Question type

Successive percent change on the SAT

For percent changes one after another, multiply the factors; never add the percents. Up 20% is $\times 1.20$ , down 25% is $\times 0.75$ , so together they are $\times 0.90$ : 90% of the original, a 10% decrease. A 40% cut followed by a 40% rise leaves 84%, not 100%.

These questions apply two or more percent increases or decreases to the same quantity, one after another, and ask for the final value or the overall percent change.

It appeared on four of the five official practice tests we analysed, five times in all, which makes it one of the most reliable Math question types on the exam.

On 4 of 5 official practice tests
29 practice questions of this type
Desmos helps

Why students miss it

Adding the percents. A 20% rise and a 25% fall do not net to $- 5%$ , because the 25% is taken from a bigger number.

Expecting equal changes to cancel. Down 40% then up 40% gives $0.60 \times 1.40 = 0.84$ , a 16% loss. The answer choice "0%" is there for students who expect them to cancel.

Working backwards by dividing by the sum. To undo two 15% falls, divide by $0.8 5^{2}$ , not by $0.70$ .

How to spot it

Two or more percent changes to the same price, population or value.
"Then", "the new price was then", "one year later".
A question about "the final price is what percent of the original".

The method

Multiply the factors

Turn each change into a factor and multiply them.

Increase by $r %$: factor $1 + r /100$ .
Decrease by $r %$: factor $1 - r /100$ .
Overall factor above 1: net increase of (factor − 1) × 100%.
Overall factor below 1: net decrease of (1 − factor) × 100%.

With Desmos

Desmos is a fast, exact calculator for the chain of factors.

Type 1.2*0.75 and read 0.9: the final price is 90% of the original.
To work backwards, divide: 14450/0.85^2 gives 20,000.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#892Data

The price of a jacket was increased by 20%. The new price was then decreased by 25%. The final price is what percent of the original price?

Pick an answer, then check it.

Show the answer and explanation

Multiply the two multipliers:

1.20 \times 0.75 = 0.90

. So the final price is

90%

of the original. Checking with

$100

: it rises to

$120

, then

25%

120

30

, leaving

$90

2Question 2 of 2#471Data

The price of a bicycle was reduced by 40%. Several weeks later, the reduced price was increased by 40%. The final price is what percent less than the original price?

Pick an answer, then check it.

Show the answer and explanation

Chain the multipliers from an original price of

100

100 \times 0.60 = 60

, then

60 \times 1.40 = 84

. The final price is

84%

of the original, so it is

16%

less. The 40% increase is taken on the reduced price, so it can't undo the 40% cut.

Questions students ask

Can you add percent changes together?

No. Each percent is taken from a different amount, so multiply the factors instead: up 20% then down 25% is 1.20 × 0.75 = 0.90, a 10% decrease overall.

If a price goes down 40% and then up 40%, is it back to the original?

No. 0.60 × 1.40 = 0.84, so it ends 16% below the original.

Practise this type until it is automatic

Five more successive-percent-change questions, checked instantly, with every wrong choice explained. Free, no card.

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