.05 = 5.00\% \end{aligned}$
Step 3: Compute Bank B's APY.

With n = 365, the compounding effect bumps the actual yield slightly above the headline 5%.
$\begin{aligned} Y_B &= \left(1 + \frac{\htmlClass{we-k1}{0.05}}{\htmlClass{we-k4}{365}}\right)^{\htmlClass{we-k4}{365}} - 1 \\ &\approx 1.05127 - 1 = 0.05127 = 5.13\% \end{aligned}$
Step 4: Compare and pick a winner.

Bank B pays 5.13% APY vs. Bank A's 5.00% APY. Same advertised 5%, but daily compounding buys you an extra 0.13 percentage points per year. On a $10,000 deposit, that's an extra $13 the first year.
The pattern: APY is always ≥ APR, and the gap grows with n. Annual: APY = APR. Continuous: $\text{APY} = e^{r} - 1$. Everything else lives in between.

AnswerBank B: 5.13% APY beats 5.00%

Watch: APR vs APY Explained: The Rate You Actually Earn · 3:50 Practise: Lesson 4 quick check Try it: Compound Interest Lab The full lesson

T3 · SavingsLesson 5 · Lump sum vs. annuity.

A typical retirement-contribution scenario worked step-by-step. The hard part isn't the formula; it's the order of operations inside it.

"You contribute $500 a month to an investment account paying 6% APR compounded monthly. After 20 years, what's the balance?"

Step 1: Identify the four pieces.

M = $500 (each monthly deposit). r = 0.06 (the annual rate, as a decimal). n = 12 (compounding frequency, monthly). t = 20 (years).
Step 2: Compute the periodic rate.

The annuity formula needs $\frac{r}{n}$ (the rate per compounding period), not just r.
$\frac{\htmlClass{we-k1}{r}}{\htmlClass{we-k4}{n}} = \frac{\htmlClass{we-k1}{0.06}}{\htmlClass{we-k4}{12}} = 0.005$
Step 3: Compute the exponent.

The exponent is n × t — total number of compounding periods.
$\htmlClass{we-k4}{n}\htmlClass{we-k3}{t} = \htmlClass{we-k4}{12} \times \htmlClass{we-k3}{20} = 240$
Step 4: Plug into the annuity formula.

Apply $A = \htmlClass{we-k2}{M}\,\frac{\left(1 + \frac{\htmlClass{we-k1}{r}}{\htmlClass{we-k4}{n}}\right)^{\htmlClass{we-k4}{n}\htmlClass{we-k3}{t}} - 1}{\frac{\htmlClass{we-k1}{r}}{\htmlClass{we-k4}{n}}}$ with the values from above. Compute the parenthesis first, then subtract 1, then divide, then multiply by M.
$\begin{aligned} (1.005)^{240} &\approx 3.31020 \\ A &= \htmlClass{we-k2}{500} \times \frac{3.31020 - 1}{0.005} \\ &= \htmlClass{we-k2}{500} \times \frac{2.31020}{0.005} = \$231{,}020 \end{aligned}$ → final balance
Step 5: Sanity check — what did the formula do?

You contributed $500 × 12 × 20 = $120,000 over 20 years. The account is at $231,020. The formula nearly doubled your money — the extra $111,020 is interest. If you'd left the same $120,000 sitting in cash, you'd still have $120,000. That's the difference compounding makes.

Answer≈ $231,020 after 20 years

Watch: What Is an Annuity? Many Payments, One Formula · 3:44 Practise: Lesson 5 quick check Try it: Retirement Planner The full lesson

T3 · SavingsLesson 6 · Build a savings plan.

Two people, same monthly contribution, same rate, same retirement age. The only difference is when they started. Watch what 10 years does.

"Person A starts contributing $300/month to a retirement account at age 25 and stops at age 65. Person B does the same thing but starts at age 35. Both earn 7% APR compounded monthly. Both retire at 65. How much does each finish with?"

Step 1: Set up Person A.

M = $300, r = 0.07, n = 12, t = 40 years. Total contributions: 300 × 12 × 40 = $144,000.
Step 2: Compute Person A's future value.

Annuity formula. The exponent is $12 \times 40 = 480$; the periodic rate is $\frac{0.07}{12} \approx 0.005833$.
$A_A = \htmlClass{we-k2}{300} \times \frac{(1.005833)^{\htmlClass{we-k3}{480}} - 1}{0.005833} \approx \$787{,}320$ → 40 years of saving
Step 3: Set up Person B.

M = $300, r = 0.07, n = 12, t = 30 years. Total contributions: 300 × 12 × 30 = $108,000. Person B put in only $36,000 less than Person A — three years' worth of contributions.
Step 4: Compute Person B's future value.

Same formula, different exponent (360 instead of 480).
$A_B = \htmlClass{we-k2}{300} \times \frac{(1.005833)^{\htmlClass{we-k3}{360}} - 1}{0.005833} \approx \$365{,}970$ → 30 years of saving
Step 5: Compare the gap.

Person A: $787,320. Person B: $365,970. The 10 extra years cost Person B over $421,000 in final balance — even though they only contributed $36,000 less in deposits.
That gap, $421,000 vs. $36,000 of skipped contributions, is the cost of starting late. Time is the biggest lever in long-term saving — bigger than the monthly amount, bigger than the rate.

AnswerA ≈ $787,320 · B ≈ $365,970

Watch: Future Value of an Annuity: $200 a Month Becomes $244,000 · 4:00 Practise: Lesson 6 quick check Try it: Retirement Planner The full lesson

T4 · LoansLesson 1 · What a loan is, and what amortization means.

Before plugging into any formula, get fluent at spotting the four pieces in a loan scenario.

"Maria buys a $30,000 car. She makes a $5,000 down payment and finances the rest with a 5-year loan at 6% APR. The bank tells her the monthly payment is $483.32. What's the total cost of the car including interest?"

Step 1: Identify the principal P (loan amount).

The sticker price is $30,000, but the loan only finances what's left after the down payment.
$\htmlClass{we-k2}{P} = \htmlClass{we-k2}{\$30{,}000} - \htmlClass{we-k4}{\$5{,}000} = \$25{,}000$
Step 2: Identify rate, term, monthly payment.

r = 0.06 (6% as a decimal). t = 5 (years). M = $483.32 (given by the bank — Lesson 2's formula will produce this from P, r, and t).
Step 3: Compute total paid.

Total paid = monthly payment × number of months. 5 years × 12 months/year = 60 monthly payments.
$\text{total paid} = M \times 12 \times \htmlClass{we-k3}{t} = 483.32 \times 60 = \$28{,}999.20$
Step 4: Compute total interest.

Total interest = total paid − loan amount. Note: this is interest on the amount financed, not the sticker price. Maria's $5,000 down payment didn't earn the bank any interest.
$\text{total interest} = \$28{,}999.20 - \$25{,}000 = \$3{,}999.20$ → ~$4,000 in interest
Step 5: Compute total cost of the car.

This is what most people don't compute: the full out-of-pocket cost including the down payment and all 60 monthly payments.
$\text{total cost} = \htmlClass{we-k4}{\text{down}} + \text{total paid} = \htmlClass{we-k4}{\$5{,}000} + \$28{,}999.20 = \$33{,}999.20$ → $4,000 over sticker, just like the hook said

AnswerTotal cost $33,999.20 ($3,999.20 interest)

Watch: What Is Amortization? Where Loan Payments Actually Go · 3:45 Practise: Lesson 1 quick check Try it: Amortization Sandbox The full lesson

T4 · LoansLesson 2 · The amortization formula, anatomized.

Five steps. The math is straightforward; the trap is typing the formula correctly. Take it slow the first time.

"You take out a $20,000 personal loan at 5% APR with a 4-year term. What's your monthly payment, total cost, and total interest paid?"

Step 1: Identify the inputs.

P = $20,000 (the loan amount). r = 0.05 (5% as a decimal). t = 4 (years).
Step 2: Compute $\tfrac{r}{12}$ (the monthly periodic rate).

The formula needs the rate per month, not per year.
$\frac{\htmlClass{we-k1}{r}}{12} = \frac{\htmlClass{we-k1}{0.05}}{12} \approx \htmlClass{we-k1}{0.004167}$
Step 3: Compute the exponent and the bracket.

The exponent is −12t. For 4 years, that's −48. The bracket $1 - \left(1 + \tfrac{r}{12}\right)^{-12t}$ evaluates as the present-value discount factor.
$\begin{aligned} (1 + \htmlClass{we-k1}{0.004167})^{-12 \times \htmlClass{we-k3}{4}} &\approx 0.81897 \\ 1 - 0.81897 &\approx 0.18103 \end{aligned}$
Step 4: Plug into the formula.

Numerator is P × $\left(\tfrac{r}{12}\right)$. Denominator is the bracket from step 3. Divide.
$M = \frac{\htmlClass{we-k2}{20{,}000} \times \htmlClass{we-k1}{0.004167}}{0.18103} = \frac{83.33}{0.18103} \approx \$460.59$ → monthly payment
Step 5: Compute total cost and total interest.

Once you have M, the rollup numbers are easy.
$\begin{aligned} \text{total paid} &= M \times 12 \times \htmlClass{we-k3}{t} = 460.59 \times 48 = \$22{,}108.12 \\ \text{total interest} &= \text{total} - \htmlClass{we-k2}{P} = \$22{,}108.12 - \htmlClass{we-k2}{\$20{,}000} = \$2{,}108.12 \end{aligned}$ → ~$2,100 of interest over 4 years

AnswerM ≈ $460.59 · ≈ $2,108 interest

Watch: How to Calculate a Loan Payment (Monthly Payment Formula) · 3:44 Practise: Lesson 2 quick check Try it: Amortization Sandbox The full lesson

T4 · LoansLesson 3 · Auto loans: the standard amortizing loan.

End-to-end: from sticker price to total out-of-pocket cost. Five steps.

"Maria buys a $35,000 SUV. She makes a 15% down payment and finances the rest with a 5-year auto loan at 5.5% APR. What's her monthly payment, total interest paid, and total cost of the SUV?"

Step 1: Compute the down payment.

15% of the sticker price.
$\htmlClass{we-k4}{\text{down payment}} = \htmlClass{we-k4}{0.15} \times \htmlClass{we-k2}{\$35{,}000} = \htmlClass{we-k4}{\$5{,}250}$
Step 2: Compute the loan amount (P).

Sticker minus down payment. This is the P that goes into the amortization formula.
$\htmlClass{we-k2}{P} = \htmlClass{we-k2}{\$35{,}000} - \htmlClass{we-k4}{\$5{,}250} = \htmlClass{we-k2}{\$29{,}750}$
Step 3: Apply the amortization formula.

r = 0.055, t = 5, so $\tfrac{r}{12} \approx 0.004583$ and n = 12t = 60. Plug in.
$M = \frac{\htmlClass{we-k2}{29{,}750} \times \htmlClass{we-k1}{0.004583}}{1 - 1.004583^{-12 \times \htmlClass{we-k3}{5}}} \approx \$568.26$ → monthly payment
Step 4: Total paid over the loan + total interest.

Sum of all 60 monthly payments, then subtract principal for interest.
$\begin{aligned} \text{total paid} &= M \times 60 = \$568.26 \times 60 \approx \$34{,}096 \\ \text{total interest} &= \$34{,}096 - \htmlClass{we-k2}{\$29{,}750} = \$4{,}346 \end{aligned}$
Step 5: Total cost (the number that matters).

Add the down payment back. This is what Maria actually pays out of pocket for the SUV.
$\text{total cost} = \htmlClass{we-k4}{\text{down}} + \text{total paid} = \htmlClass{we-k4}{\$5{,}250} + \$34{,}096 = \$39{,}346$ → ~$4,346 over sticker, all of it interest

AnswerM ≈ $568.26 · total cost ≈ $39,346

Watch: How Car Loans Work: Monthly Payment and Total Interest · 3:47 Practise: Lesson 3 quick check Try it: Amortization Sandbox The full lesson

T4 · LoansLesson 4 · Credit cards: the minimum-payment trap.

Three lines of arithmetic. ALEKS Q2 is exactly this shape with different numbers.

"Your credit card balance is $5,000. The card has a 24% APR. This statement period you make a $200 payment. What's your new balance, and how much of your payment went to interest vs. principal?"

Step 1: Convert APR to a monthly rate.

Credit cards quote APR (annual). For this month's interest charge, you need the monthly rate.
$\htmlClass{we-k1}{\text{monthly rate}} = \frac{\htmlClass{we-k1}{24\%}}{12} = 2\% = \htmlClass{we-k1}{0.02}$
Step 2: Compute this month's interest charge.

The card charges interest on the current balance, before your payment is applied.
$\text{interest} = \htmlClass{we-k2}{\$5{,}000} \times \htmlClass{we-k1}{0.02} = \$100$
Step 3: Compute the new balance.

Add the interest charge to the balance, then subtract the payment. Order matters conceptually but not arithmetically.
$\text{new balance} = \htmlClass{we-k2}{\$5{,}000} + \$100 - \htmlClass{we-k3}{\$200} = \$4{,}900$ → what ALEKS Q2 asks for
Step 4: Split your $200 payment.

Interest portion = the interest charge you just computed. Principal portion = what's left over.
$\begin{aligned} \text{interest portion} &= \$100 \\ \text{principal portion} &= \htmlClass{we-k3}{\$200} - \$100 = \$100 \end{aligned}$ → half your payment was just interest
Step 5: Read the trap.

If you'd paid only $100 (the minimum interest charge), the new balance would be $5,000 + $100 − $100 = $5,000. Same as before. The minimum payment that exactly matches the interest charge keeps the balance frozen forever — which means the bank collects interest from you forever, and you never own the thing you bought.

AnswerNew balance $4,900 · $100 interest, $100 principal

Watch: Credit Card Minimum Payments: Why You Stay in Debt · 3:46 Practise: Lesson 4 quick check Try it: Credit Card Trap The full lesson

T4 · LoansLesson 5 · Student loans: subsidized, unsubsidized, repayment plans.

Same student, same school years, same rate, same 10-year repayment. The only thing that changes is whether interest accrues during the four in-school years. Watch the gap appear.

"A student takes out a $10,000 federal loan freshman year at 6.5% APR. They're in school for 4 years, then begin a 10-year standard repayment plan. Compare the monthly payment, total paid, and total interest if the loan is subsidized versus unsubsidized."

Step 1: Subsidized: in-school years are free.

The government covers interest while the student is enrolled. P at repayment = original loan amount. No T3-style compounding needed.
$P_{\text{sub}} = \htmlClass{we-k2}{\$10{,}000}$ → same number you borrowed
Step 2: Unsubsidized: accrue simple interest.

Interest accrues for 4 years on the $10,000. ALEKS uses the simple-interest formula I = Prt; the result is added to the original principal to get P at repayment.
$\begin{aligned} I &= \htmlClass{we-k2}{P} \times \htmlClass{we-k1}{r} \times \htmlClass{we-k3}{t} = \htmlClass{we-k2}{10{,}000} \times \htmlClass{we-k1}{0.065} \times \htmlClass{we-k3}{4} = \$2{,}600 \\ P_{\text{unsub}} &= \htmlClass{we-k2}{10{,}000} + 2{,}600 = \$12{,}600 \end{aligned}$ → $2,600 of accrued interest piled on top
Step 3: Apply the L2 amortization formula to each P.

r = 0.065, t = 10, so $\tfrac{r}{12} \approx 0.005417$ and n = 120. Same r, same t for both — only P differs.
$\begin{aligned} M_{\text{sub}} &= \frac{\htmlClass{we-k2}{10{,}000} \times \htmlClass{we-k1}{0.005417}}{1 - 1.005417^{-12 \times \htmlClass{we-k4}{10}}} \approx \$113.55 \\ M_{\text{unsub}} &= \frac{12{,}600 \times \htmlClass{we-k1}{0.005417}}{1 - 1.005417^{-12 \times \htmlClass{we-k4}{10}}} \approx \$143.07 \end{aligned}$
Step 4: Total paid + total interest, side by side.

Sum of 120 monthly payments. Subtract the original $10,000 to get total interest (both students borrowed the same amount; the extra is all interest).
$\begin{aligned} \text{Sub:}\ \ \text{total} &= \$113.55 \times 120 \approx \$13{,}626 \quad\cdot\quad \text{interest} = \$3{,}626 \\ \text{Unsub:}\ \ \text{total} &= \$143.07 \times 120 \approx \$17{,}168 \quad\cdot\quad \text{interest} = \$7{,}168 \end{aligned}$
Step 5: The lifetime cost of "unsubsidized."

Subtract sub from unsub. This is what in-school interest accrual costs over the full life of the loan.
$\text{extra cost} = \$17{,}168 - \$13{,}626 \approx \$3{,}543$ → same loan, same school, same job — $3.5K more

AnswerUnsubsidized costs ≈ $3,543 more

Watch: Subsidized vs Unsubsidized Student Loans: The Difference · 3:45 Practise: Lesson 5 quick check Try it: Financial Formulas Calculator The full lesson

T4 · LoansLesson 6 · Mortgages: 30 years, dollar by dollar.

The flagship calculation: monthly payment, total interest paid over the life of the loan, then the first month's principal/interest split.

"You buy a $300,000 home with a 30-year mortgage at 7% APR (no down payment for simplicity). Compute the monthly payment, total cost over 30 years, and the principal/interest split for month 1."

Step 1: Apply the L2 amortization formula.

P = $300,000, r = 0.07, t = 30. So $\tfrac{r}{12} \approx 0.005833$ and n = 12·30 = 360.
$M = \frac{\htmlClass{we-k2}{300{,}000} \times \htmlClass{we-k1}{0.005833}}{1 - 1.005833^{-12 \times \htmlClass{we-k3}{30}}} \approx \$1{,}995.91$ → fixed monthly for the next 360 months
Step 2: Total cost + total interest over the loan.

360 payments at $1,995.91. Then subtract the original principal to isolate interest.
$\begin{aligned} \text{total cost} &= \$1{,}995.91 \times 360 \approx \$718{,}527 \\ \text{total interest} &= \$718{,}527 - \htmlClass{we-k2}{\$300{,}000} \approx \$418{,}527 \end{aligned}$ → more interest than the loan itself
Step 3: Month 1 interest portion.

The bank charges this month's interest on the current balance. At the start, balance = $300,000.
$\text{interest}_1 = \htmlClass{we-k2}{\text{balance}} \times \tfrac{\htmlClass{we-k1}{r}}{12} = \htmlClass{we-k2}{300{,}000} \times \htmlClass{we-k1}{0.005833} = \$1{,}750.00$
Step 4: Month 1 principal portion + new balance.

Whatever's left of M after interest pays down the balance.
$\begin{aligned} \text{principal}_1 &= 1{,}995.91 - 1{,}750.00 = \$245.91 \\ \text{new balance} &= \htmlClass{we-k2}{300{,}000} - 245.91 = \$299{,}754.09 \end{aligned}$ → only 12.3% of the payment built equity
Step 5: Where does it go from here?

Month 2 starts with balance $299,754.09. Run the same arithmetic to see the split begin to shift.
$\begin{aligned} \text{interest}_2 &= 299{,}754.09 \times \htmlClass{we-k1}{0.005833} \approx \$1{,}748.57 \\ \text{principal}_2 &= 1{,}995.91 - 1{,}748.57 \approx \$247.34 \end{aligned}$ → principal share grew by $1.43 in one month

AnswerM ≈ $1,995.91 · month 1: $1,750 interest, $245.91 principal

Watch: How Mortgage Payments Work: Interest vs Principal · 3:55 Practise: Lesson 6 quick check Try it: Amortization Sandbox The full lesson

T5 · StatisticsLesson 1 · What statistics is, really.

A coach measures the heights of every player on the team. The same number can power a descriptive claim or an inferential claim, depending on what we say next. Listen for the leap.

"A high-school basketball coach measures the height of every player on the team. The 15 players average 6 feet 1 inch tall. The coach posts the average on the team bulletin board."

Step 1: Identify the data set, sample, and population.

Data set: the 15 height measurements. Sample: these same 15 players (if we're thinking of them as a sample of something larger). Population: depends on what claim we want to make.
Step 2: Read the descriptive claim.

"The 15 players on this team average 6 feet 1 inch." That is exact. We measured them. The claim is about this team, no leap.
→ descriptive
Step 3: Listen for the inferential leap.

Now the coach tells a reporter: "The average high-school basketball player in our state is about 6 feet 1." Same number, different claim. The coach is using the team as a sample to infer something about the population of all high-school basketball players in the state.
→ inferential
Step 4: Spot the difference.

The descriptive claim cannot be wrong (we measured every player on the team). The inferential claim can be wrong, because the team might not be representative of the state. The honest version of the inferential claim would include a margin of error: "about 6 feet 1, give or take two inches."
→ inferential claims always carry a ±

AnswerDescriptive is exact; inferential needs a ±

Watch: Descriptive vs Inferential Statistics: The Difference · 3:20 Practise: Lesson 1 quick check The full lesson

T5 · StatisticsLesson 2 · Mean, median, mode.

An intelligence test was administered to 10 people. Their completion times in minutes are below. Compute all three measures of center for the same data set. This is the exact shape of ALEKS review Q3.

"The 10 completion times in minutes are: 38, 43, 40, 27, 45, 39, 37, 26, 27, 40. Find the median, the mean, and the mode(s)."

Step 1: Sort the data first.

Every measure benefits from sorted data, but the median requires it. Reorder from smallest to largest:
$26, 27, 27, 37, 38, 39, 40, 40, 43, 45$
Step 2: Median: average the two middle values.

With n = 10, there is no single middle entry. The two middle entries are the 5th (38) and the 6th (39). Average them:
$\htmlClass{we-k2}{\text{median}} = \dfrac{38 + 39}{2} = \htmlClass{we-k2}{38.5}$ → median
Step 3: Mean: sum, divide by count.

Add all ten values. Order doesn't matter for the mean, but a quick re-add of the sorted list reduces typos.
$\begin{aligned} \text{sum} &= 26+27+27+37+38+39+40+40+43+45 = 362 \\ \htmlClass{we-k3}{\text{mean}} &= \dfrac{362}{10} = \htmlClass{we-k3}{36.2} \end{aligned}$ → mean
Step 4: Mode: count each value's frequency.

Scan the sorted list and count repeats. Most values appear once; 27 appears twice, and 40 appears twice. The data set is bimodal.
$\htmlClass{we-k1}{\text{modes}} = \htmlClass{we-k1}{27,\ 40}$ → two modes
Step 5: Sanity check.

Mean (36.2) is below median (38.5). That tells us the smaller values (26, 27, 27) are pulling the mean down more than the larger values are pulling it up. The median, by contrast, doesn't care about the size of those outliers — only their position in the sort.

AnswerMedian 38.5 · mean 36.2 · modes 27 and 40

Watch: Mean, Median, Mode: Which Average Should You Use? · 3:59 Practise: Lesson 2 quick check Try it: Center & Spread Visualizer The full lesson

T5 · StatisticsLesson 3 · Range and standard deviation.

A veterinarian's morning data set from ALEKS Q4. Compute the range by hand in two seconds, then read off the SD that Excel produces. The contrast is the lesson.

"The veterinarian treated 7 dogs this morning. Their weights in pounds: 26, 71, 56, 3, 5, 26, 24. Find the range and the (sample) standard deviation, rounded to one decimal place."

Step 1: Sort the data (optional but helpful).

Sorting makes the max and min obvious and is harmless. The min is the first sorted value; the max is the last.
$3, 5, 24, 26, 26, 56, 71$
Step 2: Range = $\max - \min$.

Read off the two ends of the sorted list and subtract. The units are the data's units: pounds.
$\htmlClass{we-k4}{\text{range}} = 71 - 3 = \htmlClass{we-k4}{68}\ \text{pounds}$ → range
Step 3: Compute the mean (we need it for SD).

SD measures distance from the mean, so we need the mean first. Sum the seven weights, divide by 7.
$\begin{aligned} \text{sum} &= 26+71+56+3+5+26+24 = 211 \\ \htmlClass{we-k3}{\text{mean}} &= \dfrac{211}{7} \approx \htmlClass{we-k3}{30.14}\ \text{pounds} \end{aligned}$
Step 4: SD by Excel (or calculator).

Paste the seven values into a column and ask Excel for the sample standard deviation. The formula is =STDEV.S(A1:A7) if the weights live in cells A1 through A7.
$=STDEV.S(A1:A7) \approx 24.9 pounds$ → sample SD
Step 5: Sanity check.

The range (68) is much larger than the SD (24.9), as it always should be: range captures the extremes, SD captures the typical gap. A rough rule of thumb for moderate sample sizes is $\text{SD} \approx \tfrac{\text{range}}{4}$, which here gives 17 — close enough for a sanity check, though the 71-pound outlier inflates our SD past that estimate.

AnswerRange 68 · SD ≈ 24.9 lb

Watch: What Is Standard Deviation? (Explained Simply) · 4:01 Practise: Lesson 3 quick check Try it: Center & Spread Visualizer The full lesson

T5 · StatisticsLesson 4 · Reading data displays.

A small daily-commute data set lets us see how each chart type would (or would not) display it. Match each pictures' job to its data shape.

"A commuter records her drive time in minutes every weekday for two weeks: 22, 28, 25, 31, 26, 23, 27, 24, 29, 26. Which chart type best answers each of these four questions, and what would the chart show?"

Step 1: Question 1: "How does Monday's commute compare to Friday's?"

This is a comparison between categories (days of the week). Use a bar chart with one bar per day. The bars have visible gaps because the x-axis (days) is categorical, not numeric.
→ bar chart
Step 2: Question 2: "What does the distribution of commute times look like?"

This asks about shape: is the data clustered, spread, bell-shaped, skewed? Use a histogram with bins (say, 20-22, 22-24, 24-26, 26-28, 28-30, 30-32). Each bar's height is the count of commutes that fell in that bin. Bars touch because minutes are continuous.
→ histogram
Step 3: Question 3: "Where's the center and spread?"

For a quick center-and-spread summary, use a box plot. Sorted data: 22, 23, 24, 25, 26, 26, 27, 28, 29, 31. Median = (26+26)/2 = 26. Q1 = 24, Q3 = 28, min = 22, max = 31. The box runs from 24 to 28 with the median line at 26, and whiskers reach to 22 and 31. One picture, five numbers.
→ box plot
Step 4: Question 4: "What fraction of commutes were under 25 minutes?"

Three commutes (22, 23, 24) were under 25 minutes, out of 10 total. That's 30% — "parts of a whole." A pie chart with two slices (under 25 minutes = 30%, 25 minutes or more = 70%) communicates that fraction at a glance. With only two slices, however, a single sentence ("30% were under 25 minutes") arguably does the same job with less ink.
→ pie chart (or just a sentence)
Step 5: The takeaway.

The same data set produced four different pictures, each one answering a different question. The skill being tested isn't "how do I draw a bar chart?" — spreadsheets do that — but "which chart type matches this question?" That choice happens before the software ever opens.

AnswerBar chart · histogram · box plot · pie chart

Watch: How to Read Histograms, Box Plots, and Bar Charts · 3:49 Practise: Lesson 4 quick check The full lesson

T5 · StatisticsLesson 5 · The normal distribution and the empirical rule.

Read the figure to identify k (1, 2, or 3 standard deviations out), pick the matching percentage from the empirical rule, and compute U and W with two subtractions and one addition. Same five steps every time.

"Scores on a standardized test are modeled by a normal distribution with mean 72.7 and standard deviation 5.4. The figure shows the curve with V at the peak and U, W equidistant from V at the outer dashed lines, which appear to land at the third standard deviation. Find U, V, W, and the percentage of total area shaded between U and W."

Step 1: Identify V (the peak) first.

V sits at the highest point of the curve. For any normal distribution, that point is exactly the mean.
$\htmlClass{we-k3}{V} = \htmlClass{we-k3}{\mu} = \htmlClass{we-k3}{72.7}$ → V is always $\mu$
Step 2: Read k from the figure.

U and W are equidistant from V on the outer dashed lines. Count how many standard-deviation lengths each one is from V. The figure shows the outer dashed lines at the third SD — so k = 3.
Step 3: Pick the matching empirical-rule percentage.

Within $1\sigma$ ⇒ 68%. Within $2\sigma$ ⇒ 95%. Within $\htmlClass{we-k1}{3}\htmlClass{we-k4}{\sigma}$ ⇒ 99.7%. Our shaded region lies within $\htmlClass{we-k1}{3}\htmlClass{we-k4}{\sigma}$ of the mean, so the answer is 99.7%.
$\text{shaded area} \approx 99.7\%$ → k = 3 ⇒ 99.7%
Step 4: Compute $U = \mu - k\sigma$.

Three SDs below the mean. Subtract $\htmlClass{we-k1}{3}\htmlClass{we-k4}{\sigma}$ from $\htmlClass{we-k3}{\mu}$.
$\htmlClass{we-k2}{U} = \htmlClass{we-k3}{72.7} - \htmlClass{we-k1}{3}(\htmlClass{we-k4}{5.4}) = 72.7 - 16.2 = \htmlClass{we-k2}{56.5}$ → U
Step 5: Compute $W = \mu + k\sigma$.

Three SDs above the mean. Add $\htmlClass{we-k1}{3}\htmlClass{we-k4}{\sigma}$ to $\htmlClass{we-k3}{\mu}$. Sanity check: U and W should be equidistant from V (72.7), and they are — both 16.2 away.
$\htmlClass{we-k2}{W} = \htmlClass{we-k3}{72.7} + \htmlClass{we-k1}{3}(\htmlClass{we-k4}{5.4}) = 72.7 + 16.2 = \htmlClass{we-k2}{88.9}$ → W

AnswerU = 56.5 · V = 72.7 · W = 88.9 · 99.7%

Watch: The 68-95-99.7 Rule (Normal Distribution Explained) · 3:49 Practise: Lesson 5 quick check Try it: Normal Distribution Explorer The full lesson

T5 · StatisticsLesson 6 · Surveys and the margin of error.

A standard national political poll. Compute the MOE with the rule of thumb, build the confidence interval, and translate it into plain English.

"A polling firm interviews 1,000 likely voters and reports that 45% favor the candidate. The firm reports the result at 95% confidence. What is the margin of error, and what does the headline actually claim?"

Step 1: Identify the sample size and sample proportion.

Read straight from the report. n = 1,000 is the number of people interviewed; p̂ = 45% = 0.45 is the proportion of the sample favoring the candidate.
Step 2: Compute the margin of error.

Use the rule of thumb at 95% confidence: $\htmlClass{we-k4}{\text{MOE}} \approx \tfrac{1}{\sqrt{\htmlClass{we-k2}{n}}}$. With n = 1,000:
$\begin{aligned} \htmlClass{we-k4}{\text{MOE}} &\approx \dfrac{1}{\sqrt{\htmlClass{we-k2}{1{,}000}}} \approx \dfrac{1}{31.62} \approx 0.0316 \\ &\approx \htmlClass{we-k4}{3.2}\ \text{percentage points} \end{aligned}$ → MOE
Step 3: Build the confidence interval.

Take the sample proportion and add/subtract the MOE on each side.
$\text{CI} = \htmlClass{we-k3}{45\%} \pm \htmlClass{we-k4}{3.2\%} = (41.8\%,\ 48.2\%)$ → 95% confidence interval
Step 4: Translate to plain English.

The poll's claim, said carefully: "We are 95% confident that the true fraction of likely voters favoring the candidate is somewhere between about 42% and 48%." Most journalism rounds the MOE to a single digit ("±3%") and the headline reads "45% support, ±3%."
Step 5: Sanity check.

If a tighter MOE were needed (say, ±1%), the sample size would have to grow to about n = 1/(0.01)² = 10,000 voters. Most polls don't bother because the marginal cost of going from 1,000 to 10,000 interviews is huge and the additional precision rarely changes any decision based on the result.

AnswerMOE ≈ ±3.2% · 41.8% to 48.2%

Watch: What Does Margin of Error Mean in a Poll? · 3:41 Practise: Lesson 6 quick check The full lesson

T6 · ProbabilityLesson 1 · Simple probability.

Direct from the ALEKS review. A box of eight cards labeled P through W; one card is drawn at random; find the probability of drawing a card from P to T. Count favorable, count total, divide, reduce.

"A box contains eight cards labeled P, Q, R, S, T, U, V, and W. One card will be randomly chosen. What is the probability of choosing a letter from P to T? Write your answer as a fraction."

Step 1: Identify the sample space.

The sample space is the set of all eight cards: {P, Q, R, S, T, U, V, W}. Total outcomes = 8.
Step 2: Count favorable outcomes.

The event "choosing a letter from P to T" includes the cards P, Q, R, S, T. That's five outcomes:
$\htmlClass{we-k1}{\text{favorable}} = \{P, Q, R, S, T\} \rightarrow \htmlClass{we-k1}{5} \text{ cards}$ → 5 favorable
Step 3: Build the ratio.

Divide favorable by total:
$P(\text{P to T}) = \frac{\htmlClass{we-k1}{\text{favorable}}}{\htmlClass{we-k2}{\text{total}}} = \frac{\htmlClass{we-k1}{5}}{\htmlClass{we-k2}{8}}$ → probability is a fraction
Step 4: Check whether the fraction reduces.

5 and 8 share no common factors greater than 1 (5 is prime; 8 = 2×2×2). The fraction $\tfrac{5}{8}$ is already in lowest terms. That's the final answer.
→ reduce when you can

Answer5/8

Watch: What Is Probability? (Favorable Over Total) · 3:52 Practise: Lesson 1 quick check The full lesson

T6 · ProbabilityLesson 2 · Fundamental counting principle.

Direct from ALEKS Q2. The state's plate format is three letters followed by three digits, with repeats allowed in each position. Count the choices per slot, then multiply.

"Each license plate in a certain state has six characters (with repeats allowed). The first three characters are letters of the alphabet; the last three are digits 0-9. How many license plates are possible in this state?"

Step 1: Identify the steps.

Six slots to fill, one character per slot. Each slot is an independent step (repeats allowed). So six independent steps.
Step 2: Count the choices per step.

Each of the first three slots: 26 letters in the alphabet. Each of the last three slots: 10 digits (0 through 9).
$slots 1-3: 26 each slots 4-6: 10 each$ → 26 for letters, 10 for digits
Step 3: Apply the principle: multiply.

Total plates = 26 × 26 × 26 × 10 × 10 × 10. Compute in two halves:
$\begin{aligned} \htmlClass{we-k3}{26}^3 &= 17{,}576 \\ \htmlClass{we-k4}{10}^3 &= 1{,}000 \\ \text{total} &= 17{,}576 \times 1{,}000 = \htmlClass{accent}{17{,}576{,}000} \end{aligned}$
Just over 17.5 million possible plates, far more than any single state will ever issue.
→ multiply across the slots

Answer17,576,000 plates

Watch: The Fundamental Counting Principle Explained · 3:05 Practise: Lesson 2 quick check The full lesson

T6 · ProbabilityLesson 3 · Sample spaces and the addition rule.

Direct from ALEKS Q3. One card is drawn from a standard 52-card deck. Three parts, increasing in complexity: a simple count, another simple count, and a union that requires the overlap correction.

"Suppose one card is drawn at random from a standard deck of 52 cards. Find: (a) P(face), (b) P(red), (c) P(face OR red). Write your answers as fractions."

Step 1: (a) P(face) — count the face cards.

Face cards are J, Q, K in each of four suits: 3 × 4 = 12 face cards out of 52 total.
$\htmlClass{we-k1}{P(\text{face})} = \frac{\htmlClass{we-k1}{12}}{\htmlClass{we-k2}{52}} = \htmlClass{accent}{\frac{3}{13}}$ → $\tfrac{12}{52}$ reduces (GCD = 4)
Step 2: (b) P(red) — count the red cards.

Two of the four suits are red (hearts and diamonds): 2 × 13 = 26 red cards.
$\htmlClass{we-k3}{P(\text{red})} = \frac{\htmlClass{we-k3}{26}}{\htmlClass{we-k2}{52}} = \htmlClass{accent}{\frac{1}{2}}$ → exactly half the deck
Step 3: (c) P(face OR red) — addition rule.

Apply the formula, but first identify the overlap. The cards that are both face AND red are the J/Q/K of hearts and diamonds: 6 cards. So P(face AND red) = $\tfrac{6}{52}$.
$\begin{aligned} \htmlClass{we-k4}{P(\text{face or red})} &= \htmlClass{we-k1}{P(\text{face})} + \htmlClass{we-k3}{P(\text{red})} - P(\text{face and red}) \\ &= \frac{\htmlClass{we-k1}{12}}{52} + \frac{\htmlClass{we-k3}{26}}{52} - \frac{6}{52} \\ &= \htmlClass{accent}{\frac{32}{52} = \frac{8}{13}} \end{aligned}$
Sanity check: the union has 32 cards total — 26 red cards plus 6 black face cards (J♠, Q♠, K♠, J♣, Q♣, K♣) that are face but not red. $\tfrac{32}{52} = \tfrac{8}{13}$.
→ subtract the overlap once

Answer3/13 · 1/2 · 8/13

Watch: The Addition Rule in Probability: P(A or B) Explained · 3:54 Practise: Lesson 3 quick check The full lesson

T6 · ProbabilityLesson 4 · Permutations and combinations.

Direct from ALEKS Q4(a). 50 athletes, 3 distinct medals, no ties. Each medal is different, so order matters — this is a permutation.

"50 athletes are running a race. A gold medal is to be given to the winner, a silver medal to the second-place finisher, and a bronze medal to the third-place finisher. Assume that there are no ties. In how many possible ways can the 3 medals be distributed?"

Step 1: Apply the litmus test.

If athletes Alice, Bob, and Carol finish 1st-2nd-3rd, that's a different outcome than Carol-Alice-Bob finishing 1st-2nd-3rd. The order changes who gets which medal. Order matters — this is a permutation.
→ permutation, not combination
Step 2: Identify n and r.

Picking 3 medalists out of 50 athletes: n = 50, r = 3.
Step 3: Compute the permutation.

P(50, 3) is the product of r = 3 shrinking choices starting from n = 50:
$P(\htmlClass{we-k2}{50}, \htmlClass{we-k1}{3}) = \htmlClass{we-k2}{50} \times 49 \times 48$
Compute in two steps:
$\begin{aligned} 50 \times 49 &= 2{,}450 \\ 2{,}450 \times 48 &= \htmlClass{accent}{117{,}600} \end{aligned}$
Just over a hundred thousand possible medal distributions for a single 50-athlete race.
→ multiply 3 shrinking choices

Answer117,600 ways

Watch: Permutations vs Combinations: The Swap Test · 4:14 Practise: Lesson 4 quick check The full lesson

T6 · ProbabilityLesson 5 · Expected value.

Direct from ALEKS Q5. Compute E(X) for the six-slice game and interpret what it means for Debra's long-run play.

"Debra is playing a game in which she spins a spinner with 6 equal-sized slices numbered 1 through 6. She wins $1 if the spinner stops on 1, $3 if it stops on 2, $5 if it stops on 3, $7 if it stops on 4. She loses $8 if the spinner stops on 5 or 6. Find the expected value of playing the game. What can Debra expect in the long run?"

Step 1: List every outcome with its payoff and probability.

Six equal slices → each probability is $\htmlClass{we-k1}{\tfrac{1}{6}}$. The payoffs are:
$\begin{aligned} \htmlClass{we-k2}{x} &= \htmlClass{we-k2}{+\$1},\ \htmlClass{we-k2}{+\$3},\ \htmlClass{we-k2}{+\$5},\ \htmlClass{we-k2}{+\$7},\ \htmlClass{we-k2}{-\$8},\ \htmlClass{we-k2}{-\$8} \\ \htmlClass{we-k1}{P(x)} &= \htmlClass{we-k1}{\frac{1}{6}} \text{ for each} \end{aligned}$
Step 2: Multiply each payoff by its probability.

Compute the six terms:
$\begin{aligned} \htmlClass{we-k2}{(+1)}\left(\htmlClass{we-k1}{\frac{1}{6}}\right) &= +\frac{1}{6} \\ \htmlClass{we-k2}{(+3)}\left(\htmlClass{we-k1}{\frac{1}{6}}\right) &= +\frac{3}{6} \\ \htmlClass{we-k2}{(+5)}\left(\htmlClass{we-k1}{\frac{1}{6}}\right) &= +\frac{5}{6} \\ \htmlClass{we-k2}{(+7)}\left(\htmlClass{we-k1}{\frac{1}{6}}\right) &= +\frac{7}{6} \\ \htmlClass{we-k2}{(-8)}\left(\htmlClass{we-k1}{\frac{1}{6}}\right) &= -\frac{8}{6} \\ \htmlClass{we-k2}{(-8)}\left(\htmlClass{we-k1}{\frac{1}{6}}\right) &= -\frac{8}{6} \end{aligned}$ → same denominator throughout
Step 3: Sum the six terms.

All six have denominator 6, so just sum the numerators:
$\begin{aligned} E(X) &= \frac{\htmlClass{we-k2}{1 + 3 + 5 + 7 - 8 - 8}}{6} \\ &= \htmlClass{accent}{\frac{0}{6} = \$0} \end{aligned}$ → payoffs cancel exactly
Step 4: Interpret.

E(X) = $0 means the game is fair — Debra can expect to break even in the long run. On any single spin she'll win or lose money, but averaged over many spins her account balance stays approximately where she started.
→ E = 0 → break even

AnswerE(X) = $0: a fair game

Watch: Expected Value Explained (The Long-Run Average) · 3:46 Practise: Lesson 5 quick check Try it: Expected Value Lab The full lesson

T6 · ProbabilityLesson 6 · Theoretical vs experimental probability.

Direct from ALEKS Q6. A bag has 3 balls (numbered 1, 2, 3); a pack has 3 cards (lettered K, Q, J). On each trial, a ball is chosen and a card is drawn. 70 trials were run. Find the experimental and theoretical probability of both choosing a ball that is odd AND drawing a card that is either a K or a J.

"70 trials. Counts per outcome: 1K=11, 2K=8, 3K=9, 1Q=10, 2Q=5, 3Q=8, 1J=8, 2J=5, 3J=6. Find (a) experimental P(odd ball AND K or J card), (b) theoretical P, (c) what should happen as the number of trials grows."

Step 1: (a) Identify the favorable trials for the experimental probability.

Favorable: ball is odd (1 or 3) AND card is K or J. Listing those outcomes: 1K, 3K, 1J, 3J. Their counts:
$1K = \htmlClass{we-k2}{11} \quad 3K = \htmlClass{we-k2}{9} \quad 1J = \htmlClass{we-k2}{8} \quad 3J = \htmlClass{we-k2}{6}$
Step 2: (a continued) Sum and divide by total trials.

Favorable count = 11 + 9 + 8 + 6 = 34. Total trials = 70.
$\htmlClass{we-k2}{\text{experimental } P} = \frac{\htmlClass{we-k2}{34}}{\htmlClass{we-k2}{70}} \approx \htmlClass{accent}{0.486}$
Round to the nearest thousandth as ALEKS requests.
→ from the data
Step 3: (b) Compute the theoretical probability.

Two independent events → multiply. $P(\text{odd ball}) = \tfrac{2}{3}$ (numbers 1 and 3 out of {1, 2, 3}). P(K or J card) = $\tfrac{2}{3}$ (letters K and J out of {K, Q, J}).
$\htmlClass{we-k1}{\text{theoretical } P} = \left(\htmlClass{we-k1}{\frac{2}{3}}\right) \times \left(\htmlClass{we-k1}{\frac{2}{3}}\right) = \htmlClass{accent}{\frac{4}{9} \approx 0.444}$ → from the math
Step 4: (c) Interpret: as trials grow, the two get closer.

The experimental value 0.486 is close to the theoretical 0.444 — off by about 0.04. With more trials, the gap would shrink (Law of Large Numbers). The correct ALEKS answer for part (c): "As the number of trials increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal."
→ they converge, never quite equal

AnswerExperimental ≈ 0.486 · theoretical 4/9 ≈ 0.444

Watch: Theoretical vs Experimental Probability (Law of Large Numbers) · 3:40 Practise: Lesson 6 quick check Try it: Law of Large Numbers Simulator The full lesson

T7 · Taxes & StocksLesson 1 · Reading a paycheck.

Compute gross, the three standard deductions, net, and the take-home percentage for a typical entry-level paycheck. The math is straight subtraction; the work is identifying each piece.

"You earn $20/hour and worked 40 hours this week. Your federal withholding rate is 12%, state withholding is 5%, and FICA is the standard 7.65%. Compute your gross pay, each of the three deductions, your net pay, and your take-home percentage."

Step 1: Compute gross pay.

Multiply the hourly rate by the hours worked:
$\text{gross} = \htmlClass{we-k2}{\$20/\text{hr}} \times \htmlClass{we-k3}{40\ \text{hr}} = \htmlClass{accent}{\$800}$ → before any deductions
Step 2: Compute each deduction as a percent of gross.

All three deductions are percentages of gross pay:
$\begin{aligned} \text{federal}: \htmlClass{we-k1}{0.12} \times 800 &= \$96.00 \\ \text{FICA}: \htmlClass{we-k1}{0.0765} \times 800 &= \$61.20 \\ \text{state}: \htmlClass{we-k1}{0.05} \times 800 &= \$40.00 \end{aligned}$
Total deductions: $96 + $61.20 + $40 = $197.20.
→ three slices off the top
Step 3: Subtract total deductions from gross.
$\text{net} = 800 - 197.20 = \htmlClass{accent}{\$602.80}$ → what hits the account
Step 4: Compute the take-home percentage.

Divide net by gross and convert to a percent:
$\text{take-home} \% = \dfrac{602.80}{800} = 0.7535 = \htmlClass{accent}{75.35\%}$
About three-quarters of every gross dollar reaches your account; the other quarter is split among the three taxing authorities.
→ a useful single summary

AnswerNet $602.80 · 75.35% take-home

Watch: How to Read Your Paycheck: Gross Pay vs Net Pay · 4:18 Practise: Lesson 1 quick check The full lesson

T7 · Taxes & StocksLesson 2 · Tax brackets and marginal rate.

Stair-step across the brackets, summing as you go. Then read the two rates: marginal (top bracket) and effective (total / income).

"A single filer reports $60,000 in taxable income for the year. Using the 2024 brackets (10% to $11,600; 12% to $47,150; 22% to $100,525), compute the total federal income tax owed, the marginal rate, and the effective rate."

Step 1: Identify which brackets are touched.

$60,000 is between $47,150 and $100,525, so the income reaches into the 22% bracket. All three lower-or-equal brackets are touched: 10%, 12%, and 22%.
→ top bracket = 22%
Step 2: Compute the tax on the 10% bracket.

The 10% bracket runs from $0 to $11,600, a slice of width $11,600:
$\htmlClass{we-k1}{0.10} \times \htmlClass{we-k2}{11{,}600} = \htmlClass{accent}{\$1{,}160}$ → first slice
Step 3: Compute the tax on the 12% bracket.

The 12% bracket runs from $11,600 to $47,150, a slice of width 47,150 − 11,600 = 35,550:
$\htmlClass{we-k1}{0.12} \times \htmlClass{we-k2}{35{,}550} = \htmlClass{accent}{\$4{,}266}$ → second slice (full bracket)
Step 4: Compute the tax on the partial 22% bracket.

The 22% bracket runs from $47,150 upward, but our income only reaches $60,000. So the relevant slice is 60,000 − 47,150 = 12,850 (partial bracket):
$\htmlClass{we-k1}{0.22} \times \htmlClass{we-k2}{12{,}850} = \htmlClass{accent}{\$2{,}827}$ → third slice (partial)
Step 5: Sum the three pieces.
$\text{total tax} = 1{,}160 + 4{,}266 + 2{,}827 = \htmlClass{accent}{\$8{,}253}$ → total federal income tax owed
Step 6: Read the marginal and effective rates.

Marginal rate = 22% (the top bracket the income reaches). Effective rate = total tax / total income:
$\text{effective} = \dfrac{8{,}253}{\htmlClass{we-k2}{60{,}000}} \approx 0.1376 = \htmlClass{accent}{13.76\%}$
The taxpayer's headline rate ("I'm in the 22% bracket") and their actual rate of taxation (13.76%) are very different numbers. The effective rate is what you actually paid; the marginal rate is what the next dollar would cost.
→ effective < marginal, always

Answer$8,253 tax · 22% marginal · 13.76% effective

Watch: How Tax Brackets Actually Work (Marginal vs Effective Rate) · 4:23 Practise: Lesson 2 quick check Try it: Tax Bracket Calculator The full lesson

T7 · Taxes & StocksLesson 3 · Filing, refunds, and effective rate.

Walk through the complete annual cycle: gross → standard deduction → taxable → tax owed → compare to withholding → refund or owe → effective rate.

"A single filer earned $48,000 in gross income during the year. She took the standard deduction of $14,600. Across her 26 biweekly paychecks, federal withholding totaled $4,200. Using the 2024 brackets (10% to $11,600; 12% to $47,150), compute her tax owed, her refund or amount due, and her effective tax rate."

Step 1: Compute taxable income.

Subtract the standard deduction from gross income:
$\text{taxable} = \htmlClass{we-k2}{48{,}000} - \htmlClass{we-k4}{14{,}600} = \htmlClass{accent}{\$33{,}400}$ → what the brackets apply to
Step 2: Stair-step the brackets.

Taxable income $33,400 sits in the 12% bracket. Two slices:
$\begin{aligned} \htmlClass{we-k1}{10\%} \times 11{,}600 &= \$1{,}160 \\ \htmlClass{we-k1}{12\%} \times (33{,}400 - 11{,}600) &= \htmlClass{we-k1}{12\%} \times 21{,}800 = \$2{,}616 \\ \text{tax owed} &= 1{,}160 + 2{,}616 = \htmlClass{accent}{\$3{,}776} \end{aligned}$ → total federal income tax for the year
Step 3: Compare withholding to tax owed.

Withholding totaled $4,200 across the year; tax owed is $3,776:
$\text{refund} = \text{withheld} - \text{owed} = \htmlClass{we-k3}{4{,}200} - 3{,}776 = \htmlClass{accent}{\$424}$
Positive value → the filer receives a refund of $424.
→ over-withheld by $424
Step 4: Compute the effective tax rate.

Total tax owed divided by gross income:
$\text{effective} = \dfrac{3{,}776}{\htmlClass{we-k2}{48{,}000}} \approx 0.0787 = \htmlClass{accent}{7.87\%}$
Less than 8 cents on every dollar she earned actually went to federal income tax. The marginal rate (top bracket reached) was 12%, but the effective rate is lower because the first $11,600 was taxed at 10% and the first $14,600 was shielded entirely by the standard deduction.
→ effective < marginal < bracket headline

Answer$3,776 owed · $424 refund · 7.87% effective

Watch: Tax Refunds Explained: Withholding vs What You Owe · 4:17 Practise: Lesson 3 quick check Try it: Tax Bracket Calculator The full lesson

T7 · Taxes & StocksLesson 4 · Stocks: shares, dividends, return.

Decompose a single year of stock ownership into its two return components, then combine for the total and ROI.

"You buy 100 shares of XYZ Corp at $50 per share. One year later, the share price has risen to $58, and during the year XYZ paid $0.50 per share each quarter in dividends. Compute the capital gain, the total dividend income, the total return, and the return on investment."

Step 1: Compute the amount invested.
$\text{amount invested} = \htmlClass{we-k3}{100\ \text{shares}} \times \htmlClass{we-k2}{\$50} = \htmlClass{accent}{\$5{,}000}$ → purchase day
Step 2: Compute the capital gain.

Price moved from $50 to $58 — a rise of $8 per share. Multiply by share count:
$\text{capital gain} = (\htmlClass{we-k4}{\$58} - \htmlClass{we-k2}{\$50}) \times \htmlClass{we-k3}{100} = \htmlClass{accent}{\$800}$ → price-movement piece
Step 3: Compute the dividend income.

$0.50 per share per quarter, four quarters in a year, equals $2.00 per share annually. Multiply by share count:
$\text{dividends} = \htmlClass{we-k1}{\$2.00} \times \htmlClass{we-k3}{100} = \htmlClass{accent}{\$200}$ → cash-payments piece
Step 4: Sum for total return.
$\text{total return} = \text{capital gain} + \text{dividends} = \$800 + \$200 = \htmlClass{accent}{\$1{,}000}$ → both pieces combined
Step 5: Compute ROI.

Divide total return by amount invested:
$\text{ROI} = \dfrac{\$1{,}000}{\$5{,}000} = 0.20 = \htmlClass{accent}{20\%}$
A 20% one-year return is excellent, well above the long-run S&P 500 average of about 10% per year. The decomposition reveals that 80% of the return came from price movement and 20% from dividends — a typical mix for a dividend-paying stock in a good year.
→ the comparable rate

AnswerTotal return $1,000 · ROI 20%

Watch: How Stocks Make Money: Capital Gains and Dividends · 4:09 Practise: Lesson 4 quick check The full lesson

T7 · Taxes & StocksLesson 5 · Bonds: face value, coupon, yield.

Compute the annual coupon, the total interest earned over the bond's life, the total amount received, and the current yield if the bond is trading at a discount.

"You buy a corporate bond with face value $1,000, coupon rate 5%, and 10-year maturity. The bond currently trades on the secondary market at $950 (a discount). Compute the annual coupon payment, the total interest earned over 10 years if held to maturity, the total amount received, and the current yield."

Step 1: Compute the annual coupon payment.

Annual coupon = face value × coupon rate:
$\text{annual coupon} = \htmlClass{we-k2}{\$1{,}000} \times \htmlClass{we-k1}{0.05} = \htmlClass{accent}{\$50}$
Paid every year for the bond's 10-year life.
→ same every year
Step 2: Total interest earned over 10 years.
$\text{total interest} = \htmlClass{we-k3}{10} \times \$50 = \htmlClass{accent}{\$500}$ → 10 coupon payments summed
Step 3: Total amount received.

At maturity, the bondholder receives the face value back, in addition to the coupons collected over the life:
$\text{total} = \text{interest} + \text{face value} = \$500 + \htmlClass{we-k2}{\$1{,}000} = \htmlClass{accent}{\$1{,}500}$ → life-of-bond total
Step 4: Current yield at the discount price.

The bond trades at $950, not $1,000. Current yield divides the coupon by the market price:
$\text{current yield} = \dfrac{\$50}{\htmlClass{we-k4}{\$950}} \approx 0.0526 = \htmlClass{accent}{5.26\%}$
The discount price means a new buyer today earns 5.26% effective yield on their $950 investment, slightly higher than the bond's nominal 5% coupon rate. Plus, at maturity they receive $1,000 back — a $50 capital gain on top of the coupons.
→ discount → yield > coupon rate

Answer$500 interest · $1,500 total · 5.26% current yield

Watch: How Bonds Work: Face Value, Coupon Rate, and Yield · 4:55 Practise: Lesson 5 quick check The full lesson

T7 · Taxes & StocksLesson 6 · Diversification and the stock-bond split.

Apply the time-horizon framework to two investors at different life stages, and compute the expected return of each suggested mix.

"Investor A is 30 years old and saving for retirement at age 65 (35-year horizon). Investor B is 60 years old and plans to retire at 65 (5-year horizon). Using the rule of thumb percent in stocks = 110 − age, suggest an allocation for each, then compute the expected return of each portfolio assuming stocks return 10% and bonds return 5%."

Step 1: Compute Investor A's stock allocation.
$\text{stocks} \% = 110 - \htmlClass{we-k4}{30} = \htmlClass{accent}{80\%}$
So Investor A: 80% stocks, 20% bonds. The 35-year horizon gives plenty of time to ride out drawdowns.
→ stock-heavy
Step 2: Compute Investor A's expected return.

Weighted average:
$\begin{aligned} E &= 0.80 \times \htmlClass{we-k1}{10\%} + 0.20 \times \htmlClass{we-k3}{5\%} \\ &= 8\% + 1\% = \htmlClass{accent}{9\%} \end{aligned}$ → closer to the stock end
Step 3: Compute Investor B's allocation.
$\text{stocks} \% = 110 - \htmlClass{we-k4}{60} = \htmlClass{accent}{50\%}$
So Investor B: 50% stocks, 50% bonds. The 5-year horizon cannot tolerate a deep stock drawdown without delaying retirement.
→ balanced
Step 4: Compute Investor B's expected return.
$\begin{aligned} E &= 0.50 \times \htmlClass{we-k1}{10\%} + 0.50 \times \htmlClass{we-k3}{5\%} \\ &= 5\% + 2.5\% = \htmlClass{accent}{7.5\%} \end{aligned}$
Lower expected return than Investor A's, but much lower volatility — the right trade-off for the shorter horizon. Investor B sleeps better at night.
→ lower return, lower swings

AnswerA: 80/20 at 9% · B: 50/50 at 7.5%

Watch: Diversification Explained: The Stock-Bond Split · 5:13 Practise: Lesson 6 quick check Try it: Portfolio Glide Path The full lesson

Worked Example Library

A 6-month T-bill at 4.5% on $10,000.

Step 1: Identify the parts.

Step 2: Translate the words into a rule.

Step 3: Plug in different inputs.

Step 4: Read it back in English.

Step 1: Gather observations.

Step 2: Spot a pattern (induction).

Step 3: Find the actual rule.

Step 4: Predict the future (deduction).

Step 5: The takeaway.

Step 1: Pick a place value to round to.

Step 2: Round each item.

Step 3: Add the rounded numbers in your head.

Step 4: Compare to reality.

Step 1: Write the two points as ordered pairs.

Step 2: Label them so the formula doesn't trip you up.

Step 3: Plug into the formula.

Step 4: Translate it back to English.

Step 1: Identify the slope and the y-intercept.

Step 2: Plug them into y = mx + b.

Step 3: Sanity-check before predicting.

Step 4: Predict at week 24.

Step 1: Enter the data.

Step 2: Use =SLOPE() to find m.

Step 3: Use =INTERCEPT() to find b.

Step 4: Write the equation. Predict at new speeds.

Step 5: Make a scatter plot. Add a trendline.

Step 6: Interpret the slope and intercept.

Step 1: Convert the percent to a decimal.

Step 2: Multiply the rate by the base.

Step 3: Add the tax to the base.

Step 4: Sanity check.

Step 1: Fraction to decimal: divide top by bottom.

Step 2: Decimal to percent: multiply by 100.

Step 3: All three forms together.

Step 4: Sanity check.

Step 1: Compute the discount amount.

Step 2: Subtract the discount from the base.

Step 3: Compute the sales tax (on the sale price).

Step 4: Add the tax to get the total.

Step 5: Sanity check.

Step 1: Identify the conversion factor.

Step 2: Put the unwanted unit on the bottom.

Step 3: Multiply, and watch the units cancel.

Step 4: Compute the number.

Step 5: Sanity check.

Step 1: Plan the chain.

Step 2: Set up the first factor.

Step 3: Set up the second factor.

Step 4: Cancel the units along the chain.

Step 5: Multiply the numbers.

Step 1: Identify the income.

Step 2: List the expense categories.

Step 3: Sum the expenses.

Step 4: Compute savings.

Step 5: Compute the savings rate.

Step 1: Name the four pieces.

Step 2: Convert the percent to a decimal.

Step 3: Compute the interest.

Step 4: Add interest back to principal.

Step 5: Sanity check.

Step 1: Convert the time to years.

Step 2: Convert the percent to a decimal.

Step 3: Apply I = Prt.

Step 4: Find the accumulated amount.

Step 5: Sanity check.

Step 1: Annual compounding (n = 1).

Step 2: Monthly compounding (n = 12).

Step 3: Daily compounding (n = 365).

Step 4: Continuous compounding.

Step 5: Compare and sanity-check.

Step 1: Identify r and n for each.

Step 2: Compute Bank A's APY.

Step 3: Compute Bank B's APY.

Step 4: Compare and pick a winner.

Step 1: Identify the four pieces.

Step 2: Compute the periodic rate.

Step 3: Compute the exponent.

Step 4: Plug into the annuity formula.