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Math · Percent & Proportional Reasoning
Percent: Benchmarks, Percent of a Quantity, Discounts & Tax
Build percent meaning from fractions and decimals, use benchmark percents, find percents of quantities, and solve discount, tax, tip and percent-change problems.
The idea
Percent means per hundred. The same quantity can be written as a fraction, decimal or percent.
Benchmarks such as 10%, 25%, 50% and 75% make mental calculation and estimation faster.
In applications, identify the whole first: a discount, tax, tip or change is calculated relative to a particular starting amount.
Worked examples
Example 1 · Three forms
Write 3/4 as a decimal and percent.
- 3 ÷ 4 = 0.75.
- 0.75 × 100 = 75%.
3/4 = 0.75 = 75%.
Example 2 · Benchmark
Find 25% of 120.
- 25% is one quarter.
- 120 ÷ 4 = 30.
30.
Example 3 · Build 15%
Find 15% of 80.
- 10% of 80 is 8.
- 5% is half of 10%, so 4.
- 8 + 4 = 12.
12.
Example 4 · Missing percent
18 is what percent of 60?
- Divide part by whole: 18 ÷ 60 = 0.3.
- Convert 0.3 to 30%.
30%.
Example 5 · Sale price
A $90 item is 20% off.
- 20% of 90 is 18.
- 90 − 18 = 72.
$72.
Example 6 · Percent change
A quantity rises from 50 to 65.
- Increase is 15.
- Compare increase with the original: 15 ÷ 50 = 0.30.
30% increase.
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: Percent means per hundred, so divide by 100 before multiplying.
Why this fails: Percent increase or decrease normally compares the change with the original amount.
Why this fails: Follow the order given and identify the amount each percent applies to.
Why this fails: The second 20% uses a different base.
Why this fails: A quick benchmark catches impossible answers such as 35% of 40 being greater than 40.
Foundation
Foundation practice
Connect fractions, decimals and percents.
1. Write 25% as a fraction in simplest form.
2. Write 50% as a decimal.
3. Write 0.75 as a percent.
4. Write 3/5 as a percent.
5. What is 10% of 90?
6. What is 50% of 68?
7. What is 25% of 84?
8. What is 75% of 40?
9. Which is greater: 45% or 0.4?
10. Order 1/2, 60%, and 0.55 from least to greatest.
Core
Core practice
Find a percent of a quantity.
11. Find 20% of 70.
12. Find 15% of 80.
13. Find 30% of 150.
14. Find 12% of 50.
15. Find 35% of 200.
16. Find 8% of 250.
17. Find 62.5% of 32.
18. 18 is what percent of 60?
19. 24 is 40% of what number?
20. 45 is 75% of what number?
Think
Think practice
Use percents in money and everyday contexts.
21. A $50 game is 20% off. What is the discount?
22. A $50 game is 20% off. What is the sale price?
23. A $40 item has 13% tax. How much tax is added?
24. A $40 item has 13% tax. What is the total?
25. Find a 15% tip on a $60 meal.
26. A club grows from 80 to 100 members. What is the percent increase?
27. A price falls from $120 to $90. What is the percent decrease?
28. A 2 L bottle has 30% left. How much is left?
29. 27 of 30 students are present. What percent are present?
30. A reader finishes 72 of 90 pages. What percent is complete?
Challenge
Challenge practice
Combine percent reasoning with estimation and comparison.
31. A $100 item is reduced by 20%, then by another 10% of the new price. Final price?
32. After a 20% discount, a jacket costs $64. Original price?
33. Which saves more on $80: 25% off or $18 off?
34. About how much is 19% of 198?
35. A score rises from 60% to 75%. By how many percentage points?
36. In a 2:3 red-to-blue ratio, what percent of all counters are red?
37. A machine succeeds 48 times in 60 trials. Success percent?
38. A $200 bicycle is 15% off, then 13% tax is added to the sale price. Final price?
39. A student says 35% of 40 is 140. What is the error?
40. Which is closest to 67%: 2/3, 3/4, or 0.6?
Answer key
- 1. 1/4
25% means 25 out of 100, which simplifies to 1/4. - 2. 0.5
50% means 50/100 = 0.5. - 3. 75%
Multiply the decimal by 100 to express it per hundred. - 4. 60%
3/5 = 0.6 = 60%. - 5. 9
10% is one tenth, so 90 ÷ 10 = 9. - 6. 34
50% means one half. - 7. 21
25% means one quarter, so 84 ÷ 4 = 21. - 8. 30
75% is three quarters; 40 ÷ 4 × 3 = 30. - 9. 45%
0.4 = 40%, so 45% is greater. - 10. 1/2, 0.55, 60%
Convert to percents: 50%, 55%, 60%. - 11. 14
10% of 70 is 7, so 20% is 14. - 12. 12
10% is 8 and 5% is 4; together they make 12. - 13. 45
10% of 150 is 15; multiply by 3. - 14. 6
0.12 × 50 = 6. - 15. 70
35% of 200 = 0.35 × 200 = 70. - 16. 20
1% of 250 is 2.5; 8% is 20. - 17. 20
62.5% = 5/8; 32 × 5/8 = 20. - 18. 30%
18 ÷ 60 = 0.3 = 30%. - 19. 60
24 ÷ 0.4 = 60. - 20. 60
75% is 3/4; if 3/4 is 45, the whole is 60. - 21. $10
20% of $50 is $10. - 22. $40
Subtract the $10 discount from $50. - 23. $5.20
0.13 × 40 = 5.20. - 24. $45.20
Add $5.20 tax to $40. - 25. $9
10% is $6 and 5% is $3; total $9. - 26. 25%
The increase is 20; 20 ÷ 80 = 25%. - 27. 25%
The decrease is 30; 30 ÷ 120 = 25%. - 28. 0.6 L
0.30 × 2 = 0.6. - 29. 90%
27 ÷ 30 = 0.9 = 90%. - 30. 80%
72 ÷ 90 = 0.8 = 80%. - 31. $72
After 20% off it is $80; 10% of $80 is $8, leaving $72. - 32. $80
$64 is 80% of the original; 64 ÷ 0.8 = 80. - 33. 25% off
25% of $80 is $20, which is $2 more than $18. - 34. About 40
Use 20% of 200 = 40 as a close estimate. - 35. 15 percentage points
75% − 60% = 15 percentage points. - 36. 40%
There are 2 red out of 5 total; 2/5 = 40%. - 37. 80%
48 ÷ 60 = 0.8 = 80%. - 38. $192.10
15% off gives $170; 13% of $170 is $22.10; total $192.10. - 39. They forgot that 35% is 0.35, not 35.
0.35 × 40 = 14, which is less than the whole as expected. - 40. 2/3
2/3 is about 66.7%, closest to 67%.