Applying two successive percent changes

Two percent changes one after the other. They do not add, and the gap is bigger than most people expect. The single most valuable question on Exam 1.

01 Worked Example Video

Every step of Applying two successive percent changes explained on video. Pause anywhere; the embed scrolls independently of the page.

YOUTUBE

02 What This Looks Like on ALEKS

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

Two successive discounts

A $200 coat is discounted 30%, then the sale price is discounted a further 20%.

What is the final price? Type just the number.

Two markups

A part costs a supplier $85. The distributor marks it up 40%, then the retailer marks that up 25%.

What does the customer pay? Type just the number.

The same percent twice

A $1,500 item is discounted 10%, and a week later discounted another 10%.

What is the final price? Type just the number.

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

$\text{new} = \text{start} \times (1 + r_1) \times (1 + r_2)$ multiply the factors, never add the percents

increase → factor is 1 + r decrease → factor is 1 − r

Turn each percent change into a multiplier, then multiply them together. A 10% rise is ×1.10; a 30% cut is ×0.70. The second change applies to the already changed amount, which is exactly why adding the percents gives the wrong answer.

Common slips

(1) Added the percents. 10% then 70% is not 80%. This is the error the question is built to catch. (2) Applied both to the original. The second percent is of the new amount, not the starting one. (3) Used 0.30 for a 30% discount. The multiplier is 0.70, what is left, not what comes off.

04 Practice — Step by Step

Watch the two routes diverge

An item costs a shop $50. It is marked up 10%, and later marked up a further 70%.

What is the final price, and how does it compare to a single 80% markup?

Step 1

After the first markup

$50 increased by 10%.

Step 2

After the second markup

Now increase $55 by 70%.

Step 3

The single-markup answer

What would $50 marked up a single 80% come to?

Q2 Q4

Applying two successive percent changes

Where the extra $3.50 comes from