Problem-Solving and Data Analysis Practice Questions — SAT — Page 2

Question 21Question →

Last year, a gardener grew $80$ tomato plants. This year, the gardener grew $92$ tomato plants. What is the percent increase in the number of tomato plants from last year to this year?

Show answer & explanation

Answer: 15

Answer

The percent increase in the number of tomato plants is

15

.

To find the percent increase, subtract the original value from the new value to get the absolute increase:

92 - 80 = 12

. Next, divide this increase by the original value:

\frac{12}{80} = 0.15

. Finally, multiply by

100

to convert the decimal to a percentage, resulting in

15

.

Step-by-Step Solution

1

Subtract the original number of tomato plants from the new number of plants

12

plants

To find the absolute change in the quantity of plants.

2

Divide the change by the original number of plants

0.15

To find the ratio of the increase relative to the starting amount.

3

Multiply the ratio by

100

to convert it to a percentage

15

To express the ratio as a percentage value.

Key Concept

Calculating percent change from an initial value to a final value

Question 22Question →

In a certain company, the number of entry-level employees is $60\%$ of the total number of employees, and the remaining employees are managers. Over a year, the number of entry-level employees increases by $x\%$ , and the number of managers decreases by $y\%$ . As a result, the total number of employees in the company decreases by $4\%$ , and the ratio of the number of managers to the number of entry-level employees becomes $1$ to $3$ . What is the value of $x$ ?

Show answer & explanation

Answer: 20

Answer

The value of

x

is

20

.

To find the value of

x

, represent the initial and final quantities of entry-level employees and managers algebraically. Using the

4\%

decrease in total employees and the final manager-to-entry-level ratio of

1

to

3

, set up a system of linear equations:

2y - 3x = 20

and

2y = 100 - x

. Solving this system for

x

yields

20

.

Step-by-Step Solution

1

Represent the initial employee counts algebraically.

Initial entry-level employees =

0.60T

, initial managers =

0.40T

, where

T

is the initial total.

Establishing initial quantities based on the given percentages.

2

Represent the final employee counts after percent changes.

Final entry-level =

0.60T(1 + 0.01x)

, final managers =

0.40T(1 - 0.01y)

.

Applying the percent increase and decrease to the respective initial quantities.

3

Use the total percent change to create a linear equation.

2y - 3x = 20

Setting the sum of the final counts equal to

0.96T

and simplifying.

4

Use the final ratio of managers to entry-level employees to create a second equation.

2y = 100 - x

Setting the ratio of final managers to final entry-level employees equal to

1/3

and simplifying.

5

Solve the system of equations for

x

.

x = 20

Substituting the second equation into the first to solve for the target variable.

Key Concept

Percents and Percent Change

Question 23Question →

A city's annual budget is divided into education, public safety, and other expenses. In 2025, the budget allocated for education was $12$ million dollars. For 2026, the education budget is increased by $15\%$ , and the public safety budget is decreased by $8\%$ from its 2025 value. If the public safety budget in 2026 is equal to the education budget in 2026, what was the public safety budget, in millions of dollars, in 2025?

Show answer & explanation

Answer: 15

Answer

The public safety budget in 2025 was

15

million dollars.

To find the 2025 public safety budget, we first determine the 2026 education budget. A

15\%

increase on the 2025 education budget of

12

million yields

12 \times 1.15 = 13.8

million. Since the 2026 public safety budget is equal to this amount, it is also

13.8

million. Given that the 2026 public safety budget is an

8\%

decrease from its 2025 value (

P

), we set up the equation

P \times (1 - 0.08) = 13.8

. Solving for

P

gives

P = \frac{13.8}{0.92} = 15

million.

Step-by-Step Solution

1

Calculate the education budget for 2026.

The education budget in 2026 is

13.8

million dollars.

The education budget in 2026 is a

15\%

increase from the 2025 education budget of

12

million dollars:

12 \times (1 + 0.15) = 12 \times 1.15 = 13.8

million dollars.

2

Express the relationship between the 2025 and 2026 public safety budgets.

0.92 \times P = 13.8

, where

P

is the 2025 public safety budget.

The public safety budget in 2026 is equal to the education budget in 2026 (

13.8

million dollars) and represents an

8\%

decrease from the 2025 public safety budget

P

:

P \times (1 - 0.08) = 13.8

.

3

Solve for the 2025 public safety budget

P

.

P = 15

million dollars.

Dividing both sides of the equation by

0.92

yields

P = \frac{13.8}{0.92} = 15

million dollars.

Key Concept

Calculating initial values after a percentage change

Alternative Method

Instead of converting to decimals immediately, you can use fractions:

12 \times \frac{115}{100} = 12 \times \frac{23}{20} = \frac{69}{5} = 13.8

. Let

P

be the 2025 public safety budget. Then

\frac{92}{100} P = 13.8

, so

P = 13.8 \times \frac{100}{92} = \frac{1380}{92} = 15

.

Estimated Time:1m 30s

Question 24Question →

During a weekend sale, a store reduced the price of a bicycle by $20\%$ . If the sale price of the bicycle is $\$160$ , what was the original price, in dollars, of the bicycle?

\$128

\$180

\$192

\$200

Show answer & explanation

Answer: $\$200$

Answer

The original price of the bicycle was

\$200

.

The correct answer is

\$200

. A discount of

20\%

means that the sale price of

\$160

is

80\%

of the original price (

100\% - 20\% = 80\%

). Setting up the equation

0.80 \times P = 160

, where

P

is the original price, and solving for

P

gives

P = \frac{160}{0.80} = 200

.

Step-by-Step Solution

1

Express the relationship between the original price and the sale price using the discount percentage.

The sale price represents

80\%

of the original price because a

20\%

discount was applied (

100\% - 20\% = 80\%

).

Establishing the percentage value of the sale price relative to the original price is necessary to set up the equation.

2

Set up an equation with the original price, represented by

P

.

0.80 \times P = 160

Since the sale price is

80\%

of the original price, multiplying the original price by

0.80

must equal the sale price of

\$160

.

3

Solve for the original price,

P

, by dividing both sides of the equation by

0.80

.

P = \frac{160}{0.80} = 200

Dividing the sale price by the decimal representation of its percentage of the original price isolates the variable and calculates the original price.

Key Concept

Percents and Percent Change

Question 25Question →

A laboratory technician prepared a chemistry solution containing only liquid A and liquid B. Initially, liquid A made up $35\%$ of the total volume of the solution. After $150$ milliliters of liquid B was removed from the solution, the volume of liquid A became $42\%$ of the total volume of the remaining solution. What was the initial volume of liquid A, in milliliters?

315

378

750

900

Show answer & explanation

Answer: 315

Answer

315

The correct answer is 315. Let

V

represent the initial total volume of the solution. Initially, liquid A's volume is

0.35V

. After removing

150

milliliters of liquid B, the total volume becomes

V - 150

. Since only liquid B was removed, the volume of liquid A remains unchanged. This volume is now

42\%

of the new total volume, which gives the equation

0.35V = 0.42(V - 150)

. Expanding the right side yields

0.35V = 0.42V - 63

. Subtracting

0.35V

and adding

63

to both sides results in

0.07V = 63

, which simplifies to

V = 900

milliliters. To find the initial volume of liquid A, we calculate

35\%

of

900

, which is

315

milliliters.

Step-by-Step Solution

1

Let

V

represent the initial total volume of the solution in milliliters. Express the initial volume of liquid A in terms of

V

.

The initial volume of liquid A is

0.35V

milliliters.

Liquid A constitutes

35\%

of the initial total volume

V

.

2

Express the final total volume of the solution after removing

150

milliliters of liquid B.

The final total volume is

V - 150

milliliters.

Subtracting the volume of liquid B that was removed from the initial total volume gives the remaining total volume.

3

Set up an equation representing the volume of liquid A in the remaining solution and solve for

V

.

0.35V = 0.42(V - 150) \implies 0.35V = 0.42V - 63 \implies 0.07V = 63 \implies V = 900

milliliters.

The volume of liquid A remains constant, but it now represents

42\%

of the remaining total volume

V - 150

.

4

Calculate the initial volume of liquid A.

0.35 \times 900 = 315

milliliters.

Substitute the value of

V

back into the expression for the initial volume of liquid A.

Key Concept

Solving multi-step percent problems involving a change in the baseline total volume.

Estimated Time:2m 0s

Question 26Question →

A researcher is drying a sample of wet soil in a laboratory. Initially, water makes up $40\%$ of the total weight of the sample. During the first stage of drying, $25\%$ of the water in the sample evaporates. During the second stage of drying, an additional quantity of water evaporates such that the water in the sample now makes up $20\%$ of the sample's total weight. What percent of the water that remained after the first stage evaporated during the second stage?

20\%

37.5\%

50\%

33.3\%

Show answer & explanation

Answer: $50\%$

Answer

50\%

The correct answer is the option stating

50\%

. After the first stage of drying, the sample contains

30

grams of water and has a total weight of

90

grams. In the second stage,

15

grams of water evaporates, leaving

15

grams of water and a total weight of

75

grams. Since

15

is exactly

20\%

of

75

, the final condition is met. The

15

grams of water that evaporated in the second stage represents exactly

50\%

of the

30

grams of water that remained after the first stage.

Step-by-Step Solution

1

Define the initial weights using a convenient total weight.

Let the initial total weight of the wet soil sample be

100

grams. Since water makes up

40\%

of this weight, the initial water weight is

40

grams and the dry soil weight is

60

grams.

Choosing a concrete value of

100

grams simplifies the percent calculations.

2

Calculate the water remaining after the first stage of drying.

In the first stage,

25\%

of the water evaporates. The evaporated water is

0.25 \times 40 = 10

grams. The remaining water weight is

40 - 10 = 30

grams. The total weight of the sample is now

60 \text{ (dry soil)} + 30 \text{ (water)} = 90

grams.

We must establish the new baseline amount of water and total sample weight before the second drying stage.

3

Set up an equation for the second stage of drying.

Let

x

be the weight of water in grams that evaporates during the second stage. The final water weight is

30 - x

grams, and the final total weight of the sample is

90 - x

grams. Since the final water weight is

20\%

of the final total weight, we write:

\frac{30 - x}{90 - x} = 0.20

.

This relates the unknown quantity of water evaporated to the given final water percentage.

4

Solve the equation for the evaporated water weight

x

.

Multiplying both sides by

90 - x

gives

30 - x = 0.20(90 - x) \Rightarrow 30 - x = 18 - 0.20x

. Rearranging terms yields

12 = 0.80x

, which gives

x = 15

grams.

Solving the algebraic equation finds the exact amount of water that evaporated during the second stage.

5

Calculate the percent of the remaining water that evaporated.

Divide the water evaporated in the second stage (

15

grams) by the water remaining after the first stage (

30

grams):

\frac{15}{30} \times 100\% = 50\%

.

This answers the specific question asked using the correct base weight.

Key Concept

Multi-step percent change with a shifting base

Alternative Method

Instead of assuming a weight of

100

grams, the problem can be solved using variables. Let the initial total weight of the wet soil sample be

W

. The initial water weight is

0.40W

and the dry soil is

0.60W

. After the first stage, remaining water is

0.75 \times 0.40W = 0.30W

, and total weight is

0.90W

. Let

E

be the water evaporated in the second stage. The final water content is

0.30W - E

, and the final total weight is

0.90W - E

. The equation is

0.30W - E = 0.20(0.90W - E)

, which solves to

E = 0.15W

. The ratio of evaporated water to remaining water is

\frac{0.15W}{0.30W} = 0.50

, or

50\%

.

Estimated Time:3m 0s

Question 27Question →

A garden hose discharges water at a constant rate of $8$ quarts per minute. What is this rate, in gallons per hour? (Given that $1\text{ gallon} = 4\text{ quarts}$ )

Show answer & explanation

Answer: 120

Answer

120

To convert the discharge rate from quarts per minute to gallons per hour, we first convert quarts to gallons. Since

1\text{ gallon} = 4\text{ quarts}

, we divide the rate of

8

quarts per minute by

4

to get

2

gallons per minute. Next, to convert minutes to hours, we multiply this rate by

60

(since there are

60

minutes in

1\text{ hour}

). This gives

2 \times 60 = 120

gallons per hour.

Step-by-Step Solution

1

Convert quarts per minute to gallons per minute

2

gallons per minute

Since

1\text{ gallon} = 4\text{ quarts}

, divide the flow rate of

8

quarts per minute by the conversion factor of

4

to find the rate in gallons per minute:

\frac{8\text{ quarts/min}}{4\text{ quarts/gallon}} = 2\text{ gallons/min}

.

2

Convert gallons per minute to gallons per hour

120

gallons per hour

Since there are

60

minutes in an hour, multiply the rate of

2

gallons per minute by

60

to find the total gallons discharged in one hour:

2\text{ gallons/min} \times 60\text{ min/hour} = 120\text{ gallons/hour}

.

Key Concept

Unit Conversions

Question 28Question →

A retailer purchased a shipment of coats for a wholesale price of $x$ dollars each. The retailer marked up the wholesale price by $40\%$ to determine the retail price. During a winter sale, the retailer offered a discount of $20\%$ off the retail price of the coats. If the sale price of a coat was $84$ dollars, what was the wholesale price, $x$ , of the coat?

63

70

75

80

Show answer & explanation

Answer: $75$

Answer

75

The correct answer is

75

. The retail price is determined by increasing the wholesale price,

x

, by

40\%

, which can be represented as

1.40x

. The sale price is determined by decreasing the retail price by

20\%

, which is

0.80(1.40x) = 1.12x

. Since the sale price of the coat was

84

dollars, we set up the equation

1.12x = 84

. Solving for

x

yields

x = \frac{84}{1.12} = 75

.

Step-by-Step Solution

1

Represent the retail price in terms of the wholesale price

x

.

Retail price =

1.40x

A markup of

40\%

on

x

is equivalent to multiplying

x

by

1 + 0.40 = 1.40

.

2

Represent the sale price in terms of the retail price.

Sale price =

0.80(1.40x) = 1.12x

A discount of

20\%

off the retail price is equivalent to multiplying the retail price by

1 - 0.20 = 0.80

.

3

Set up the equation with the given sale price and solve for

x

.

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

The sale price of the coat is given as

84

dollars, so we set the algebraic representation equal to

84

and solve for

x

.

Key Concept

Applying consecutive percentage increases and decreases to find initial values.

Question 29Question →

A retailer purchases an item at a wholesale price. The retailer marks up the wholesale price by $p\%$ to establish the retail price. During a clearance sale, the retailer discounts the retail price by $(p - 10)\%$ . If the clearance sale price of the item is $8\%$ greater than the original wholesale price, and $p > 10$ , what is the value of $p$ ?

Show answer & explanation

Answer: 20

Answer

The value of

p

is

20

.

The correct answer is

20

. By representing the markup and discount as decimal multipliers, we can write the equation for the final price as a function of the wholesale price:

W \left(1 + \frac{p}{100}\right)\left(1 - \frac{p - 10}{100}\right) = 1.08W

. Dividing by

W

and letting

y = \frac{p}{100}

, we get

(1 + y)(1.1 - y) = 1.08

. Expanding this gives

1.1 + 0.1y - y^2 = 1.08

, which rearranges to the quadratic equation

y^2 - 0.1y - 0.02 = 0

. Factoring this equation yields

(y - 0.2)(y + 0.1) = 0

. Since

p > 10

,

y

must be positive, which means

y = 0.2

. Therefore,

p = 20

.

Step-by-Step Solution

1

Express the retail price in terms of the wholesale price

W

and the markup percentage

p\%

.

Retail Price =

W \left(1 + \frac{p}{100}\right)

A markup of

p\%

increases the base price

W

by a factor of

\left(1 + \frac{p}{100}\right)

.

2

Express the clearance sale price after applying a discount of

(p - 10)\%

to the retail price.

Clearance Price =

W \left(1 + \frac{p}{100}\right)\left(1 - \frac{p - 10}{100}\right)

A discount of

(p - 10)\%

decreases the retail price by a factor of

\left(1 - \frac{p - 10}{100}\right)

.

3

Set the clearance price equal to

1.08W

, which represents an

8\%

increase over the wholesale price, and simplify the equation by dividing both sides by

W

.

\left(1 + \frac{p}{100}\right)\left(1 - \frac{p - 10}{100}\right) = 1.08

The final clearance price is

8\%

greater than the wholesale price

W

, so we equate it to

1.08W

and divide both sides by

W

to eliminate the variable.

4

Substitute

y = \frac{p}{100}

into the simplified equation and expand the terms.

(1 + y)(1.1 - y) = 1.08 \Rightarrow 1.1 + 0.1y - y^2 = 1.08

Writing the equation in terms of

y

simplifies the algebraic expansion. The term

1 - \frac{p - 10}{100}

becomes

1 - (y - 0.1) = 1.1 - y

.

5

Rearrange the quadratic equation into standard form, factor it, and solve for

y

.

y^2 - 0.1y - 0.02 = 0 \Rightarrow (y - 0.2)(y + 0.1) = 0 \Rightarrow y = 0.2

(since

p > 10

,

y > 0.1

)

Factoring the quadratic yields

y = 0.2

and

y = -0.1

. Since

p > 10

,

y

must be positive, which leaves

y = 0.2

as the only valid solution.

6

Convert the value of

y

back to

p

.

p = 20

Since

y = \frac{p}{100} = 0.2

, multiplying both sides by

100

gives

p = 20

.

Key Concept

Compounding percent changes algebraically using variable markups and discounts.

Question 30Question →

A 3D printer extrudes plastic filament at a constant rate of $8$ millimeters per second. What is this rate, in meters per hour?

0.48

2.88

28.8

28,800

Show answer & explanation

Answer: $28.8$

Answer

The rate is

28.8

meters per hour.

To convert the rate from millimeters per second to meters per hour, we apply the unit conversion factors. We convert millimeters to meters by dividing by

1000

, and convert seconds to hours by multiplying by

3600

. This gives:

8\text{ mm/s} \times \frac{1\text{ m}}{1000\text{ mm}} \times \frac{3600\text{ s}}{1\text{ hour}} = 8 \times 3.6 = 28.8\text{ meters per hour}

.

Step-by-Step Solution

1

Convert the length unit from millimeters to meters.

Since

1\text{ meter} = 1000\text{ millimeters}

, a rate of

8\text{ millimeters per second}

is equal to

\frac{8}{1000} = 0.008\text{ meters per second}

.

To convert from a smaller unit (millimeters) to a larger unit (meters), divide by the conversion factor of

1000

.

2

Convert the time unit from seconds to hours.

Since

1\text{ hour} = 3600\text{ seconds}

, a rate of

0.008\text{ meters per second}

is equal to

0.008 \times 3600 = 28.8\text{ meters per hour}

.

To find the distance printed in one hour, multiply the distance printed in one second by the number of seconds in one hour (

3600

).

Key Concept

Unit Conversions

Alternative Method

We can combine the conversion steps into a single conversion factor:

\frac{3600\text{ seconds/hour}}{1000\text{ millimeters/meter}} = 3.6\text{ (meters \cdot seconds) / (millimeters \cdot hour)}

. Multiplying the rate of

8\text{ mm/s}

by

3.6

gives the rate directly as

28.8\text{ m/h}

.

Estimated Time:1m 0s

Question 31Question →

A digital photography archive consists of RAW files and JPEG files. Initially, the RAW files account for $80\%$ of the total storage space used by the archive. To reduce the storage space, the photographer compresses $40\%$ of the RAW files into JPEGs, which reduces the storage space of those specific files by $62.5\%$ . If the remaining uncompressed RAW files now account for $x\%$ of the updated total storage space of the archive, what is the value of $x$ ?

Show answer & explanation

Answer: 60

Answer

The remaining uncompressed RAW files occupy 48% of the initial total storage space, and the updated total storage space is 80% of the initial storage space. The percentage of the updated total storage space occupied by the uncompressed RAW files is therefore 60%.

The remaining uncompressed RAW files occupy

48\%

of the initial total storage space, and the updated total storage space is

80\%

of the initial storage space. The percentage of the updated total storage space occupied by the uncompressed RAW files is therefore 60%.

Step-by-Step Solution

1

Define variables for initial storage space.

Initial RAW space is

0.80T

; initial JPEG space is

0.20T

.

Establishes the baseline values relative to the initial total storage space

T

.

2

Calculate the space occupied by the portion of RAW files to be compressed and the remaining uncompressed RAW files.

Compressed RAW space is

0.32T

; remaining uncompressed RAW space is

0.48T

.

Splits the RAW files into the group that changes size and the group that remains unchanged.

3

Determine the new space occupied by the compressed files after the

62.5\%

reduction.

Reduction is

0.20T

; new space of compressed files is

0.12T

.

Calculates the impact of the compression percent change on the target subpopulation.

4

Calculate the updated total storage space of the archive.

Updated total storage space is

0.80T

.

Summing all updated file spaces (

0.48T

uncompressed RAW +

0.20T

original JPEG +

0.12T

newly compressed files) provides the new base for the final percentage calculation.

5

Compute the final percentage of the updated total storage space occupied by the uncompressed RAW files.

x = 60

.

Dividing the remaining uncompressed RAW space (

0.48T

) by the new total space (

0.80T

) gives the updated percentage.

Key Concept

Multi-step percent change and relative base calculations.

Question 32Question →

A bakery recipe requires $12$ ounces of butter to make one batch of bread. How many pounds of butter are needed to make $8$ batches of bread? (Given that $1\text{ pound} = 16\text{ ounces}$ )

Show answer & explanation

Answer: 6

Answer

The bakery needs

6

pounds of butter.

The total amount of butter needed for

8

batches is calculated by multiplying the butter required for one batch by the number of batches:

8 \times 12 = 96

ounces. Given that

1\text{ pound} = 16\text{ ounces}

, we convert ounces to pounds by dividing the total ounces by

16

. Therefore,

96 \div 16 = 6

pounds of butter are needed.

Step-by-Step Solution

1

Calculate the total ounces of butter needed.

96

ounces

Since each batch requires

12

ounces,

8

batches require

8 \times 12 = 96

ounces.

2

Convert the total ounces to pounds.

6

pounds

Divide the total number of ounces by the conversion factor (

16

ounces per pound):

\frac{96}{16} = 6

.

Key Concept

To convert a quantity from a smaller unit to a larger unit, multiply the initial quantity by the number of groups to find the total in the smaller unit, and then divide by the conversion factor that relates the two units.

Question 33Question →

A medical study compared the effectiveness of two headache treatments, Treatment X and Treatment Y. The results of the study are shown in the table below.

Treatment	Improved	No Improvement	Total
Treatment X	40	10	50
Treatment Y	35	15	50
Total	75	25	100

If a participant who received Treatment X is selected at random, what is the probability that the participant's headache improved?

\frac{1}{5}

\frac{2}{5}

\frac{8}{15}

\frac{4}{5}

Show answer & explanation

Answer: $\frac{4}{5}$

Answer

The correct answer is

\frac{4}{5}

.

The correct answer is the fraction expressing the number of participants who received Treatment X and improved (40) out of the total number of participants who received Treatment X (50), which simplifies to

\frac{4}{5}

.

Step-by-Step Solution

1

Identify the total number of participants in the conditional group.

The total number of participants who received Treatment X is

50

.

The question specifies that a participant is selected from those who received Treatment X, which restricts the sample space to this row.

2

Identify the number of participants in that conditional group who experienced improvement.

There are

40

participants who received Treatment X and experienced improvement.

We need to find the number of favorable outcomes within the restricted sample space.

3

Calculate the conditional probability.

The probability is

\frac{40}{50} = \frac{4}{5}

.

Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes in the restricted group.

Key Concept

Calculating conditional probability from a two-way table by identifying the restricted sample space.

Estimated Time:45s

Question 34Question →

A map has a scale where $2$ centimeters represents $15$ kilometers. If the distance between two cities on the map is $8$ centimeters, what is the actual distance between the two cities, in kilometers?

Show answer & explanation

Answer: 60

Answer

The actual distance between the two cities is

60

kilometers.

The correct answer is

60

. The map scale is

2

centimeters to

15

kilometers. The map distance between the cities is

8

centimeters, which is

4

times the scale distance of

2

centimeters (since

8 \div 2 = 4

). Therefore, the actual distance between the cities is

4

times the scale distance of

15

kilometers, which is

15 \times 4 = 60

kilometers.

Step-by-Step Solution

1

Set up a proportion using the map scale ratio of centimeters to kilometers.

\frac{2}{15} = \frac{8}{x}

This establishes that the ratio of map distance to actual distance remains constant.

2

Solve for the unknown actual distance,

x

, by cross-multiplying.

2x = 120

Multiplying the numerator of each fraction by the denominator of the other solves the proportion.

3

Divide by the coefficient of

x

to find the final actual distance.

x = 60

Dividing

120

by

2

isolates

x

and gives the actual distance in kilometers.

Key Concept

Setting up and solving proportions using scale factors.

Question 35Question →

In 2018, a forestry service recorded $400$ pine trees in a specific section of a nature reserve. Due to a reforestation initiative, the number of pine trees in this section increased by $15\%$ from 2018 to 2021. From 2021 to 2024, the number of pine trees increased by $20\%$ of the number of pine trees in 2021. What was the number of pine trees in this section of the reserve in 2024?

480

540

542

552

Show answer & explanation

Answer: 552

Answer

The correct number of pine trees in the reserve in 2024 is 552.

To find the number of pine trees in 2024, first calculate the population in 2021. A

15\%

increase on the initial

400

trees results in

400 \times 1.15 = 460

trees. Next, find the population in 2024 by applying a

20\%

increase to the 2021 population of

460

trees, which yields

460 \times 1.20 = 552

trees.

Step-by-Step Solution

1

Calculate the number of pine trees in 2021 after the

15\%

increase.

400 \times (1 + 0.15) = 460

trees

The initial number of trees in 2018 was

400

, and it increased by

15\%

, so we multiply

400

by

1.15

.

2

Calculate the number of pine trees in 2024 after the

20\%

increase.

460 \times (1 + 0.20) = 552

trees

The percentage increase of

20\%

from 2021 to 2024 is based on the 2021 tree population of

460

, so we multiply

460

by

1.20

.

Key Concept

Calculating successive percentage increases by applying each percentage change to the correct sequential baseline value.

Alternative Method

Alternatively, you can compute the final value in a single expression by multiplying the initial value by both growth factors:

400 \times 1.15 \times 1.20 = 552

.

Estimated Time:1m 30s

Question 36Question →

At the beginning of an experiment, the number of bacteria in Population X is $50\%$ greater than the number of bacteria in Population Y. During the experiment, Population X increases by $10\%$ and Population Y increases by $r\%$ . At the end of the experiment, the number of bacteria in Population X is $25\%$ greater than the number of bacteria in Population Y. What is the value of $r$ ?

24

32

35

40

Show answer & explanation

Answer: 32

Answer

32

The correct answer is

32

. If the initial population of Y is assumed to be

100

, then the initial population of X is

150

because it is

50\%

greater. A

10\%

increase of X results in a final population of

150 \times 1.10 = 165

. Since the final population of X is

25\%

greater than the final population of Y, we have

165 = 1.25 \times Y_{\text{final}}

, which gives

Y_{\text{final}} = 132

. The growth from

100

to

132

represents a

32\%

increase, so

r = 32

.

Step-by-Step Solution

1

Set up initial values for both populations.

Let the initial size of Population Y be

100

. Since Population X is

50\%

greater than Y, the initial size of Population X is

150

.

Choosing a concrete initial value like

100

simplifies the percentage calculations.

2

Calculate the final population of X.

The final population of X is

150 \times (1 + 0.10) = 165

.

Population X increased by

10\%

during the experiment.

3

Calculate the final population of Y.

Since the final population of X is

25\%

greater than the final population of Y, we write

165 = 1.25 \times Y_{\text{final}}

, which yields

Y_{\text{final}} = 132

.

We must divide the final population of X by

1.25

to find the final population of Y, because X is

25\%

greater than Y.

4

Find the percentage increase of Population Y.

The percent increase from

100

to

132

is

\frac{132 - 100}{100} \times 100\% = 32\%

. Therefore,

r = 32

.

This determines the value of

r

representing the percentage growth of Population Y.

Key Concept

Multi-step percentage change and identifying the correct baseline value.

Question 37Question →

At the beginning of the year, a library had $800$ history books. In the first half of the year, the number of history books increased by $15\%$ . In the second half of the year, the library acquired more history books, representing a percent increase of $x\%$ over the number of history books at the midyear point. If the library had a total of $1{}012$ history books at the end of the year, what is the value of $x$ ?

Show answer & explanation

Answer: 10

Answer

The value of

x

is

10

.

To find the second percent increase, we must first calculate the intermediate midyear value. A

15\%

increase on

800

is

800 \times 1.15 = 920

. The subsequent increase of

x\%

is based on this midyear value of

920

. Setting up the equation

920 \times (1 + \frac{x}{100}) = 1{}012

and solving for

x

yields

1 + \frac{x}{100} = 1.10

, which gives

x = 10

.

Step-by-Step Solution

1

Calculate the number of history books at the midyear point after the first increase of

15\%

920

books

An increase of

15\%

on the initial

800

books is calculated as

800 \times (1 + 0.15) = 920

.

2

Set up an equation for the second percent increase of

x\%

from the midyear value to the final value of

1{}012

920 \times (1 + \frac{x}{100}) = 1{}012

The second increase is

x\%

of the midyear value of

920

, resulting in the final value of

1{}012

.

3

Solve the equation for

x

x = 10

Divide both sides of the equation by

920

to get

1 + \frac{x}{100} = 1.10

, subtract

1

to get

\frac{x}{100} = 0.10

, and multiply by

100

to find

x = 10

.

Key Concept

Calculating successive percent increases by applying the percent change to the intermediate base value.

Alternative Method

Instead of calculating the intermediate number of books, we can express the final number of books as

800 \times 1.15 \times (1 + \frac{x}{100}) = 1{}012

. Simplifying

800 \times 1.15

gives

920

, so

920 \times (1 + \frac{x}{100}) = 1{}012

, leading to the same result of

x = 10

.

Estimated Time:1m 30s

Question 38Question →

A survey of 80 commuters asked about their primary method of transportation to work and their commute distance. The results are summarized in the table below.

Commute Distance	Bicycle	Walk	Total
Under 1 mile	15	25	40
1 mile or more	35	5	40
Total	50	30	80

Based on the table, what fraction of the commuters who commute a distance of under 1 mile choose walking as their primary transportation method?

Show answer & explanation

Answer: 0.625

Answer

5/8 or 0.625

To find the fraction of commuters who commute a distance of under 1 mile that choose walking, limit the sample space to the row representing 'Under 1 mile', which has a subtotal of 40 commuters. Within this row, 25 commuters walk. Thus, the fraction is 25/40, which simplifies to 5/8 (or 0.625).

Step-by-Step Solution

1

Identify the conditional group (denominator).

The row 'Under 1 mile' has a total of 40 commuters.

The phrase 'of the commuters who commute a distance of under 1 mile' restricts the group of interest to only those in this row.

2

Identify the target group within the restricted group (numerator).

25 commuters choose walking.

Within the 'Under 1 mile' row, we look at the 'Walk' column.

3

Calculate the conditional fraction.

25/40 = 5/8 (or 0.625).

Divide the target count by the conditional group total to find the fraction.

Key Concept

Conditional Probability from a Two-Way Table

Question 39Question →

The table below shows the initial number of tomato plants and pepper plants in two agricultural fields, Field A and Field B, at the start of a crop monitoring study.

Field	Tomato plants	Pepper plants
Field A	$150$	$100$
Field B	$200$	$300$

During the study:
- The total number of plants in Field A increased by $20\%$ , and the number of tomato plants in Field A increased by $30\%$ .
- The total number of plants in Field B increased by $10\%$ , and the number of pepper plants in Field B increased by $15\%$ .

What was the percent increase in the total number of pepper plants across both fields combined from the start to the end of the study?

10.0%

11.1%

12.5%

16.7%

Show answer & explanation

Answer: 12.5%

Answer

12.5%

The correct option is 12.5%. The initial combined number of pepper plants is

100 + 300 = 400

. In Field A, the total number of plants increases by 20% from 250 to 300, while the tomato plants increase by 30% from 150 to 195. This leaves

300 - 195 = 105

pepper plants in Field A at the end of the study. In Field B, the number of pepper plants increases by 15% from 300 to 345. Combining the final counts gives

105 + 345 = 450

pepper plants. The total increase of 50 pepper plants relative to the initial combined count of 400 represents a

\frac{50}{400} \times 100\% = 12.5\%

increase.

Step-by-Step Solution

1

Calculate the initial total number of plants in Field A and the initial combined number of pepper plants.

Field A initial total is

150 + 100 = 250

plants. The initial combined number of pepper plants across both fields is

100 + 300 = 400

plants.

Establishing the initial baseline values is necessary to compute final changes.

2

Determine the final number of pepper plants in Field A.

The final total in Field A is

250 \times 1.20 = 300

plants. The final number of tomato plants in Field A is

150 \times 1.30 = 195

plants. Therefore, the final number of pepper plants in Field A is

300 - 195 = 105

plants.

This isolates the final number of pepper plants contributed by Field A since the percent increase was given for the total and for tomato plants.

3

Determine the final number of pepper plants in Field B.

The final number of pepper plants in Field B is

300 \times 1.15 = 345

plants.

Applying the direct percentage increase of 15% to the initial Field B pepper plants gives their final count.

4

Calculate the combined final number of pepper plants and the percent increase.

The final combined number of pepper plants is

105 + 345 = 450

plants. The net increase is

450 - 400 = 50

plants. The percent increase is

\frac{50}{400} \times 100\% = 12.5\%

.

Dividing the change in the target quantity by its original combined base yields the combined percent change.

Key Concept

Weighted percent change and finding missing values using totals and subcategory percentages

Question 40Question →

A solar panel generates $120$ watt-hours of electricity every $3$ hours. At this rate, how many watt-hours of electricity will the solar panel generate in $8$ hours?

Show answer & explanation

Answer: 320

Answer

The solar panel will generate

320

watt-hours of electricity in

8

hours.

To find the total electricity generated, we determine the unit rate of energy generation:

\frac{120 \text{ watt-hours}}{3 \text{ hours}} = 40

watt-hours per hour. Multiplying this rate by the new duration of

8

hours gives

40 \times 8 = 320

watt-hours. Proportional reasoning also works:

\frac{120}{3} = \frac{x}{8}

, which yields

x = 320

.

Step-by-Step Solution

1

Calculate the hourly rate of electricity generation.

The rate is

40

watt-hours per hour.

Dividing the energy produced (

120

watt-hours) by the elapsed time (

3

hours) gives the unit rate.

2

Calculate the total energy generated over the new duration of

8

hours.

The total energy is

320

watt-hours.

Multiplying the unit rate (

40

watt-hours per hour) by the target duration (

8

hours) gives the total energy generated.

Key Concept

Calculating and applying unit rates to solve proportional relationships.

Alternative Method

Set up a proportion where the ratio of energy to time remains constant:

\frac{120}{3} = \frac{x}{8}

. Cross-multiply to solve for

x

:

3x = 120 \times 8

, which simplifies to

3x = 960

. Divide by

3

to find

x = 320

.

Estimated Time:45s

Problem-Solving and Data Analysis

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