Percent increase and decrease

Learning goals

Uses a multiplier for a percent increase or decrease.
Computes percent increase or decrease from an original value.
Distinguishes the amount of change from the new value.
Identifies the correct reference value for the percent change.
Checks a percent change using the starting and ending values.

Difficulty: 1 – 5

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Percent increase and decrease 1

A concert ticket price changes from $\$50$ to $\$62$. Identify: a) the original value b) the new value c) the amount of change d) whether the change is an increase or a decrease

Hints

- The original value is the value before the change. - The new value is the value after the change. - Compare the two values to determine both the size and direction of the change.

Solution

1. The starting price is the original value, so it is $\$50$. 2. The ending price is the new value, so it is $\$62$. 3. The amount of change is $\$62-\$50=\$12$. 4. Because the new value is greater than the original value, the change is an increase.

Answer

a) $\$50$ b) $\$62$ c) $\$12$ d) increase

Percent increase and decrease 2

A reading club has $50$ members. Membership grows by $10\%$. How many new members join?

Hints

- The question asks for new members, not the club's new total. - How could you write the given percent as a fraction? - Check whether the added members make sense compared with the starting group.

Solution

1. Ten percent is one tenth of the original amount. 2. One tenth of $50$ members is $50\div10=5$ members.

Answer

$5$ new members

Percent increase and decrease 3

A store lowers a hat's price from $\$18$ to $\$9$. What percent of the original price was taken off? Explain why you compare with the old price.

Hints

- Find how much the price dropped. - Which price was there before the change? - Compare the drop with that earlier price.

Solution

1. The drop is $18-9=\$9$. 2. The amount taken off is half of the original $\$18$, so the decrease is $50\%$.

Answer

$50\%$ was taken off. The old $\$18$ price is the whole that the $\$9$ decrease is compared with.

Percent increase and decrease 4

A pair of shoes cost $\$40$ last week and $\$30$ during this week's sale. Which price represents $100\%$ when you find the percent decrease? Explain.

Hints

- Ask when the change happened. - Does a discount refer to the starting price or the final price?

Solution

1. A percent decrease compares an amount taken away with the original whole. 2. The original price, $\$40$, represents $100\%$. The sale price is the amount left after the decrease.

Answer

$\$40$ represents $100\%$, because it is the price before the sale.

Percent increase and decrease 5

A sod invoice is $\$816.00$ before sales tax and $\$881.28$ after sales tax. By what percent did the tax increase the price?

Hints

- Find the dollar amount of the increase first. - Percent increase compares the increase with the original price, not with the final price. - Convert the resulting decimal rate to a percent.

Solution

1. Find the increase: $\$881.28 - \$816.00 = \$65.28$. 2. Compare the increase with the original pretax price: $65.28 \div 816 = 0.08$. 3. Convert to a percent: $0.08 = 8\%$.

Answer

The tax increased the price by $8\%$.

Percent increase and decrease 6

A school club has $125$ members. Its membership increases by $12\%$. How many members does the club have after the increase?

Hints

- Use the original membership as the base for the percent increase. - Find the amount represented by $12\%$ of the original membership. - Add the increase amount to the original number of members.

Solution

1. Find the increase: $0.12 \times 125 = 15$ members. 2. Add the increase to the original membership: $125 + 15 = 140$.

Answer

$140$ members

Percent increase and decrease 7

A backpack costs $\$80$. The price is decreased by $25\%$. What is the new price?

Hints

- Use the original price as the base for the percent decrease. - Find the amount represented by $25\%$ of the original price. - A decrease means subtracting that amount from the original price.

Solution

1. Find the decrease: $0.25 \times \$80 = \$20$. 2. Subtract the decrease from the original price: $\$80 - \$20 = \$60$.

Answer

$\$60$

Percent increase and decrease 8

A garden club had $40$ seedlings at the start of a project and $50$ seedlings one week later. Was this an increase or a decrease, and what was the percent change based on the starting number of seedlings?

Hints

- Compare the later amount with the starting amount to decide the direction of change. - Find the size of the change before finding a percent. - For percent change, compare the change with the starting amount.

Solution

1. The number of seedlings increased by $50 - 40 = 10$. 2. Use the starting number, $40$, as the base: $\frac{10}{40} = 0.25 = 25\%$. 3. The change was a $25\%$ increase.

Answer

A $25\%$ increase

Percent increase and decrease 9

The number line shows an original value marked O and a new value marked N. What is the percent increase from O to N? Use the original value as the reference whole.

Hints

- Read the numerical values at O and N from the number line. - Find the amount of increase before forming a percent. - For percent increase, compare the change with the original value, not the new value.

Solution

1. The number line shows an original value of $40$ and a new value of $46$. 2. The increase is $46-40=6$. 3. Compare the increase with the original value: $\frac{6}{40}=0.15=15\%$.

Answer

$15\%$

Percent increase and decrease 10

A skate rental rises from $\$12$ to $\$15$. Find the percent increase. Explain which price you used as the whole.

Hints

- Find how much the price changed. - Which price makes sense as the whole when finding an increase from the old price?

Solution

1. The price increases by $\$15-\$12=\$3$. 2. Compare the increase with the original $\$12$: $\frac{3}{12}=25\%$.

Answer

$25\%$; the whole is the original price of $\$12$.

Percent increase and decrease 11

Two community gardens have $25$ and $50$ tomato plants. Each garden adds $8\%$ more plants. Do they add the same number? Find how many plants each garden adds and explain.

Hints

- What number of plants is the starting whole for each garden? - Compare the added plants only after finding each garden's share.

Solution

1. The smaller garden adds $0.08\times25=2$ plants. 2. The larger garden adds $0.08\times50=4$ plants. 3. Both changes are $8\%$, but the numbers added differ because the starting counts differ.

Answer

No. The gardens add $2$ and $4$ plants. Equal percent increases can give different counts when the starting counts differ.

Percent increase and decrease 12

What percent of the original shaded amount was lost from a) to b)?

Hints

- Compare the same-length bars, starting with a). - Count how many shaded parts disappeared. - Compare the lost part with the original shaded amount.

Solution

1. The first bar has $10$ of $10$ parts shaded; the second has $8$. 2. Two out of the original ten parts were lost. 3. The decrease is $\frac{2}{10}=20\%$.

Answer

A $20\%$ decrease

Percent increase and decrease 13

Two aquariums each get $5$ new fish. One aquarium starts with $50$ fish, and the other with $20$. Did their fish counts rise by the same percent? Find both percent increases.

Hints

- Compare the five new fish with each aquarium's starting number. - Does each aquarium have the same starting number of fish? - Can equal-sized additions make different fractions of two groups?

Solution

1. The first aquarium's increase is $\frac{5}{50}=10\%$. 2. The second aquarium's increase is $\frac{5}{20}=25\%$. 3. The same number of new fish makes a different percent increase when the starting counts differ.

Answer

No. The first aquarium increases by $10\%$ and the second by $25\%$.

Percent increase and decrease 14

A jacket costs $\$30$ after a $25\%$ discount. What was its original price? Explain what percent of the original price was paid.

Hints

- The amount paid and the discount together make the full original price. - Can you describe the paid part as a fraction of the original? - How could a known part help you recover the whole?

Solution

1. A $25\%$ discount leaves $100\%-25\%=75\%$ of the original price to pay. 2. Three fourths of the original price are $\$30$. One fourth is $\$30\div3=\$10$. 3. Four fourths cost $4\times\$10=\$40$.

Answer

$\$40$; the sale price is $75\%$ of the original.

Percent increase and decrease 15

A $\$100$ bike helmet goes up in price by $10\%$. Later, the new price is reduced by $10\%$. Is the final price back to $\$100$? Explain using the two different starting amounts.

Hints

- What amount does the first percentage describe? - What amount does the second percentage describe? - Compare the two dollar changes, not just the percent labels.

Solution

1. The increase is $10\%$ of $\$100$, or $\$10$, so the new price is $\$110$. 2. The later decrease is $10\%$ of $\$110$, or $\$11$. 3. The final price is $\$110-\$11=\$99$, not $\$100$.

Answer

No. The final price is $\$99$; the decrease is based on $\$110$, not $\$100$.

Percent increase and decrease 16

A backpack's price drops from $\$80$ to $\$60$. Kai says the decrease is $\frac{20}{60}$ of the price. Find the correct percent decrease and explain Kai's error.

Hints

- What amount did the price lose? - A percentage change compares the change with which price? - Does Kai's fraction use that amount as its whole?

Solution

1. The decrease is $\$80-\$60=\$20$. 2. Compare the decrease with the original price: $\frac{20}{80}=\frac{1}{4}=25\%$. 3. Kai used the new price as the whole instead of the original price.

Answer

A $25\%$ decrease. Kai divided by $\$60$ instead of the original $\$80$.

Percent increase and decrease 17

A science club has $55$ members after membership increases by $10\%$. How many members did it have before the increase? Explain.

Hints

- Does the new membership represent exactly $100\%$ of the old membership, or more? - Can the new total be split into equal tenths of the old total? - How could you check an earlier membership count?

Solution

1. The new membership is $110\%$ of the old membership, or eleven equal tenths. 2. One tenth of the old membership is $55\div11=5$ members. 3. Ten tenths make the old membership: $10\times5=50$ members. 4. Check: ten percent of $50$ is $5$, making the new total $55$.

Answer

$50$ members; adding $10\%$ of $50$, or $5$ members, gives $55$.

Percent increase and decrease 18

After $40\%$ of the water in a rain barrel is used, $36\,\text{gal}$ remains. How many gallons were in the barrel at first? Explain what percent of the starting amount remains.

Hints

- Is the water left the whole starting amount, or only a part of it? - Could a bar split into equal parts help show what was used and what remains? - How could you check a starting amount against both the amount used and the amount left?

Solution

1. Since $40\%$ was used, $60\%$ of the original water remains. 2. This is $\frac{3}{5}$ of the starting amount, so one fifth is $36\div3=12\,\text{gal}$. 3. The full starting amount was $5\times12=60\,\text{gal}$. 4. Check: $40\%$ of $60$ is $24$ gallons, leaving $36$ gallons.

Answer

$60\,\text{gal}$ at first; $60\%$ remained after the water was used.

Percent increase and decrease 19

Read the news report and evaluate its claim. “Bicycle theft has risen dramatically in our town! The number of reported thefts increased by $100\%$ from last year.” One bicycle theft was reported last year, and two were reported this year. Explain why the headline could create a misleading impression.

Hints

- Find the actual increase in the number of reports. - Compare the effect of saying “one more case” with saying “a $100\%$ increase.” - Consider how a small starting value affects percent change.

Solution

1. The number increased by $2 - 1 = 1$ theft. 2. Relative to last year’s $1$ theft, the percent increase is $\frac{1}{1} \times 100\% = 100\%$, so the numerical claim is correct. 3. However, the absolute number only changed from $1$ to $2$. With a very small starting value, a small absolute change can produce a large percent increase. The word “dramatically” may therefore exaggerate the scale of the situation.

Answer

The $100\%$ increase is mathematically correct, but the headline is potentially misleading because the reported count rose by only one case, from $1$ to $2$.

Percent increase and decrease 20

Decrease a duration of $1\,\text{h}\ 40\,\text{min}$ by $15\%$. Give the result in hours and minutes.

Hints

- Would it be easier to convert the entire duration to minutes first? - How many minutes are in one hour? - A decrease means the percent amount must be subtracted.

Solution

1. Convert the duration to minutes: $1\,\text{h}\ 40\,\text{min} = 60\,\text{min} + 40\,\text{min} = 100\,\text{min}$. 2. Find the decrease: $15\%$ of $100\,\text{min}$ is $100\,\text{min} \times 0.15 = 15\,\text{min}$. 3. Subtract the decrease: $100\,\text{min} - 15\,\text{min} = 85\,\text{min}$. 4. Convert back to hours and minutes: $85\,\text{min} = 1\,\text{h}\ 25\,\text{min}$.

Answer

$1\,\text{h}\ 25\,\text{min}$

Percent increase and decrease 21

A smartphone battery has a capacity of $4500\,\text{mAh}$, where mAh means milliamp-hours. a) After one hour of heavy use, $3330\,\text{mAh}$ remains. What percent of the original charge remains? b) After one more hour in power-saving mode, $2830.5\,\text{mAh}$ remains. By what percent did the charge decrease during the second hour, based on the charge after the first hour?

Hints

- Identify the whole used in each part. - For part a), compare the remaining charge with the original capacity. - For part b), first find the change in charge, then compare it with the charge at the start of the second hour.

Solution

1. For part a), the remaining portion is $\frac{3330}{4500} = 0.74 = 74\%$. 2. During the second hour, the charge decreases by $3330\,\text{mAh} - 2830.5\,\text{mAh} = 499.5\,\text{mAh}$. 3. Relative to the $3330\,\text{mAh}$ starting charge for that hour, the percent decrease is $\frac{499.5}{3330} \times 100\% = 15\%$.

Answer

a) $74\%$ of the original charge remains. b) The charge decreased by $15\%$ during the second hour.

Percent increase and decrease 22

Lakeview registered $120$ new electric vehicles last year and $168$ this year. a) Find the percent increase in new electric-vehicle registrations. b) Lakeview has $2400$ registered vehicles altogether. What percent of all registered vehicles is represented by this year’s $168$ new electric vehicles? c) Last year, electric vehicles made up exactly $25\%$ of all $480$ newly registered vehicles. Verify that this is consistent with the stated $120$ electric vehicles.

Hints

- Identify the base value in each part. - For percent increase, compare the increase with last year’s number. - To verify part c), find $25\%$ of the stated total.

Solution

1. For part a), the increase is $168 - 120 = 48$. The percent increase is $\frac{48}{120} \times 100\% = 40\%$. 2. For part b), $\frac{168}{2400} \times 100\% = 7\%$. 3. For part c), $480 \times 0.25 = 120$, so the statement is consistent.

Answer

a) Registrations increased by $40\%$. b) This year’s new electric vehicles represent $7\%$ of all registered vehicles. c) Yes. $25\%$ of $480$ is $120$.

Percent increase and decrease 23

A company has two offices with different numbers of apprentices. Harbor Office: $80$ employees, including $12$ apprentices. Ridge Office: $120$ employees, including $15$ apprentices. a) Find the percent of employees who are apprentices at each office. Which office has the greater percent? b) Harbor Office plans to increase its number of apprentices by $25\%$. How many apprentices will it then have?

Hints

- Compare the number of apprentices with the total number of employees at each office. - A percent increase is added to the original amount.

Solution

1. At Harbor Office, the apprentice percent is $\frac{12}{80} \times 100\% = 15\%$. 2. At Ridge Office, the apprentice percent is $\frac{15}{120} \times 100\% = 12.5\%$. Harbor Office has the greater percent. 3. A $25\%$ increase in $12$ apprentices is $12 \times 0.25 = 3$. The new number is $12 + 3 = 15$.

Answer

a) Harbor Office: $15\%$; Ridge Office: $12.5\%$. Harbor Office has the greater percent. b) Harbor Office will have $15$ apprentices.

Percent increase and decrease 24

For a household electricity-use model, use a total annual use of $3500\,\text{kWh}$, where kWh means kilowatt-hours, divided among the categories shown in the pie chart. a) How many kilowatt-hours are used for cooking? b) Replacing the household’s lights with efficient LEDs reduces the lighting electricity use by $60\%$. How many kilowatt-hours are saved per year? c) What is the household’s new total annual electricity use after this savings?

Hints

- Read the current category percentages from the pie chart. - The $60\%$ reduction applies only to lighting use, not to the total electricity use. - Subtract the saved amount from the original total.

Solution

1. For part a), cooking uses $15\%$ of the total: $3500\,\text{kWh} \times 0.15 = 525\,\text{kWh}$. 2. Lighting currently uses $10\%$ of the total: $3500\,\text{kWh} \times 0.10 = 350\,\text{kWh}$. 3. For part b), the savings is $60\%$ of the lighting use: $350\,\text{kWh} \times 0.60 = 210\,\text{kWh}$. 4. For part c), the new total use is $3500\,\text{kWh} - 210\,\text{kWh} = 3290\,\text{kWh}$.

Answer

a) $525\,\text{kWh}$ b) $210\,\text{kWh}$ c) $3290\,\text{kWh}$

Percent increase and decrease 25

A paint purchase costs $\$54.00$ after an $8\%$ sales tax is added. What was the price before tax?

Hints

- The final price includes the whole original price plus an additional $8\%$ of that original price. - Think about what percent of the original price the final amount represents. - Work backward from the final amount to the original amount.

Solution

1. After an $8\%$ increase, the final price is $108\%$ of the original price. 2. Let the original price be $x$. Then $1.08x = 54$. 3. Divide by $1.08$: $x = 54 \div 1.08 = 50$. 4. The price before tax was $\$50.00$.

Answer

$\$50.00$

Percent increase and decrease 26

A desk price increases from $\$80$ to $\$100$. Elena says the percent increase is $20\%$ because the increase is $\$20$ and $\frac{20}{100}=20\%$. Explain Elena’s error and find the correct percent increase.

Hints

- First identify which price is the starting value. - The change amount and the reference whole play different roles in a percent-change calculation. - Ask which value the phrase “increase from” makes the reference whole.

Solution

1. The amount of increase is $\$100-\$80=\$20$. 2. Elena divided by the new price, but percent increase uses the original price as the reference whole. 3. The correct percent increase is $\frac{20}{80}=\frac{1}{4}=25\%$.

Answer

Elena used the new value as the denominator. The original value is $\$80$, so the correct percent increase is $25\%$.

Percent increase and decrease 27

A game controller costs $\$72$ after its original price is decreased by $20\%$. What was the original price?

Hints

- Determine what percent of the original price remains after the decrease. - The given $\$72$ is the new value, not the original whole. - Work backward from the remaining percent to recover the whole.

Solution

1. After a $20\%$ decrease, the new price is $80\%$ of the original price. 2. Let the original price be represented by the whole. Since $\$72$ is $80\%$ of that whole, divide by $0.80$: $72 \div 0.80=90$. 3. Check: $20\%$ of $\$90$ is $\$18$, and $\$90-\$18=\$72$.

Answer

$\$90$

Percent increase and decrease 28

Orion Robotics Club grows from $40$ members to $50$ members. Maple Robotics Club grows from $80$ members to $90$ members. Both clubs gain $10$ members. Which club has the greater percent increase, and how many times as large is that percent increase as the other club’s?

Hints

- Equal absolute changes do not guarantee equal percent changes. - For each club, compare the gain with that club’s own original membership. - After finding both percentages, compare the percentages multiplicatively.

Solution

1. Orion’s increase is $10$ members from an original $40$, so its percent increase is $\frac{10}{40}=25\%$. 2. Maple’s increase is $10$ members from an original $80$, so its percent increase is $\frac{10}{80}=12.5\%$. 3. Compare the percentages: $25 \div 12.5=2$. 4. Orion’s percent increase is therefore twice Maple’s, even though the absolute increases are equal.

Answer

Orion Robotics Club. Its percent increase is $25\%$, which is $2$ times Maple Robotics Club’s $12.5\%$ increase.

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