Problem-Solving and Data Analysis Practice Questions — SAT

Question 1Question →

A laboratory dispensing pump releases a liquid solution at a constant rate of $0.75\text{ deciliters per minute}$ . If $1\text{ deciliter} = 100\text{ milliliters}$ , what is the pump's dispensing rate, in milliliters per second?

Show answer & explanation

Answer: 1.25

Answer

The correct answer is 1.25. The pump dispenses 1.25 milliliters per second.

To find the dispensing rate in milliliters per second, we first convert the volume from deciliters to milliliters. Multiplying

0.75\text{ deciliters}

by

100\text{ milliliters per deciliter}

gives

75\text{ milliliters per minute}

. Next, to convert minutes to seconds, we divide the rate by

60

(since

1\text{ minute} = 60\text{ seconds}

), which yields

\frac{75}{60} = 1.25\text{ milliliters per second}

.

Step-by-Step Solution

1

Convert the dispensing rate from deciliters per minute to milliliters per minute using the conversion factor

1\text{ deciliter} = 100\text{ milliliters}

.

0.75\text{ deciliters/minute} \times 100\text{ milliliters/deciliter} = 75\text{ milliliters/minute}

This converts the volume unit from deciliters to milliliters.

2

Convert the rate from milliliters per minute to milliliters per second by dividing by

60

, since there are

60\text{ seconds}

in

1\text{ minute}

.

75\text{ milliliters/minute} \div 60\text{ seconds/minute} = 1.25\text{ milliliters/second}

This converts the time unit from minutes to seconds, yielding the final rate.

Key Concept

Unit Conversions

Alternative Method

Alternatively, you can convert the units in a single dimensional analysis step:

0.75 \text{ dL/min} \times \frac{100 \text{ mL}}{1 \text{ dL}} \times \frac{1 \text{ min}}{60 \text{ s}} = \frac{75}{60} \text{ mL/s} = 1.25 \text{ mL/s}

.

Estimated Time:1m 30s

Question 2Question →

A solar energy farm has two arrays of solar panels, Array A and Array B. At the beginning of 2024 ( $t = 0$ ), the generating capacity of Array A is $50$ megawatts and increases by $11$ megawatts each year. At the beginning of 2024, the generating capacity of Array B is $32$ megawatts and increases exponentially by a constant percent each year. If the generating capacities of the two arrays are equal at the beginning of 2026 ( $t = 2$ ), what is the generating capacity, in megawatts, of Array B at the beginning of 2027 ( $t = 3$ )?

Show answer & explanation

Answer: 108

Answer

The correct answer is 108.

The capacity of Array A at

t = 2

is calculated using the linear growth model

A(t) = 50 + 11t

, which yields

A(2) = 50 + 11(2) = 72

megawatts. Since the capacity of Array B grows exponentially and starts at

32

megawatts, its model is

B(t) = 32 \cdot b^t

, where

b

is the annual growth factor. At

t = 2

, their capacities are equal, so

32 \cdot b^2 = 72

. Dividing both sides by

32

yields

b^2 = 2.25

. Taking the positive square root gives

b = 1.5

. To find the capacity of Array B at

t = 3

, we compute

B(3) = 32 \cdot (1.5)^3 = 108

megawatts.

Step-by-Step Solution

1

Write the linear model for the capacity of Array A.

A(t) = 50 + 11t

, where

t

is the number of years since the beginning of 2024.

Array A increases by a constant number of megawatts each year, which represents linear growth.

2

Calculate the capacity of Array A at

t = 2

.

A(2) = 50 + 11(2) = 72

megawatts.

We need to find the capacity of Array A at the beginning of 2026 to determine the capacity of Array B at that same time.

3

Represent the exponential model for the capacity of Array B.

B(t) = 32 \cdot b^t

, where

b

is the constant annual growth factor.

Array B increases exponentially by a constant percent each year, which is represented by a geometric sequence or exponential function.

4

Solve for the growth factor

b

using the capacity at

t = 2

.

32 \cdot b^2 = 72 \implies b^2 = 2.25 \implies b = 1.5

.

Since the capacities of both arrays are equal at the beginning of 2026 (

t = 2

), we set

B(2) = A(2) = 72

to find the positive base

b

.

5

Find the capacity of Array B at

t = 3

.

B(3) = 32 \cdot (1.5)^3 = 108

megawatts.

We substitute

t = 3

into the exponential model for Array B to find its capacity at the beginning of 2027.

Key Concept

Modeling linear and exponential growth, finding parameters of functions, and calculating values at specific time steps.

Alternative Method

Once the growth factor

b = 1.5

is found, the capacity of Array B at

t = 3

can be calculated by multiplying the capacity at

t = 2

by the growth factor:

B(3) = B(2) \cdot b = 72 \cdot 1.5 = 108

megawatts.

Estimated Time:1m 30s

Question 3Question →

A toy train moves along a track at a constant rate of $15$ feet per second. What is the train's speed, in yards per second? (Given that $1\text{ yard} = 3\text{ feet}$ )

5

12

18

45

Show answer & explanation

Answer: 5

Answer

5

The correct answer is 5. To convert the speed from feet per second to yards per second, divide the rate of

15

feet per second by the conversion factor of

3

feet per yard. This yields

5

yards per second.

Step-by-Step Solution

1

Identify the given rate and the conversion factor.

The given rate is

15\text{ feet per second}

, and the conversion factor is

1\text{ yard} = 3\text{ feet}

.

This establishes the relationship between the current unit (feet) and the target unit (yards).

2

Convert the speed by dividing by the conversion factor.

\frac{15\text{ feet}}{1\text{ second}} \times \frac{1\text{ yard}}{3\text{ feet}} = 5\text{ yards per second}

.

Dividing the speed in feet per second by 3 converts the distance unit from feet to yards, resulting in the correct rate of 5 yards per second.

Key Concept

Unit Conversions

Estimated Time:45s

Question 4Question →

At the beginning of a school year, a bookstore had $250$ copies of a textbook in stock. During the first week of classes, the bookstore sold $30\%$ of the textbooks in stock. During the second week, a new shipment increased the number of textbooks in stock by $40\%$ of the number of textbooks in stock at the end of the first week. How many copies of the textbook were in stock at the end of the second week?

Show answer & explanation

Answer: 245

Answer

245

To find the final quantity, we must calculate the successive percentage changes sequentially. First, a

30\%

decrease on the initial

250

textbooks leaves

70\%

of the stock:

250 \times 0.70 = 175

. Second, a

40\%

increase on this new amount of

175

textbooks results in

175 \times 1.40 = 245

textbooks in stock at the end of the second week.

Step-by-Step Solution

1

Find the number of textbooks remaining after the first week's sales.

175

Selling

30\%

of the

250

textbooks leaves

70\%

of them:

250 \times (1 - 0.30) = 175

.

2

Find the total number of textbooks in stock after the second week's shipment increase.

245

An increase of

40\%

on the

175

remaining textbooks results in

140\%

of that amount:

175 \times (1 + 0.40) = 245

.

Key Concept

Applying consecutive percent decreases and increases to changing base values.

Question 5Question →

A survey of 140 students in the 9th and 10th grades asked about their preferred after-school activity. The results are summarized in the table below.

Grade	Sports	Arts	Total
9th Grade	40	20	60
10th Grade	30	50	80
Total	70	70	140

If a student is selected at random from the 9th-grade students, what is the probability that the selected student preferred sports?

\frac{2}{7}

\frac{1}{3}

\frac{2}{3}

\frac{4}{7}

Show answer & explanation

Answer: $\frac{2}{3}$

Answer

\frac{2}{3}

The correct answer is

\frac{2}{3}

. The question asks for the probability that a student preferred sports, given that the student is in the 9th grade. This means we only consider the 60 students in the 9th grade. Of those 60 students, 40 preferred sports. Therefore, the probability is

\frac{40}{60}

, which simplifies to

\frac{2}{3}

.

Step-by-Step Solution

1

Identify the group of interest from which the student is selected.

The selection is made from the 9th-grade students, so the denominator for the probability is the total number of 9th-grade students, which is

60

.

The problem states: 'If a student is selected at random from the 9th-grade students,' establishing this as a conditional probability where the sample space is restricted to 9th graders.

2

Identify the number of students within this group who satisfy the target condition.

The number of 9th-grade students who preferred sports is

40

.

We look at the intersection of the '9th Grade' row and the 'Sports' column in the table.

3

Calculate the probability by dividing the number of target outcomes by the total outcomes in the conditional sample space, and simplify the fraction.

The probability is

\frac{40}{60}

, which simplifies to

\frac{2}{3}

.

Dividing both the numerator and the denominator by their greatest common divisor, 20, gives the simplified probability.

Key Concept

Calculating conditional probability from a two-way frequency table by restricting the sample space to a specific row or column.

Question 6Question →

A transit agency surveyed a sample of 160 commuters about their primary mode of transportation and whether they travel during peak commuting hours. The results are summarized in the table below.

Commute Mode	Peak Hours	Off-Peak Hours	Total
Bus	35	45	80
Train	55	25	80
Total	90	70	160

If a commuter who travels during peak hours is selected at random from those surveyed, what is the probability that the commuter's primary mode of transportation is the train?

\frac{11}{32}

\frac{11}{18}

\frac{11}{16}

\frac{7}{18}

Show answer & explanation

Answer: $\frac{11}{18}$

Answer

The probability that the commuter's primary mode of transportation is the train, given they travel during peak hours, is

\frac{11}{18}

.

The correct answer is

\frac{11}{18}

. To find the probability that a selected commuter's primary mode of transportation is the train, given that they travel during peak hours, we must look only at the column representing peak-hour commuters. The total number of peak-hour commuters is

35 + 55 = 90

. Among these commuters, the number whose primary mode of transportation is the train is

55

. Therefore, the probability is

\frac{55}{90}

, which simplifies to

\frac{11}{18}

.

Step-by-Step Solution

1

Identify the total number of commuters who meet the condition of traveling during peak hours.

The total number of peak-hour commuters is

90

.

The question asks for the probability given that the commuter travels during peak hours, restricting our sample space to this column total.

2

Identify the number of commuters in the peak hours group whose primary mode of transportation is the train.

There are

55

commuters who travel by train during peak hours.

This represents the favorable outcomes within our restricted sample space.

3

Calculate the conditional probability by dividing the number of favorable outcomes by the size of the restricted sample space.

\frac{55}{90} = \frac{11}{18}

Probability is the ratio of favorable outcomes to the total possible outcomes under the given condition.

Key Concept

Conditional Probability from a Two-Way Table

Question 7Question →

In 2024, a city recycled a certain number of tons of plastic. In 2025, the number of tons of plastic recycled was $15\%$ greater than the number of tons recycled in 2024. In 2026, the number of tons of plastic recycled was $20\%$ less than the number of tons recycled in 2025. If the city recycled $5,520$ tons of plastic in 2026, how many tons of plastic did the city recycle in 2024?

Show answer & explanation

Answer: 6000

Answer

The city recycled

6,000

tons of plastic in 2024.

To find the number of tons of plastic recycled in 2024, we must work backward from the 2026 value of 5,520 tons. Let

x

represent the tons recycled in 2025. Since the 2026 value represents a

20\%

decrease from 2025, we have

0.80x = 5,520

, which simplifies to

x = 6,900

tons. Next, let

y

represent the tons recycled in 2024. Since the 2025 value represents a

15\%

increase from 2024, we have

1.15y = 6,900

. Solving for

y

yields

y = \frac{6,900}{1.15} = 6,000

tons.

Step-by-Step Solution

1

Set up an equation to find the amount of plastic recycled in 2025 using the 2026 amount of 5,520 tons.

The amount of plastic recycled in 2025 was

6,900

tons.

The 2026 amount is a

20\%

decrease from the 2025 amount, so

5,520

is equal to

80\%

of the 2025 amount. Solving

0.80x = 5,520

gives

x = 6,900

.

2

Set up an equation to find the amount of plastic recycled in 2024 using the 2025 amount of 6,900 tons.

The amount of plastic recycled in 2024 was

6,000

tons.

The 2025 amount is a

15\%

increase from the 2024 amount, so

6,900

is equal to

115\%

of the 2024 amount. Solving

1.15y = 6,900

gives

y = 6,000

.

Key Concept

Working backward through multiple sequential percentage changes.

Question 8Question →

A pharmaceutical company produces two strengths of a certain medication, Strength P and Strength Q, by mixing an active ingredient with a liquid base. In Strength P, the ratio of the volume of the active ingredient to the volume of the liquid base is $2$ to $5$ . In Strength Q, the ratio of the volume of the active ingredient to the volume of the liquid base is $3$ to $8$ . A technician prepares a batch of Strength P and a batch of Strength Q such that the volume of the liquid base used in the batch of Strength P is equal to the volume of the liquid base used in the batch of Strength Q. If the total volume of the active ingredient used in both batches combined is $62$ liters, what is the total volume of the liquid base, in liters, used in both batches combined?

Show answer & explanation

Answer: 160

Answer

The total volume of the liquid base used in both batches combined is 160 liters.

The total volume of the liquid base used in both batches combined is 160 liters. To find this, let the volume of the liquid base in each batch be

L

liters. The volume of the active ingredient in Strength P is

\frac{2}{5}L

liters, and the volume of the active ingredient in Strength Q is

\frac{3}{8}L

liters. The sum of the active ingredients is

\frac{2}{5}L + \frac{3}{8}L = 62

liters. Finding a common denominator gives

\frac{16}{40}L + \frac{15}{40}L = 62

, which simplifies to

\frac{31}{40}L = 62

. Solving for

L

yields

L = 62 \times \frac{40}{31} = 80

liters. Since both batches contain equal volumes of liquid base, the combined total volume of the liquid base is

2L = 2 \times 80 = 160

liters.

Step-by-Step Solution

1

Represent the volumes of the ingredients in terms of a single variable.

Let the volume of the liquid base used in each batch be

40x

liters, where 40 is chosen as the least common multiple of 5 and 8. The volume of the active ingredient in Strength P is

\frac{2}{5} \times 40x = 16x

liters, and the volume of the active ingredient in Strength Q is

\frac{3}{8} \times 40x = 15x

liters.

This establishes a common algebraic scale for both mixtures, utilizing the fact that the volume of the liquid base is identical in both batches.

2

Formulate an equation using the total volume of the active ingredient.

16x + 15x = 62

, which simplifies to

31x = 62

.

The problem states that the combined volume of the active ingredient from both batches is 62 liters.

3

Solve for the variable

x

.

x = 2

Dividing both sides of the equation by 31 yields the scaling factor.

4

Calculate the total volume of the liquid base used in both batches combined.

Total liquid base =

40x + 40x = 80x = 80(2) = 160

liters.

Since each batch uses

40x

liters of liquid base, the total volume for both batches is the sum of their individual base volumes, which is

80x

liters.

Key Concept

Solving ratio and proportion problems with a shared quantity

Estimated Time:2m 30s

Question 9Question →

In 2024, a solar power system generated $250$ megawatt-hours of electricity. In 2025, the system's electricity generation was $12\%$ greater than in 2024. In 2026, the electricity generation was $5\%$ less than in 2025. How many megawatt-hours of electricity did the solar power system generate in 2026?

Show answer & explanation

Answer: 266

Answer

266

To find the electricity generation in 2026, we first determine the generation in 2025. Since the 2025 generation was

12\%

greater than the 2024 generation of

250

megawatt-hours, the 2025 generation is

250 \times (1 + 0.12) = 250 \times 1.12 = 280

megawatt-hours. Next, the 2026 generation was

5\%

less than the 2025 generation of

280

megawatt-hours. Therefore, the 2026 generation is

280 \times (1 - 0.05) = 280 \times 0.95 = 266

megawatt-hours.

Step-by-Step Solution

1

Calculate the electricity generation in 2025.

280 megawatt-hours

The 2025 generation is

12\%

greater than the 2024 generation of

250

megawatt-hours. So, we multiply

250

by

1 + 0.12 = 1.12

to get

280

.

2

Calculate the electricity generation in 2026.

266 megawatt-hours

The 2026 generation is

5\%

less than the 2025 generation of

280

megawatt-hours. So, we multiply

280

by

1 - 0.05 = 0.95

to get

266

.

Key Concept

Calculating consecutive percent changes by applying each percentage change to the correct sequential base value.

Alternative Method

We can set up a single equation representing the sequential changes:

\text{2026 Generation} = 250 \times (1 + 0.12) \times (1 - 0.05) = 250 \times 1.12 \times 0.95 = 266

megawatt-hours.

Estimated Time:1m 15s

Question 10Question →

The price of a book was increased by $20\%$ to $\$30$ . What was the original price, in dollars, of the book?

24

25

36

6

Show answer & explanation

Answer: 25

Answer

The original price of the book was $25.

The correct answer is the price of

25. Let

x

represent the original price. Since the price was increased by

20\%

, the new price is

120\%

of

x

, or

1.20x

. Setting up the equation

1.20x = 30

and dividing both sides by

1.20

gives

x = 25$.

Step-by-Step Solution

1

Set up an equation representing the percent increase. Let

x

represent the original price of the book. A

20\%

increase means the final price is

120\%

of the original price.

1.20x = 30

To find the original price, we must relate the original price, the percent increase, and the final price.

2

Solve for

x

by dividing both sides of the equation by

1.20

.

x = \frac{30}{1.20} = 25

Dividing isolates the variable

x

to find the original price.

Key Concept

Solving for the initial value after a percentage increase.

Question 11Question →

The table below shows the number of employees in Division A and Division B of a company in 2024.

Division	Number of employees (2024)
Division A	160
Division B	240

In 2025, the number of employees in Division A increased by 35%, and the total number of employees in the company increased by 2%. What was the percent decrease in the number of employees in Division B from 2024 to 2025?

12%

20%

25%

80%

Show answer & explanation

Answer: 20%

Answer

The correct answer is 20%.

The correct answer is 20%. To find this, first calculate the total number of employees in 2024, which is

160 + 240 = 400

. In 2025, Division A's employees increase by 35% to

160 \times 1.35 = 216

, and the total number of employees increases by 2% to

400 \times 1.02 = 408

. The number of employees in Division B in 2025 is the difference:

408 - 216 = 192

. The decrease in Division B is

240 - 192 = 48

employees. Finally, the percent decrease is calculated relative to Division B's original count of 240:

\frac{48}{240} \times 100\% = 20\%

.

Step-by-Step Solution

1

Find the total number of employees in the company in 2024.

The total number of employees in 2024 is

160 + 240 = 400

.

To find changes in the total company size, we need to establish the baseline total.

2

Calculate the number of employees in Division A in 2025.

The number of employees in Division A in 2025 is

160 \times 1.35 = 216

.

Division A's workforce increased by 35% from its 2024 value of 160.

3

Calculate the total number of employees in the company in 2025.

The total number of employees in 2025 is

400 \times 1.02 = 408

.

The total company workforce increased by 2% from its 2024 value of 400.

4

Determine the number of employees in Division B in 2025.

The number of employees in Division B in 2025 is

408 - 216 = 192

.

The employees in Division B are the remaining employees after subtracting Division A's employees from the company total.

5

Calculate the percent decrease in Division B's employees from 2024 to 2025.

The decrease in Division B is

240 - 192 = 48

employees. The percent decrease is

\frac{48}{240} \times 100\% = 20\%

.

The percent decrease is the change divided by the initial value of Division B (240), multiplied by 100.

Key Concept

Solving multi-step percent problems with varying base values.

Alternative Method

Instead of calculating the absolute number of employees, you can use the weighted average of the percent changes. Since Division A represents

\frac{160}{400} = 40\%

of the company and Division B represents

\frac{240}{400} = 60\%

, the weighted percent change equation is:

0.40(35\%) + 0.60(x) = 2\%

, where

x

is the percent change for Division B. Solving for

x

gives:

14\% + 0.60x = 2\% \Rightarrow 0.60x = -12\% \Rightarrow x = -20\%

, representing a

20\%

decrease.

Estimated Time:2m 0s

Question 12Question →

The price of a ticket to a museum is $25. If a$ 6\%$ sales tax is added to the price of the ticket, what is the total cost of the ticket, in dollars?

Show answer & explanation

Answer: 26.5

Answer

The correct answer is 26.5.

To find the total cost of the museum ticket, the

6\%

sales tax must be added to the original price. The tax amount is calculated by converting the percent to a decimal and multiplying it by the original price:

25 \times 0.06 = 1.5

dollars. Adding this tax to the original price of

25

gives the total cost:

25 + 1.5 = 26.5

dollars. Alternatively, this can be solved by multiplying the original price by

1.06

:

25 \times 1.06 = 26.5

dollars.

Step-by-Step Solution

1

Calculate the amount of sales tax.

1.5

dollars

To find

6\%

of

25

, convert the percentage to a decimal by dividing by 100 (

6\% = 0.06

) and multiply by the ticket price:

25 \times 0.06 = 1.5

.

2

Add the sales tax to the original ticket price to get the total cost.

26.5

dollars

The total cost is the sum of the original price and the tax:

25 + 1.5 = 26.5

.

Key Concept

Calculating a final value after a percentage increase

Alternative Method

The total cost of the ticket including a

6\%

tax can be found in one step by multiplying the original price by

1.06

:

25 \times 1.06 = 26.5

.

Estimated Time:45s

Question 13Question →

An online learning platform has $500$ active subscribers. If the number of active subscribers increases by $12\%$ next month, what is the total number of active subscribers the platform will have next month?

440

512

560

600

Show answer & explanation

Answer: $560$

Answer

The total number of active subscribers the platform will have next month is

560

.

To find the total number of active subscribers after a

12\%

increase, first calculate

12\%

of

500

. This is done by converting

12\%

to its decimal equivalent,

0.12

, and multiplying by the base:

500 \times 0.12 = 60

. Then, add the increase to the initial value:

500 + 60 = 560

. Alternatively, multiply the initial value by

1.12

to directly get the new total:

500 \times 1.12 = 560

. Therefore, the total number of active subscribers next month is

560

.

Step-by-Step Solution

1

Calculate the amount of increase by finding

12\%

of the initial

500

subscribers.

500 \times 0.12 = 60

To find the number of new subscribers, the percentage of increase must be multiplied by the initial amount.

2

Add the increase to the initial number of subscribers to find the new total.

500 + 60 = 560

An increase means adding the calculated amount of change to the baseline value.

Key Concept

Calculating a percentage increase involves finding the specified percent of the original value and adding it to that original value.

Question 14Question →

A store owner purchases an item from a wholesaler. She increases the wholesaler's cost by $40\%$ to set the retail price of the item. During a clearance sale, the retail price is discounted by $20\%$ . If the clearance sale price of the item is $84.00$ dollars, what was the wholesaler's cost of the item, in dollars?

70

75

63

72

Show answer & explanation

Answer: 75

Answer

75

The correct answer is

75

. The relationship between the wholesaler's cost

C

, the retail price, and the clearance price is

C \times 1.40 \times 0.80 = 84

. Simplifying the expression yields

1.12C = 84

, which solves to

C = 75

.

Step-by-Step Solution

1

Represent the retail price in terms of the wholesaler's cost

C

.

Retail price =

1.40C

The retail price represents a

40\%

markup on the original wholesaler's cost.

2

Represent the clearance sale price in terms of the retail price.

Clearance price =

0.80 \times (1.40C) = 1.12C

A

20\%

discount reduces the price to

80\%

of the retail price.

3

Set up the equation using the given clearance sale price of

84.00

dollars and solve for

C

.

1.12C = 84 \implies C = \frac{84}{1.12} = 75

Dividing the sale price by the net percentage factor isolates the initial cost.

Key Concept

Applying consecutive percentage changes to find an unknown initial value.

Estimated Time:1m 30s

Question 15Question →

In 2015, the population of a certain species of owl in a nature reserve was $x$ . By 2020, the population had increased by $35\%$ . Between 2020 and 2025, the population decreased by $20\%$ of its 2020 value. If the population of the owl species in 2025 was $270$ , what was the value of $x$ ?

Show answer & explanation

Answer: 250

Answer

The population in 2015 was 250.

To find the initial population

x

, we apply the percent changes sequentially. A

35\%

increase is represented by multiplying by

1.35

, and a

20\%

decrease is represented by multiplying by

0.80

. The final population in terms of

x

is

x \times 1.35 \times 0.80 = 1.08x

. Given that the population in 2025 is

270

, we set up the equation

1.08x = 270

. Dividing both sides by

1.08

yields

x = 250

.

Step-by-Step Solution

1

Write an expression for the owl population in 2020 in terms of

x

.

The population in 2020 was

1.35x

.

An increase of

35\%

is represented by multiplying the initial amount by

1 + 0.35 = 1.35

.

2

Write an expression for the owl population in 2025 in terms of

x

.

The population in 2025 was

1.08x

.

A decrease of

20\%

from the 2020 population is represented by multiplying the 2020 population by

1 - 0.20 = 0.80

. Thus, the 2025 population is

1.35x \times 0.80 = 1.08x

.

3

Set the expression for the 2025 population equal to the given value of 270 and solve for

x

.

x = 250

Solving the equation

1.08x = 270

by dividing both sides by

1.08

gives

x = \frac{270}{1.08} = 250

.

Key Concept

Consecutive percent changes

Estimated Time:1m 30s

Question 16Question →

A software update reduced the storage space used by an application on a smartphone by $30\%$ . If the application originally used $400$ megabytes ( $\text{MB}$ ) of storage space, how many megabytes of storage space does the application use after the update?

Show answer & explanation

Answer: 280

Answer

The application uses

280

megabytes of storage space after the update.

The correct answer is

280

. The reduction in storage space is

30\%

of the original

400

megabytes, which is calculated as

0.30 \times 400 = 120

megabytes. Subtracting this reduction from the original storage space gives

400 - 120 = 280

megabytes. Alternatively, a reduction of

30\%

means the remaining storage is

70\%

of the original storage, which is

0.70 \times 400 = 280

megabytes.

Step-by-Step Solution

1

Calculate the reduction in storage space.

The reduction is

120

megabytes.

Multiply the original storage space of

400

megabytes by the percent reduction expressed as a decimal (

0.30

).

2

Subtract the reduction from the original storage space.

The new storage space is

280

megabytes.

Subtracting the amount of space saved (

120

megabytes) from the original space (

400

megabytes) gives the remaining space used.

Key Concept

Calculating a decreased amount after a percentage reduction.

Question 17Question →

In a certain school, the number of students in the science club was $25\%$ of the number of students in the math club. During a membership drive, the number of students in the science club increased by $30\%$ , and the number of students in the math club increased by $10\%$ . After the drive, the total number of students in both clubs combined was $228$ . If no student is a member of both clubs, what was the total number of students in both clubs combined before the membership drive?

Show answer & explanation

Answer: 200

Answer

The total number of students in both clubs combined before the membership drive was 200.

To find the initial total number of students, we define the initial math club members as

M

. Since the science club has

25\%

of the size of the math club, its initial size is

0.25M

. After a

30\%

increase, the science club size is

1.30(0.25M) = 0.325M

. After a

10\%

increase, the math club size is

1.10M

. The new combined total is

0.325M + 1.10M = 228

. Simplifying gives

1.425M = 228

, which yields

M = 160

. The initial science club size is

0.25(160) = 40

. The total number of students before the drive was

160 + 40 = 200

.

Step-by-Step Solution

1

Define variables for the initial number of students in each club.

Let

M

be the initial number of students in the math club, and

S = 0.25M

be the initial number of students in the science club.

This establishes a relationship between the sizes of the two clubs based on the given percentage.

2

Express the new number of students in each club after the percent increases.

Science club:

1.30 \times S = 1.30(0.25M) = 0.325M

. Math club:

1.10M

.

An increase of

30\%

is represented by multiplying by

1.30

, and an increase of

10\%

is represented by multiplying by

1.10

.

3

Set up and solve an equation using the new combined total.

0.325M + 1.10M = 228 \implies 1.425M = 228 \implies M = 160

.

The sum of the new club sizes equals the new total of

228

students.

4

Calculate the initial science club members and the initial total combined members.

Initial science members:

S = 0.25 \times 160 = 40

. Total initial members:

M + S = 160 + 40 = 200

.

This answers the final question asking for the combined total before the membership drive.

Key Concept

Applying percent changes to individual components to determine initial values in a combined total system.

Alternative Method

Instead of setting up equations using

M

, you can choose an arbitrary starting value for the math club that is easily divisible, such as

100

. If the math club starts with

100

students, the science club starts with

25

students. After the increases, the math club has

110

students and the science club has

25 \times 1.3 = 32.5

students, for a total of

142.5

students. Since the actual final total is

228

, we scale our initial values by a factor of

\frac{228}{142.5} = 1.6

. The initial combined total is therefore

125 \times 1.6 = 200

.

Estimated Time:2m 0s

Question 18Question →

The table below shows the total number of registered voters in two districts, District X and District Y, in 2020 and 2024.

District	2020 registered voters	2024 registered voters
District X	$10,000$	$15,000$
District Y	$15,000$	$18,000$

In 2020, $12\%$ of the registered voters in District X and $24\%$ of the registered voters in District Y were between the ages of 18 and 25. In 2024, $16\%$ of the registered voters in District X and $20\%$ of the registered voters in District Y were between the ages of 18 and 25. From 2020 to 2024, what was the percent increase in the total number of registered voters between the ages of 18 and 25 across both districts combined?

20%

25%

32%

50%

Show answer & explanation

Answer: 25%

Answer

The percent increase in the total number of registered voters between the ages of 18 and 25 across both districts combined is 25%.

To find the percent increase in the total number of registered voters between the ages of 18 and 25, we first determine the number of voters in this age group for both years. For 2020, District X had

10,000 \times 0.12 = 1,200

voters and District Y had

15,000 \times 0.24 = 3,600

voters, giving a combined total of

4,800

voters aged 18-25. For 2024, District X had

15,000 \times 0.16 = 2,400

voters and District Y had

18,000 \times 0.20 = 3,600

voters, giving a combined total of

6,000

voters aged 18-25. The percent increase is computed by dividing the change in this subpopulation by the initial 2020 value:

\frac{6,000 - 4,800}{4,800} \times 100\% = 25\%

. Therefore, the total number of registered voters in this age group increased by 25%.

Step-by-Step Solution

1

Calculate the number of registered voters aged 18-25 in both districts in 2020.

In District X:

10,000 \times 0.12 = 1,200

. In District Y:

15,000 \times 0.24 = 3,600

. Total 2020 voters aged 18-25 =

1,200 + 3,600 = 4,800

.

To find the overall percent change of the subpopulation, we need to know its initial size in 2020.

2

Calculate the number of registered voters aged 18-25 in both districts in 2024.

In District X:

15,000 \times 0.16 = 2,400

. In District Y:

18,000 \times 0.20 = 3,600

. Total 2024 voters aged 18-25 =

2,400 + 3,600 = 6,000

.

To find the overall percent change of the subpopulation, we need to know its final size in 2024.

3

Calculate the absolute increase in the registered voters aged 18-25.

6,000 - 4,800 = 1,200

voters.

The percent change formula requires the difference between the final and initial values.

4

Calculate the percent increase relative to the 2020 initial value.

\frac{1,200}{4,800} \times 100\% = 25\%

.

The percent increase is defined as the change divided by the initial value, multiplied by 100 to get a percentage.

Key Concept

Calculating percent change for a combined population by computing the absolute initial and final counts of the subpopulation first, rather than manipulating percent rates directly.

Estimated Time:3m 0s

Question 19Question →

A small company had $150$ employees. If the number of employees decreased by $8\%$ , what is the current number of employees at the company?

12

138

142

162

Show answer & explanation

Answer: $138$

Answer

The correct answer is the option stating

138

.

The correct answer is the option stating

138

. To find the current number of employees, first find

8\%

of

150

, which is

0.08 \times 150 = 12

. Since the number of employees decreased, subtract

12

from the original count of

150

to get

150 - 12 = 138

.

Step-by-Step Solution

1

Calculate the number of employees represented by the

8\%

decrease.

0.08 \times 150 = 12

employees

To find the decrease in the number of employees, multiply the initial number of employees by the decimal equivalent of the percentage decrease.

2

Subtract the decrease from the initial number of employees to find the current count.

150 - 12 = 138

employees

Since the employee count decreased, subtracting the change from the original count yields the current number of employees.

Key Concept

Calculating a percentage decrease from an initial value.

Estimated Time:45s

Question 20Question →

The table below shows the population of a certain plant species in two conservation areas in 2024 and 2025.

Conservation Area	2024 Population	2025 Population
Area P	$120$	$150$
Area Q	$160$	$180$

The percent increase in the plant population of Area P from 2024 to 2025 is how many percentage points greater than the percent increase in the plant population of Area Q from 2024 to 2025?

7.5\%

10.0\%

12.5\%

8.9\%

Show answer & explanation

Answer: $12.5\%$

Answer

The percent increase in the plant population of Area P is

12.5\%

greater than that of Area Q.

To find the difference in the percent increases, we first calculate the percent increase for each conservation area. The percent increase is calculated by dividing the change in population by the original 2024 population, then multiplying by 100. For Area P, the percent increase is

\frac{150 - 120}{120} \times 100 = 25\%

. For Area Q, the percent increase is

\frac{180 - 160}{160} \times 100 = 12.5\%

. Subtracting these two percentages yields a difference of

25\% - 12.5\% = 12.5\%

.

Step-by-Step Solution

1

Calculate the percent increase in the plant population for Area P from 2024 to 2025.

25\%

The initial population in 2024 was

120

and the population in 2025 was

150

. The increase is

150 - 120 = 30

. The percent increase is calculated relative to the 2024 base:

\frac{30}{120} \times 100 = 25\%

.

2

Calculate the percent increase in the plant population for Area Q from 2024 to 2025.

12.5\%

The initial population in 2024 was

160

and the population in 2025 was

180

. The increase is

180 - 160 = 20

. The percent increase is calculated relative to the 2024 base:

\frac{20}{160} \times 100 = 12.5\%

.

3

Calculate the difference between the two percent increases in percentage points.

12.5

percentage points

Subtract the percent increase of Area Q from the percent increase of Area P:

25\% - 12.5\% = 12.5\%

.

Key Concept

Percent change calculations require identifying the correct baseline (initial) value.

Problem-Solving and Data Analysis

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