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Percent problems
Discounts, tax, tips, raises, “what percent of”: they look like ten kinds of problem. They are one. Learn the one idea, the shortcut that replaces the calculator, and the four traps waiting in the answer choices.
Percent means “out of 100”
Three ways of writing the same amount. You will switch between them all the time: the percent is how the question talks, the decimal is how you calculate.
A percent is always a percent of something
On its own, 35% is not an amount. 35% of 300 people is 105 people; 35% of 20 dollars is 7 dollars. The whole you take the percent of is the base. Finding the base is the first job in every percent problem, and most wrong answers come from picking the wrong one.
How to solve them
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1
One equation behind every problem
part = percent × base
Every percent question gives you two of these three and asks for the third. Before you calculate anything, label what you have. The word “of” points at the base: the number right after “of” is the whole.
Next step: No calculatorFind the partWhat is 15% of 64?
0.15 × 64 = 9.6
Find the percent18 is what percent of 72?
18 ÷ 72 = 0.25 = 25%
Find the base27 is 30% of what number?
27 ÷ 0.30 = 90
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2
No calculator: build any percent from 10% and 1%
The ASVAB does not allow a calculator. You don’t need one, because three percents are free. Every other percent is built from those pieces.
10%Move the decimal one place left.
10% of 64 = 6.45%Half of 10%.
5% of 64 = 3.21%Move the decimal two places left.
1% of 260 = 2.6Next step: Discounts & taxExample35% of 260
- 35% = 10% + 10% + 10% + 5%.
- 10% of 260 = 26, so 30% = 3 × 26 = 78.
- 5% is half of 26, which is 13.
- 78 + 13 = 91
Answer 91
ShortcutFlip it
x% of y is the same as y% of x. 8% of 25 is hard; 25% of 8 is a quarter of 8, which is 2.
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3
Discounts, tax and tips: one multiplication
Students find the discount, then subtract: two steps, two chances to slip. Ask instead: what percent do I actually pay?
25% offyou pay 75%× 0.758% sales taxyou pay 108%× 1.0820% tipyou pay 120%× 1.20Next step: Percent changeExampleA $36 shirt, 25% off
- You pay 75% of the price.
- 36 × 0.75 is three quarters of 36. A quarter is 9, so three quarters is 27.
Answer $27
Example$45 with 8% sales tax
- The tax is 8% of 45: 1% is 0.45, so 8% is 8 × 0.45 = 3.60.
- 45 + 3.60 = 48.60, the same as 45 × 1.08.
Answer $48.60
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4
Percent change: divide by where you started
percent change = change ÷ original
The base is always the starting value, never the new one. That is why going up and coming back down are not the same percent.
Next step: Two in a rowExampleFrom $25 to $30, and back
- Up: the change is 5 and the start is 25. 5 ÷ 25 = 0.20, a 20% increase.
- Down: the change is still 5, but now the start is 30. 5 ÷ 30 ≈ 0.167, about a 16.7% decrease.
Answer Same $5, two different percents
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5
Two percents in a row: multiply, never add
The second percent is taken from the new amount, not from the original. So you multiply the two multipliers.
Next step: Original priceExampleA 10% raise, then a 10% cut, on $500 a week
- After the raise: 500 × 1.10 = 550.
- The cut is 10% of 550, not of 500: 550 × 0.90 = 495.
- In one step: 1.10 × 0.90 = 0.99. You end at 99%, 1% below where you started.
Answer $495, not $500
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6
Finding the original price: divide
When the problem gives the price after the change and asks for the price before, the unknown is the base. Divide by the multiplier.
Example$63 after 30% off. What was the price?
- The sale price is 70% of the original: original × 0.70 = 63.
- 63 ÷ 0.70 = 630 ÷ 7 = 90.
- Check: 30% of 90 is 27, and 90 − 27 = 63.
Answer $90
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
The wrong base
In percent change, dividing by the new value. From 80 to 100:
Wrong20 ÷ 100 = 20%
Right20 ÷ 80 = 25%
Adding percents
20% off, then another 10% off:
Wrong20% + 10% = 30% off
Right0.80 × 0.90 = 0.72 → 28% off
Adding the percent back
$56 after 30% off. The original price:
Wrong56 + 30% of 56 = $72.80
Right56 ÷ 0.70 = $80
“Of” versus “more than”
Compare 150 with 100:
Wrong150 is 50% of 100
Right150 is 50% more than 100
5 questions at test level
Pick an answer, then read why. No timer, no account.
A $64 pair of boots is marked 25% off. What is the sale price?
Show the answer
C. You pay 75% of the price. A quarter of 64 is 16, so three quarters is 48. $16 is only the discount, $39 takes off 25 dollars instead of 25 percent, and $80 adds the 25%. The sale price is $48.
A restaurant bill is $45. Sam leaves a 20% tip. How much does Sam pay in total?
Show the answer
D. 10% of 45 is 4.50, so 20% is 9. The total is 45 + 9 = 54. $9.00 is only the tip and $49.50 is a 10% tip. Sam pays $54.00.
A town’s population grew from 2,400 to 3,000. By what percent did it grow?
Show the answer
B. The change is 600. Divide by where it started, 2,400: 600 ÷ 2,400 = 1/4. 20% divides by the new value, and 80% and 125% compare the two populations instead of the change. It grew 25%.
After a 20% discount, a jacket costs $72. What was the original price?
Show the answer
C. $72 is 80% of the original, so divide: 72 ÷ 0.80 = 720 ÷ 8 = 90. Adding 20% of 72 gives $86.40, which takes the percent from the wrong base. The original price was $90.00.
A $200 bike goes up 15% in price, then goes down 15%. What does it cost now?
Show the answer
B. Up first: 200 × 1.15 = 230. The 15% cut is taken from 230: 15% of 230 is 34.50, and 230 − 34.50 = 195.50. The percents do not cancel. It costs $195.50.
11 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 11 cards known.
You finished percent problems.
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