A percentage describes a comparison with a particular whole. Identify that whole before multiplying or dividing: most difficult percentage problems change the base somewhere in the story.
Suggested pace: 10 minutes on examples · 10–15 minutes on questions · 5 minutes on review. Take a second session if a method needs more work.
Keep in mind: part = decimal rate × whole
Method 1 of 4
Translate “percent” as “out of 100.” You can use a decimal, a fraction, or a familiar benchmark. Choose the form that makes the arithmetic easiest, then check whether you were asked for the amount of a change or the final amount.
A $64 tool kit is discounted by 25%. What is the sale price before tax?
Answer: $48
Check it: A quarter off leaves three quarters of the original price. Three quarters of $64 is $48.
A $40 item is 25% off. What is the sale price?
$30. The discount is one quarter of $40, or $10; subtract it from the original price.
Common mistake: Stopping at $16 answers “How much is saved?” Read the last sentence again before choosing an answer.
Method 2 of 4
When a final amount and its percentage are known, the unknown is the whole. Dividing by the decimal rate reverses multiplication. A price after a 20% discount is 80% of its original value, so use 0.80 rather than 0.20.
A jacket costs $72 after a 20% discount. What was its original price?
Answer: $90
Check it: 20% of $90 is $18. Subtracting $18 gives the stated $72 sale price.
A sale price of $48 is 80% of the original. What was the original price?
$60, because 48 ÷ 0.80 = 60. Check: 80% of $60 is $48.
Common mistake: Adding 20% of $72 gives $86.40. That uses the sale price as the base, but the discount was taken from the original price.
Method 3 of 4
Percentage change compares the amount gained or lost with the starting amount. The subtraction finds the change; the division tells you how large it is relative to the original. If two rates themselves change, distinguish percentage points from relative percentage change.
Weekly production rises from 80 units to 100 units. By what percentage does production increase?
Answer: 25% increase
Check it: A 25% increase from 80 adds 20. Dividing by 100 instead would measure the increase against the ending value.
A quantity falls from 50 to 40. What is the percentage decrease?
20%. The decrease is 10, and the starting value is 50: 10/50 × 100% = 20%.
Common mistake: A pass rate moving from 60% to 75% increases by 15 percentage points, but its relative increase is 15/60 = 25%. Those are different questions.
Method 4 of 4
After the first change, the whole for the second change is different. Multipliers keep track of that: an increase of 10% multiplies by 1.10, while a decrease of 10% multiplies by 0.90. Opposite percentage changes therefore do not usually cancel.
A $200 item increases in price by 10%, then receives a 10% discount. What is the final price?
Answer: $198
Check it: The increase was $20, but the later discount was $22 because it used a larger base.
A $100 price rises by 20%, then falls by 20%. Where does it finish?
$96: 100 × 1.20 × 0.80. The $24 decrease is larger than the $20 increase.
Common mistake: Adding +10% and −10% assumes both percentages refer to the same starting amount.
Before each practice question, underline the amount that represents 100%. If the problem changes the group or price, write a new base for the next step.
Work the first six questions with the explanations closed. Review any missed or guessed answers before continuing with the other six. If you cannot set up a problem, return to its method above and try again from a blank page.
Work through one question at a time. Explain your approach, check the worked answer, then move on. Use scratch paper for calculations.
0 of 12 checked · 0 correct.
Progress is saved in this browser. These original study questions do not produce an official AFQT score.
Answer: C. 30
Red-capped bottles make up the remaining 60%. If 60% of the total is 18, the total is 18 ÷ 0.60 = 30. Dividing by 40% gives 45, but 18 counts the red group, not the blue group. Twelve is the number with blue caps.
Answer: A. $40
Undo the last change first: $56 − $6 = $50. That $50 is 125% of the original price, so the original is $50 ÷ 1.25 = $40. Subtracting 25% from $56 uses the wrong base and fails to remove the separate fee correctly.
Answer: A. 12
There are 80 × 0.60 = 48 manually checked forms. One quarter of that group is 48 ÷ 4 = 12. The 20 choice applies one quarter to all 80 forms, but the error fraction applies only to the manual subgroup.
Answer: D. 14%
The 6-percentage-point decrease is 30% of the old rate, so the old rate was 6 ÷ 0.30 = 20%. Subtracting 6 percentage points gives a new rate of 14%. Applying 70% to 6 to obtain 4.2 uses the decrease as though it were the old rate; 1.8 likewise applies 30% to the wrong base.
Answer: B. 40%
If the original capacities are a, b and c, the liquid totals 0.40a + 0.40b + 0.40c = 0.40(a + b + c). The new capacity is a + b + c, so it is still 40% full. Adding the three percentages to get 120% ignores that the total capacity also combines all three containers.
Answer: D. 30
Garden A has 150 × 0.80 = 120 survivors, and Garden B has 100 × 0.90 = 90. The difference is 30 seedlings. A lower percentage can still produce more survivors with a larger starting group. Applying the 10-percentage-point difference to the combined 250 seedlings gives 25, but the percentages refer to separate groups.
Answer: D. 45%
Use helmets, rather than the entire shipment, as the denominator: 18/40 = 0.45, or 45%. For every 100 items, there are 40 helmets and 18 junior helmets. Multiplying 18% by 40% gives 7.2% but does not answer the within-helmet question.
Answer: D. 6.25%
Second-week production is 800 × 1.25 = 1,000 parts, with 1,000 × 0.04 = 40 defective. Forty is 80% of the earlier defective count, so that count was 40 ÷ 0.80 = 50. The first-week rate is 50 ÷ 800 × 100% = 6.25%. Five percent uses the later defective count with the earlier production total.
Answer: B. 100 milliliters
The added capacity is 20% of 500 milliliters: 0.20 × 500 = 100 milliliters. The replacement’s total capacity is 600 milliliters, but the question asks only for the additional amount. The percentage number 20 is not itself a volume.
Answer: C. 33 1/3%
Hourly output changes from 60/32 to 60/24 parts. The rate ratio is (60/24)/(60/32) = 32/24 = 4/3, an increase of 1/3 or 33 1/3%. Twenty-five percent is the decrease in time per part, 8/32, but output is the reciprocal of time per part.
Answer: A. 17.5%
The groups contain 50 × 0.40 = 20 and 150 × 0.10 = 15 errors. There are 35 errors among 200 forms, giving 35/200 × 100% = 17.5%. Taking the simple average of 40% and 10% gives 25%, but that wrongly weights the unequal batch sizes equally.
Answer: C. 78 liters
Removing 18 liters increases the empty share by 35% − 20% = 15% of capacity. Capacity is therefore 18/0.15 = 120 liters. The tank is now 65% full, leaving 0.65 × 120 = 78 liters. Forty-two liters is the empty space after removal; 96 liters is the amount before removal.
A remembered choice letter is different from a method you can reconstruct. Keep one correction in your error log and use it on your next attempt.