SAT Math Review · Problem-Solving and Data Analysis
Percentages, percent change and reverse percent
Every percent question is one multiplier. Knowing which multiplier, and which number it was applied to, is the whole topic.
What is the value of the quantity described?
The price of a gym membership increased by 20% this year. The new price is $96. What was the price before the increase?
The 20% was applied to the old price, so old × 1.20 = 96 and the old price is 96 ÷ 1.20 = $80. Check forwards: 80 × 1.20 = 96 ✓. A takes 20% off the new price, 96 × 0.80 = 76.80 — the most popular wrong answer in this topic, because 20% of 96 is not the same as 20% of 80. C increases the new price again, 96 × 1.20. D divides by 0.80, undoing a decrease that never happened.
1
Percent of a number: turn the percent into a decimal and multiply.
“Percent” means hundredths, so 18% is 0.18, and the word of means multiply.
part =
rate × whole
18% of 250 = 0.18 × 250 = 45
the whole follows “of”
What percent of 160 is 36?
now the rate is the unknown
36 ÷ 160 = 0.225 = 22.5%
part ÷ whole, then × 100
54 is 40% of what number? → 54 ÷ 0.40 = 135
unknown whole: divide instead
Two words do all the work
Whatever follows of is the whole; whatever follows is is the part. Label those two, and the third quantity is the one you solve for. Finding a part multiplies; finding the whole divides.
2
An increase or a decrease is a single multiplier.
Up 15% means keep all of it and add 15% more: × 1.15. Down 15% means keep the other 85%: × 0.85.
Wording
Multiplier
Applied to 180
15% of it
0.15
27
increased by 15%
1.15
207
decreased by 15%
0.85
153
is 15% of it
÷ 0.15
1,200
“15% of” is not “15% more than”
15% of 180 is 27, a small piece of it. 15% more than 180 is 207, the whole thing plus that piece. The SAT puts both numbers in the choices and lets the wording decide, so read the preposition, not just the percent.
3
Percent change is measured against where you started.
A tablet is worth $480 when new and $384 one year later. By what percent did its value decrease?
percent change =
new − oldold
× 100
change = 384 − 480 = −96
negative, so it is a decrease
96 ÷ 480 = 0.20
divide by the old value
= 20% decrease
× 100, and name the direction
check: 480 × 0.80 = 384 ✓
the multiplier reproduces the new value
The base is always the original
Dividing by 384 gives 25%, and 25% is a real number about these two prices — it is how far the new value would have to rise to get back to the old one. It is not the percent decrease, and it is always in the choices.
4
Successive changes multiply — they do not add.
A $250 coat is marked 20% off, and then a weekend sale takes a further 10% off the reduced price.
250 × 0.80 = 200
the first cut, off the original
200 × 0.90 = 180
the second cut, off the reduced price
250 × 0.80 × 0.90 = 250 × 0.72 = 180
one combined multiplier does both
1 − 0.72 = 0.28 → 28% off in total
not 30%
Why 20% off then 10% off is not 30% off
30% off would be 250 × 0.70 = $175, and the store charges $180. The second discount is 10% of the reduced $200, which is $20, not 10% of the original $250, which is $25. That missing $5 is the whole difference — and the same logic works the other way, since a 20% rise followed by a 20% fall gives 1.20 × 0.80 = 0.96, a 4% net loss rather than a return to where you began.
Where the calculator earns its keep
Chain the multipliers in one Desmos line — 250 · 0.8 · 0.9 — instead of rounding at each stage. For a net percent, type 0.8 · 0.9 and read 0.72 straight off: anything under 1 is a net decrease of (1 − that) and anything over 1 is a net increase.
5
Reverse percent: divide by the multiplier to go backwards.
Forwards is old × multiplier = new. So backwards is old = new ÷ multiplier. Never “take the same percent off the new number”.
The opening problem: a 20% increase leaves the price at $96.
up 20% → multiplier 1.20
name the multiplier first, always
old × 1.20 = 96
write the forward sentence
old = 96 ÷ 1.20 = $80
÷ 1.20 both sides
check: 80 × 1.20 = 96 ✓
one line, and it settles the answer
A decrease works the same way. “After a 25% discount a jacket costs $60” means old × 0.75 = 60, so the jacket was 60 ÷ 0.75 = $80. Note that both of these started at $80 and that neither arrived there by touching the percent of the new price.
correct
96 ÷ 1.20 = 80
undo the multiplier
wrong
96 × 0.80 = 76.80
20% of the wrong number
Why 76.80 is wrong, in one sentence
The 20% was 20% of the old price, $16, and 20% of the new price is $19.20 — a different amount of money entirely. Taking a percent off the new number subtracts the wrong one, which is why the two answers miss each other by $3.20.
6
When the base is a letter, use the same multipliers.
The SAT's harder percent items drop the numbers and chain the relationships instead.
x is 30% less than y, and z is 20% greater than x. z is what percent of y?
x = 0.70y
30% less than y keeps 70% of y
z = 1.20x
20% greater than x, so the base is x
z = 1.20(0.70y) = 0.84y
substitute, then multiply
z is 84% of y
and so 16% less than y
If letters make it abstract, set y = 100 and follow the money: x = 70, z = 1.20 × 70 = 84, and 84 out of 100 is 84%. Because every step is a multiplication, whatever y you choose gives the same percent — so choose the easiest one.
“less than y” → base is y“greater than x” → base is x“what percent of y” → divide by y
Each percent has its own base
Reading the 20% as 20% of y gives z = 0.70y + 0.20y = 0.90y, and 90% is waiting in the choices. The phrase after “greater than” names the base, and here that phrase is x, not y.
✓
Check: run the multiplier forwards.
the membership80 × 1.20 = 96back to the new price ✓
the tablet480 × 0.80 = 384a 20% decrease ✓
the coat250 × 0.72 = 18028% off, not 30% ✓
the lettersy = 100 → z = 8484% of y ✓
Remember the flow
Name the multiplier · Name the base it acts on · Multiply forwards, divide backwards
Then reverse the answer through the story to see it land on the given number.
Now you try
Practice problem
What is the value of the quantity described?
A store increased the price of a coat by 30%, and later sold it at 15% off the increased price. The final selling price was $110.50. What was the price of the coat before the increase?
Two multipliers act in turn: 1.30 for the rise, then 0.85 for the discount, so old × 1.30 × 0.85 = 110.50. Since 1.30 × 0.85 = 1.105, the old price is 110.50 ÷ 1.105 = $100. Check forwards: 100 → 130 → 110.50 ✓. A undoes only the increase, 110.50 ÷ 1.30 = 85, and leaves the discount in place. B adds the percents instead of multiplying them, using 1.30 − 0.15 = 1.15 to get 96.09. D multiplies by 1.105 instead of dividing, running the story forwards again from the wrong end.
Hint: write the forward sentence first, with the unknown at the start — old × 1.30 × 0.85 = 110.50 — and only then solve it. Every reverse-percent item becomes a one-step division once that sentence is on the page.
Every lesson in the SAT Math Review has one of these. 10 questions, marked as you answer, with the working shown for each.
Question 1 of 10
What percentage of 80 is 28?
28/80 = 0.35 = 35%.
Question 2 of 10
A pair of shoes is priced at 65 dollars. During a sale, the price is reduced by 30%. What is the sale price of the shoes, in dollars?
A 30% reduction leaves 70% of the price: 0.70 × 65 = 45.50 dollars.
Question 3 of 10
The number of employees at a company increased from 120 to 150. What was the percent increase in the number of employees?
The increase is 30 employees, and 30/120 = 0.25 = 25%.
Question 4 of 10
When a positive quantity x is increased by 15%, the result is 138. What is the value of x?
1.15x = 138, so x = 138/1.15 = 120.
Question 5 of 10
Of the 480 students at a school, 35% take Spanish. Of the students who take Spanish, 25% also take French. How many students at the school take both Spanish and French?
0.35 × 480 = 168 students take Spanish, and 0.25 × 168 = 42 of them also take French.
Question 6 of 10
Which expression represents the result of decreasing a positive quantity y by 18%?
Decreasing by 18% leaves 100% − 18% = 82% of the quantity, which is 0.82y.
Question 7 of 10
The number b is 25% less than 40. The number a is 60% greater than b. What is the value of a?
b = 0.75 × 40 = 30, and a = 1.6 × 30 = 48.
Question 8 of 10
The length of a rectangle is increased by 20% and its width is decreased by 20%. The area of the new rectangle is what percent of the area of the original rectangle?
The new area is (1.2L)(0.8W) = 0.96LW, which is 96% of the original area.
Question 9 of 10
In 2015 a factory produced 8% more units than in 2014, and in 2016 it produced 5% fewer units than in 2015. If the number of units produced in 2016 was k times the number produced in 2014, what is the value of k?
The two changes multiply: k = 1.08 × 0.95 = 1.026. Percent changes do not simply add to give 3%.
Question 10 of 10
210 is p% greater than 60. What is the value of p?
210/60 = 3.5, so 210 is 350% of 60, which is 250% greater than 60. Thus p = 250.
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