Worked Example Library

Your phone bill, written as 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.

1. → x = overage minutes
4. → that's what makes it a function

The case of the disappearing Wi-Fi.

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.

1. → this is your data
2. → specific cases → general pattern
4. → rule + case → certain conclusion

Predicting your grocery total, in your head.

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.

1. → round to the nearest $1
4. → if the receipt had said $89, you'd know to look closer

How fast is your savings account growing?

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

1. → x = days, y = dollars
4. → slope = rate, with units attached

Building a savings model, start to finish.

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.

1. → x = weeks, y = total savings
2. → that's your linear model
4. → that's a prediction, not a guess

Build a stopping-distance model, from real data.

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.

1. → s is the input, d is the output
2. → Excel returns m ≈ 2.43
6. → slope: makes sense. intercept: does not.

Buy a $40 shirt with 7% sales tax.

Three steps. Same three steps every percent problem uses.

3. → that's what you pay

Convert 3/4 to a decimal and a percent.

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

3. → same value, three masks

$80 jacket, 25% off, plus 7% sales tax.

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

4. → what you actually pay

Convert 4 miles to feet.

Same five steps for every single-factor conversion.

4. → that's how far you ran

Convert 5 miles to inches.

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

5. → 5 miles in inches

Build a $3,000 monthly budget.

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

4. → left over each month

Lend a friend $500 at 6% for two years.

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

4. → that's the full payback

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

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

4. → payout at maturity

$1,000 at 5% for 10 years, four ways.

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

4. → the ceiling

Two banks at "5%". Which actually wins?

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

$500/month for 20 years at 6%.

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

4. → final balance

Time is the biggest lever.

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

2. → 40 years of saving
4. → 30 years of saving

Maria's auto loan: name every piece.

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

4. → ~$4,000 in interest
5. → $4,000 over sticker, just like the hook said

$20,000 at 5% APR for 4 years.

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

4. → monthly payment
5. → ~$2,100 of interest over 4 years

$35K SUV, 15% down, 5 years at 5.5%.

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

3. → monthly payment
5. → ~$4,346 over sticker, all of it interest

$5,000 balance at 24% APR, $200 payment.

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

3. → what ALEKS Q2 asks for
4. → half your payment was just interest

$10,000 loan, sub vs unsub, head-to-head.

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.

1. → same number you borrowed
2. → $2,600 of accrued interest piled on top
5. → same loan, same school, same job — $3.5K more

$300K @ 7%, 30 years, month 1.

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

1. → fixed monthly for the next 360 months
2. → more interest than the loan itself
4. → only 12.3% of the payment built equity
5. → principal share grew by $1.43 in one month

Same number, two claims, two kinds of statistics.

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.

2. → descriptive
3. → inferential
4. → inferential claims always carry a ±

Compute mean, median, and modes of one data set.

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.

2. → median
3. → mean
4. → two modes

Range and SD on the same data set.

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.

2. → range
4. → sample SD

Same data, four different pictures.

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.

1. → bar chart
2. → histogram
3. → box plot
4. → pie chart (or just a sentence)

ALEKS Q5 head-on: μ = 72.7, σ = 5.4.

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.

1. → V is always μ
3. → k = 3 ⇒ 99.7%
4. → U
5. → W

Read the poll: 45%, n = 1,000, 95% confidence.

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

2. → MOE
3. → 95% confidence interval

Eight cards, five favorable.

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.

2. → 5 favorable
3. → probability is a fraction
4. → reduce when you can

Counting license plates.

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.

2. → 26 for letters, 10 for digits
3. → multiply across the slots

P(face), P(red), P(face or red).

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.

1. → 12/52 reduces (GCD = 4)
2. → exactly half the deck
3. → subtract the overlap once

Distributing three medals to 50 athletes.

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

1. → permutation, not combination
3. → multiply 3 shrinking choices

Debra's spinner — full E(X) calculation.

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

2. → same denominator throughout
3. → payoffs cancel exactly
4. → E = 0 → break even

Balls and cards: both probabilities, then compare.

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.

2. → from the data
3. → from the math
4. → they converge, never quite equal

Reading a 40-hour weekly 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.

1. → before any deductions
2. → three slices off the top
3. → what hits the account
4. → a useful single summary

Computing tax on $60,000 taxable income.

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

1. → top bracket = 22%
2. → first slice
3. → second slice (full bracket)
4. → third slice (partial)
5. → total federal income tax owed
6. → effective < marginal, always

A year of tax: gross to refund.

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

1. → what the brackets apply to
2. → total federal income tax for the year
3. → over-withheld by $424
4. → effective < marginal < bracket headline

$5,000 in XYZ stock: one year of return.

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

1. → purchase day
2. → price-movement piece
3. → cash-payments piece
4. → both pieces combined
5. → the comparable rate

A standard $1,000 corporate bond.

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.

1. → same every year
2. → 10 coupon payments summed
3. → life-of-bond total
4. → discount → yield > coupon rate

Choosing an allocation for two investors.

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

1. → stock-heavy
2. → closer to the stock end
3. → balanced
4. → lower return, lower swings