Aimathic – Percent increase and decrease

Percent increase and decrease

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Difficulty: 1 – 5

5542116

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

5358286

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 $8\%$.

5512056

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

5512066

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

$\$60$

5512076

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

5542126

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

$15\%$

5115666

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 $2$.

5117486

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

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

5127786

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.

5127796

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 $120$.

5127806

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.

5354956

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) $3290\,\text{kWh}$

5358296

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

$\$50.00$

5542136

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 $25\%$.

5542146

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

$\$90$

5542156

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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