Percentages | Math Coach Amy

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Key Ideas Percent Change Successive Changes Percent of a Percent Original Value Mixed

What to Remember

The multiplier

A percent increase multiplies by 1 plus the percent.

A 20% increase uses 1 + 0.20 = 1.20.

A percent decrease multiplies by 1 minus the percent.

A 20% decrease uses 1 − 0.20 = 0.80.

Percent change

percent change = changeoriginal × 100%

A price goes from $20 to $24.

change = 24 − 20

change = 4

percent change = 420 × 100%

percent change = 20% increase

Successive percents do not add

A 50% increase then a 50% decrease is not back to the start.

$80 × 1.50 = $120

$120 × 0.50 = $60

The second percent acts on the new amount, not the original.

Percent of a percent

Take the first percent of the whole, then the next percent of that part.

25% of 40 is 10.

50% of those 10 is 5.

0.50 × 0.25 × 40 = 5

Working backwards

10% off. The price is $36.

That means you PAID 90% of the original price.

Translate into an equation.

.9x = 36

x = 36.9

x = 40

The original price was $40.

Percent Change (Increase and Decrease)

Find how much a value grew or shrank, then write that change as a percent of the original.

Example

A jacket costs $80. The sale price is $60. What is the percent change?

Original = $80. New = $60.

change = original − new

change = $80 − $60

change = $20

percent change = changeoriginal × 100%

percent change = 2080 × 100%

percent change = 14 × 100%

percent change = 25%

Answer: 25% decrease

Level 1

Easy

One increase or one decrease. The numbers divide evenly.

A shirt costs $40. Later it costs $50. What is the percent change?

Show Answer

Original = $40. New = $50.

change = new − original

change = $50 − $40

change = $10

percent change = changeoriginal × 100%

percent change = 1040 × 100%

percent change = 14 × 100%

percent change = 25%

Answer: 25% increase

A test score goes from 80 to 72. What is the percent change?

Show Answer

Original = $80. New = $72.

change = original − new

change = $80 − $72

change = $8

percent change = changeoriginal × 100%

percent change = 880 × 100%

percent change = 110 × 100%

percent change = 10%

Answer: 10% decrease

Level 2

Medium

Larger numbers. Reduce the fraction before you multiply by 100%.

A town’s population goes from 2400 to 2760. What is the percent change?

Show Answer

Original = 2400. New = 2760.

change = new − original

change = 2760 − 2400

change = 360

percent change = changeoriginal × 100%

percent change = 3602400 × 100%

percent change = 320 × 100%

percent change = 15%

Answer: 15% increase

A $90 item is marked down to $63. What is the percent change?

Show Answer

Original = $90. New = $63.

change = original − new

change = $90 − $63

change = $27

percent change = changeoriginal × 100%

percent change = 2790 × 100%

percent change = 310 × 100%

percent change = 30%

Answer: 30% decrease

Level 3

Hard

The percent may not be a whole number.

A count goes from 48 to 66. What is the percent change?

Show Answer

Original = 48. New = 66.

change = new − original

change = 66 − 48

change = 18

percent change = changeoriginal × 100%

percent change = 1848 × 100%

percent change = 38 × 100%

percent change = 37.5%

Answer: 37.5% increase

A stock price falls from $250 to $175. What is the percent change?

Show Answer

Original = $250. New = $175.

change = original − new

change = $250 − $175

change = $75

percent change = changeoriginal × 100%

percent change = 75250 × 100%

percent change = 310 × 100%

percent change = 30%

Answer: 30% decrease

A value rises from 80 to 98. What is the percent change?

Show Answer

Original = $80. New = $98.

change = new − original

change = $98 − $80

change = $18

percent change = changeoriginal × 100%

percent change = 1880 × 100%

percent change = 940 × 100%

percent change = 22.5%

Answer: 22.5% increase

Successive Percent Changes (The Non-Additive Trap)

Two percent changes in a row do not add. Each percent hits the amount after the last change.

Example

A price of $100 rises 10%, then falls 10%. What is the final price?

Start with $100.

Do not add the percents. Multiply one change at a time.

A 10% increase multiplies by 1.1.

$100 × 1.1 = $110

A 10% decrease multiplies by 0.9.

$110 × 0.9 = $99

Adding the percents would give 0%, which is $100. That is the trap.

Answer: $99

Level 1

Easy

Two changes. Compare the true result with adding the percents.

A $50 item increases 20%, then increases 10%. What is the final price?

Show Answer

Start with $50.

Do not add the percents. Multiply one change at a time.

A 20% increase multiplies by 1.2.

$50 × 1.2 = $60

A 10% increase multiplies by 1.1.

$60 × 1.1 = $66

Adding the percents would give 30%, which is $65. That is the trap.

Answer: $66

A $200 item decreases 10%, then decreases 10% again. What is the final price?

Show Answer

Start with $200.

Do not add the percents. Multiply one change at a time.

A 10% decrease multiplies by 0.9.

$200 × 0.9 = $180

A 10% decrease multiplies by 0.9.

$180 × 0.9 = $162

Adding the percents would give -20%, which is $160. That is the trap.

Answer: $162

Level 2

Medium

A common sale trap: extra percent off the already-reduced price.

A $80 coat is 25% off, then an extra 20% off the sale price. What do you pay?

Show Answer

Start with $80.

Do not add the percents. Multiply one change at a time.

A 25% decrease multiplies by 0.75.

$80 × 0.75 = $60

A 20% decrease multiplies by 0.8.

$60 × 0.8 = $48

Adding the percents would give -45%, which is $44. That is the trap.

Answer: $48

A $40 value increases 50%, then decreases 50%. What is the final value?

Show Answer

Start with $40.

Do not add the percents. Multiply one change at a time.

A 50% increase multiplies by 1.5.

$40 × 1.5 = $60

A 50% decrease multiplies by 0.5.

$60 × 0.5 = $30

Adding the percents would give 0%, which is $40. That is the trap.

Answer: $30

Level 3

Hard

Three changes, or matching percents that do not cancel.

Start at $125. Increase 20%, increase 20% again, then decrease 20%. What is left?

Show Answer

Start with $125.

Do not add the percents. Multiply one change at a time.

A 20% increase multiplies by 1.2.

$125 × 1.2 = $150

A 20% increase multiplies by 1.2.

$150 × 1.2 = $180

A 20% decrease multiplies by 0.8.

$180 × 0.8 = $144

Adding the percents would give 20%, which is $150. That is the trap.

Answer: $144

Start at $200. Increase 10%, decrease 10%, then increase 10%. What is the final price?

Show Answer

Start with $200.

Do not add the percents. Multiply one change at a time.

A 10% increase multiplies by 1.1.

$200 × 1.1 = $220

A 10% decrease multiplies by 0.9.

$220 × 0.9 = $198

A 10% increase multiplies by 1.1.

$198 × 1.1 = $217.80

Adding the percents would give 10%, which is $220. That is the trap.

Answer: $217.80

Start at $400. Increase 15%, then decrease 15%. What is the final value?

Show Answer

Start with $400.

Do not add the percents. Multiply one change at a time.

A 15% increase multiplies by 1.15.

$400 × 1.15 = $460

A 15% decrease multiplies by 0.85.

$460 × 0.85 = $391

Adding the percents would give 0%, which is $400. That is the trap.

Answer: $391

Percent of a Percent (Subgroups)

Take a percent of the whole, then a percent of that subgroup.

Example

A school has 250 students. 40% are in band. 20% of the band students also play orchestra. How many students play both?

Start with 250.

band students = 40% of 250

band students = 0.4 × 250

band students = 100

both = 20% of 100

both = 0.2 × 100

both = 20

You can also multiply the percents first.

0.4 × 0.2 × 250 = 20

Answer: 20

Level 1

Easy

Two percents. The first percent of the whole, then the second percent of that part.

A crate has 80 apples. 50% are red. 25% of the red apples are large. How many large red apples are there?

Show Answer

Start with 80.

red apples = 50% of 80

red apples = 0.5 × 80

red apples = 40

large red apples = 25% of 40

large red apples = 0.25 × 40

large red apples = 10

You can also multiply the percents first.

0.5 × 0.25 × 80 = 10

Answer: 10

200 tickets are printed. 30% are sold. 10% of the sold tickets are student tickets. How many student tickets were sold?

Show Answer

Start with 200.

tickets sold = 30% of 200

tickets sold = 0.3 × 200

tickets sold = 60

student tickets = 10% of 60

student tickets = 0.1 × 60

student tickets = 6

You can also multiply the percents first.

0.3 × 0.1 × 200 = 6

Answer: 6

Level 2

Medium

Larger totals. Keep the second percent attached to the subgroup, not the original whole.

A company has 450 employees. 60% work full time. 40% of the full-time employees take the bus. How many full-time employees take the bus?

Show Answer

Start with 450.

full-time employees = 60% of 450

full-time employees = 0.6 × 450

full-time employees = 270

bus riders = 40% of 270

bus riders = 0.4 × 270

bus riders = 108

You can also multiply the percents first.

0.6 × 0.4 × 450 = 108

Answer: 108

A school has 800 students. 25% play a sport. Of those athletes, 35% play soccer. How many students play soccer?

Show Answer

Start with 800.

athletes = 25% of 800

athletes = 0.25 × 800

athletes = 200

soccer players = 35% of 200

soccer players = 0.35 × 200

soccer players = 70

You can also multiply the percents first.

0.25 × 0.35 × 800 = 70

Answer: 70

Level 3

Hard

Bigger numbers, or three percents stacked.

1250 people are registered to vote. 80% turn out. 55% of those who turn out vote yes. How many yes votes are there?

Show Answer

Start with 1250.

voters who turned out = 80% of 1250

voters who turned out = 0.8 × 1250

voters who turned out = 1000

yes votes = 55% of 1000

yes votes = 0.55 × 1000

yes votes = 550

You can also multiply the percents first.

0.8 × 0.55 × 1250 = 550

Answer: 550

A warehouse ships 2400 packages. 15% are delayed. 20% of the delayed packages are lost. How many packages are lost?

Show Answer

Start with 2400.

delayed packages = 15% of 2400

delayed packages = 0.15 × 2400

delayed packages = 360

lost packages = 20% of 360

lost packages = 0.2 × 360

lost packages = 72

You can also multiply the percents first.

0.15 × 0.2 × 2400 = 72

Answer: 72

Start with 360. Take 50% of that amount, then 40% of the result, then 25% of that. What number is left?

Show Answer

Start with 360.

first group = 50% of 360

first group = 0.5 × 360

first group = 180

second group = 40% of 180

second group = 0.4 × 180

second group = 72

final group = 25% of 72

final group = 0.25 × 72

final group = 18

You can also multiply the percents first.

0.5 × 0.4 × 0.25 × 360 = 18

Answer: 18

Working Backwards (Finding the Original Value)

The number you see is after the percent change. Write the percent you paid as an equation, then solve for x.

Example

After a 25% increase, a price is $50. What was the original price?

After a 25% increase, the price is $50.

That means you PAID 125% of the original price.

Translate into an equation.

1.25x = 50

x = 501.25

x = 40

The original price was $40.

Answer: $40

Level 1

Easy

Write the percent you paid, then solve .8x = 80 or 1.1x = 55.

After a 10% increase, a price is $55. What was the original price?

Show Answer

After a 10% increase, the price is $55.

That means you PAID 110% of the original price.

Translate into an equation.

1.1x = 55

x = 551.1

x = 50

The original price was $50.

Answer: $50

After a 20% decrease, a price is $80. What was the original price?

Show Answer

After a 20% decrease, the price is $80.

That means you PAID 80% of the original price.

Translate into an equation.

.8x = 80

x = 80.8

x = 100

The original price was $100.

Answer: $100

After a 50% increase, a price is $45. What was the original price?

Show Answer

After a 50% increase, the price is $45.

That means you PAID 150% of the original price.

Translate into an equation.

1.5x = 45

x = 451.5

x = 30

The original price was $30.

Answer: $30

After a 25% decrease, a price is $60. What was the original price?

Show Answer

After a 25% decrease, the price is $60.

That means you PAID 75% of the original price.

Translate into an equation.

.75x = 60

x = 60.75

x = 80

The original price was $80.

Answer: $80

After a 5% increase, a price is $210. What was the original price?

Show Answer

After a 5% increase, the price is $210.

That means you PAID 105% of the original price.

Translate into an equation.

1.05x = 210

x = 2101.05

x = 200

The original price was $200.

Answer: $200

Level 2

Medium

Same method. The percent you paid may be 115%, 60%, 112%, 65%, or 80%.

After a 15% increase, a price is $92. What was the original price?

Show Answer

After a 15% increase, the price is $92.

That means you PAID 115% of the original price.

Translate into an equation.

1.15x = 92

x = 921.15

x = 80

The original price was $80.

Answer: $80

After a 40% decrease, a price is $54. What was the original price?

Show Answer

After a 40% decrease, the price is $54.

That means you PAID 60% of the original price.

Translate into an equation.

.6x = 54

x = 54.6

x = 90

The original price was $90.

Answer: $90

After a 12% increase, a price is $224. What was the original price?

Show Answer

After a 12% increase, the price is $224.

That means you PAID 112% of the original price.

Translate into an equation.

1.12x = 224

x = 2241.12

x = 200

The original price was $200.

Answer: $200

After a 35% decrease, a price is $130. What was the original price?

Show Answer

After a 35% decrease, the price is $130.

That means you PAID 65% of the original price.

Translate into an equation.

.65x = 130

x = 130.65

x = 200

The original price was $200.

Answer: $200

A sale price of $48 is 20% off. What was the original price?

Show Answer

The sale price is $48 after 20% off.

That means you PAID 80% of the original price.

Translate into an equation.

.8x = 48

x = 48.8

x = 60

The original price was $60.

Answer: $60

Level 3

Hard

Same method. The percent you paid may have a decimal, like 112.5% or 102.5%.

After a 12.5% increase, a value is $90. What was the original value?

Show Answer

After a 12.5% increase, the price is $90.

That means you PAID 112.5% of the original price.

Translate into an equation.

1.125x = 90

x = 901.125

x = 80

The original price was $80.

Answer: $80

After 8% tax, a bill is $54. What was the price before tax?

Show Answer

After 8% tax, the bill is $54.

That means you PAID 108% of the original price.

Translate into an equation.

1.08x = 54

x = 541.08

x = 50

The original price was $50.

Answer: $50

After an 18% decrease, a price is $328. What was the original price?

Show Answer

After an 18% decrease, the price is $328.

That means you PAID 82% of the original price.

Translate into an equation.

.82x = 328

x = 328.82

x = 400

The original price was $400.

Answer: $400

A number increased by 60% is 128. What was the original number?

Show Answer

After a 60% increase, the number is 128.

That means you PAID 160% of the original number.

Translate into an equation.

1.6x = 128

x = 1281.6

x = 80

The original number was 80.

Answer: 80

After a 2.5% increase, an account is $1640. What was the original amount?

Show Answer

After a 2.5% increase, the price is $1640.

That means you PAID 102.5% of the original price.

Translate into an equation.

1.025x = 1640

x = 16401.025

x = 1600

The original price was $1600.

Answer: $1600

Mixed Practice

Ten problems that mix the four topics and all three levels. The tag tells you the type.

Percent Change · Easy

A value goes from 20 to 25. What is the percent change?

Show Answer

Original = 20. New = 25.

change = new − original

change = 25 − 20

change = 5

percent change = changeoriginal × 100%

percent change = 520 × 100%

percent change = 14 × 100%

percent change = 25%

Answer: 25% increase

Percent Change · Medium

A price falls from $160 to $136. What is the percent change?

Show Answer

Original = $160. New = $136.

change = original − new

change = $160 − $136

change = $24

percent change = changeoriginal × 100%

percent change = 24160 × 100%

percent change = 320 × 100%

percent change = 15%

Answer: 15% decrease

Successive Changes · Easy

A $100 item increases 10%, then increases 10% again. What is the final price?

Show Answer

Start with $100.

Do not add the percents. Multiply one change at a time.

A 10% increase multiplies by 1.1.

$100 × 1.1 = $110

A 10% increase multiplies by 1.1.

$110 × 1.1 = $121

Adding the percents would give 20%, which is $120. That is the trap.

Answer: $121

Successive Changes · Medium

A $50 item is 30% off, then an extra 10% off the sale price. What do you pay?

Show Answer

Start with $50.

Do not add the percents. Multiply one change at a time.

A 30% decrease multiplies by 0.7.

$50 × 0.7 = $35

A 10% decrease multiplies by 0.9.

$35 × 0.9 = $31.50

Adding the percents would give -40%, which is $30. That is the trap.

Answer: $31.50

Percent of a Percent · Easy

A club has 150 members. 40% go on the trip. 50% of those travelers are first-years. How many first-years go on the trip?

Show Answer

Start with 150.

travelers = 40% of 150

travelers = 0.4 × 150

travelers = 60

first-years = 50% of 60

first-years = 0.5 × 60

first-years = 30

You can also multiply the percents first.

0.4 × 0.5 × 150 = 30

Answer: 30

Percent of a Percent · Hard

A factory makes 2500 parts. 12% fail inspection. 40% of the failed parts are sent back to be remade. How many parts are sent back?

Show Answer

Start with 2500.

failed parts = 12% of 2500

failed parts = 0.12 × 2500

failed parts = 300

parts sent back = 40% of 300

parts sent back = 0.4 × 300

parts sent back = 120

You can also multiply the percents first.

0.12 × 0.4 × 2500 = 120

Answer: 120

Original Value · Easy

After a 20% increase, a price is $72. What was the original price?

Show Answer

After a 20% increase, the price is $72.

That means you PAID 120% of the original price.

Translate into an equation.

1.2x = 72

x = 721.2

x = 60

The original price was $60.

Answer: $60

Original Value · Medium

After a 25% decrease, a price is $75. What was the original price?

Show Answer

After a 25% decrease, the price is $75.

That means you PAID 75% of the original price.

Translate into an equation.

.75x = 75

x = 75.75

x = 100

The original price was $100.

Answer: $100

Original Value · Hard

After a 7.5% increase, a value is $215. What was the original value?

Show Answer

After a 7.5% increase, the price is $215.

That means you PAID 107.5% of the original price.

Translate into an equation.

1.075x = 215

x = 2151.075

x = 200

The original price was $200.

Answer: $200

Successive Changes · Hard

Start at $200. Increase 40%, then decrease 25%. What is the final value?

Show Answer

Start with $200.

Do not add the percents. Multiply one change at a time.

A 40% increase multiplies by 1.4.

$200 × 1.4 = $280

A 25% decrease multiplies by 0.75.

$280 × 0.75 = $210

Adding the percents would give 15%, which is $230. That is the trap.

Answer: $210

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