Word Problems and Applied Math Practice Questions — GMAT — Page 10

Question 181Question →

A commercial real estate developer purchases a building. In the first year after purchase, the value of the building increases by $40\%$ . In the second year, the value decreases by $25\%$ from its first-year value. In the third year, the value increases by $20\%$ from its second-year value. If the value of the building at the end of the third year is $\$630,000$ , what was the original purchase price of the building?

\$375,000

\$378,000

\$450,000

\$500,000

\$525,000

Show answer & explanation

Answer: $\$500,000$

Answer

The original purchase price of the building was

\$500,000

.

To find the original price, express each sequential percent change as a multiplier applied to the preceding value. A

40\%

increase corresponds to a factor of

1.40

, a

25\%

decrease corresponds to a factor of

0.75

, and a

20\%

increase corresponds to a factor of

1.20

. Combining these gives a overall multiplier of

1.40 \times 0.75 \times 1.20 = 1.26

. Setting

1.26 \times P = \$630,000

yields

P = \$500,000

.

Step-by-Step Solution

1

Express each annual percentage change as a multiplier factor.

Year 1 increase of

40\% \rightarrow 1 + 0.40 = 1.40 = \frac{7}{5}

. Year 2 decrease of

25\% \rightarrow 1 - 0.25 = 0.75 = \frac{3}{4}

. Year 3 increase of

20\% \rightarrow 1 + 0.20 = 1.20 = \frac{6}{5}

.

Successive percentage changes compound sequentially on the updated intermediate values.

2

Calculate the cumulative multiplier over the three-year period.

\text{Overall Multiplier} = 1.40 \times 0.75 \times 1.20 = \frac{7}{5} \times \frac{3}{4} \times \frac{6}{5} = \frac{126}{100} = 1.26.

Multiplying individual year multipliers yields the net scale factor from the initial price to the final value.

3

Set up the linear equation for the initial price

P

and solve.

1.26 \times P = 630,000 \implies P = \frac{630,000}{1.26} = 500,000.

Dividing the final valuation by the overall multiplier determines the original purchase price.

Key Concept

Successive Percent Change

Question 182Question →

A cyclist travels along a scenic mountain road from a base lodge to a summit point at a constant speed of $12$ miles per hour. The cyclist then returns along the exact same route from the summit point back to the base lodge at a constant speed of $24$ miles per hour. If the entire round-trip ride takes a total of $3$ hours, what is the total distance, in miles, for the entire round trip?

Show answer & explanation

Answer: 48

Answer

The total distance for the entire round trip is 48 miles.

To find the total round-trip distance, let

d

be the one-way distance. The time spent going uphill is

\frac{d}{12}

hours and downhill is

\frac{d}{24}

hours. Combining these gives

\frac{d}{12} + \frac{d}{24} = 3

, which simplifies to

\frac{3d}{24} = 3

or

d = 24

miles. Since the trip covers both legs, the total distance is

2d = 48

miles.

Step-by-Step Solution

1

Define the variable for distance.

Let

d

represent the one-way distance in miles between the base lodge and the summit point.

Establishing a variable for the unknown leg distance allows setting up rate-time-distance relationships.

2

Express time in terms of distance and rate for both legs.

Uphill time

t_1 = \frac{d}{12}

hours; Downhill time

t_2 = \frac{d}{24}

hours.

Using

\text{Time} = \frac{\text{Distance}}{\text{Rate}}

converts given speeds into expressions for duration.

3

Formulate and solve the total time equation.

\frac{d}{12} + \frac{d}{24} = 3 \implies \frac{3d}{24} = 3 \implies d = 24 \text{ miles}.

The sum of the time taken for both legs equals the given total round-trip time of 3 hours.

4

Compute the total round-trip distance.

\text{Total distance} = 2 \times d = 2 \times 24 = 48 \text{ miles}.

The question specifically asks for the total distance of the entire round trip, which consists of two equal legs of length

d

.

Key Concept

Harmonic Mean and Round-Trip Rate Problems

Estimated Time:2m 0s

Question 183Question →

At the beginning of 2023, a manufacturing plant had an initial annual production capacity of $C$ units. Over the next three years, its annual production capacity experienced the following changes: it increased by $10\%$ in 2023, decreased by $20\%$ in 2024, and increased by $25\%$ in 2025. If the plant's production capacity at the end of 2025 was $22,000$ units, what was the initial production capacity $C$ , in units, at the beginning of 2023?

Show answer & explanation

Answer: 20000

Answer

The initial production capacity at the beginning of 2023 was 20,000 units.

Each annual percentage change applies to the value at the end of the previous period. Representing the changes as multipliers gives

1.10

,

0.80

, and

1.25

. Multiplying these together gives a net change factor of

1.10 \times 0.80 \times 1.25 = 1.10

. Setting

1.10 C = 22,000

and solving for

C

yields

C = 20,000

.

Step-by-Step Solution

1

Convert each percentage change into a decimal multiplier

Increase of

10\% \rightarrow 1.10

; decrease of

20\% \rightarrow 0.80

; increase of

25\% \rightarrow 1.25

.

Successive percentage changes compound sequentially by multiplying their respective growth or decay factors.

2

Compute the overall compound multiplier factor

Overall multiplier =

1.10 \times 0.80 \times 1.25 = 1.10 \times (0.80 \times 1.25) = 1.10 \times 1 = 1.10

.

Multiplying

0.80 \times 1.25

yields

1

, simplifying the combined factor to

1.10

times the original capacity.

3

Solve for the initial capacity

C

1.10 \times C = 22,000 \implies C = \frac{22,000}{1.10} = 20,000

.

Dividing the final capacity by the combined multiplier yields the initial capacity.

Key Concept

Successive Percent Change

Question 184Question →

An electric shuttle travels along a straight $240$ -mile route from Facility A to Facility B. For the first $120$ miles of the journey, the shuttle travels at a constant average speed of $60$ miles per hour. For the remaining $120$ miles, due to traffic congestion, the shuttle travels at a constant average speed of $30$ miles per hour. What was the average speed of the electric shuttle, in miles per hour, for the entire $240$ -mile trip?

36

40

42.5

45

50

Show answer & explanation

Answer: 40

Answer

40 miles per hour

To find the average speed over a journey with varying rates, divide total distance by total time. The shuttle covers

120

miles at

60

mph in

2

hours, and

120

miles at

30

mph in

4

hours. The total travel time is

6

hours for a total distance of

240

miles. Dividing

240

miles by

6

hours yields an average speed of

40

miles per hour.

Step-by-Step Solution

1

Calculate the time spent on the first leg of the trip.

Time for first leg =

\frac{120 \text{ miles}}{60 \text{ mph}} = 2 \text{ hours}

.

Time is equal to distance divided by speed.

2

Calculate the time spent on the second leg of the trip.

Time for second leg =

\frac{120 \text{ miles}}{30 \text{ mph}} = 4 \text{ hours}

.

Time is equal to distance divided by speed.

3

Calculate the total time for the entire journey.

Total time =

2 \text{ hours} + 4 \text{ hours} = 6 \text{ hours}

.

The total duration is the sum of the time spent on each leg.

4

Compute the overall average speed using total distance divided by total time.

Average speed =

\frac{240 \text{ miles}}{6 \text{ hours}} = 40 \text{ miles per hour}

.

Average speed is defined as total distance divided by total elapsed time.

Key Concept

Average Speed for a Multi-Leg Trip

Question 185Question →

An investor distributed a total principal of $\$20,000$ between two accounts, Account X and Account Y. Account X earns simple annual interest at a rate of $8\%$ , while Account Y earns compound annual interest at a rate of $10\%$ compounded annually. If no further deposits or withdrawals were made and the total interest earned from both accounts combined at the end of $2$ years was $\$3,600$ , how much money was initially invested in Account X?

\$8,000

\$10,000

\$12,000

\$14,000

\$15,000

Show answer & explanation

Answer: $\$12,000$

Answer

\$12,000

Account X earns

16\%

simple interest over 2 years (

8\% \times 2

), and Account Y earns

21\%

compound interest over 2 years (

(1.10)^2 - 1 = 0.21

). Letting the amount in Account X be

P_X

, the total interest equation is

0.16 P_X + 0.21(20,000 - P_X) = 3,600

. Expanding yields

4,200 - 0.05 P_X = 3,600

, which simplifies to

0.05 P_X = 600

, giving

P_X = 12,000

.

Step-by-Step Solution

1

Express the interest rate for Account X over 2 years.

Account X earns simple interest at

8\%

per year for

2

years, giving a total interest multiplier of

0.08 \times 2 = 0.16

or

16\%

.

Simple interest accumulates linearly as Principal

\times

Rate

\times

Time.

2

Express the interest rate for Account Y over 2 years.

Account Y earns compound interest at

10\%

per year compounded annually for

2

years. The total growth factor is

(1 + 0.10)^2 = 1.21

, meaning the interest earned is

1.21 - 1 = 0.21

or

21\%

of the principal.

Compound interest multiplies the balance each period.

3

Set up an equation using the total principal of

\$20,000

and total interest of

\$3,600

.

Let

P_X

be the principal in Account X. Then the principal in Account Y is

20,000 - P_X

. The total interest equation is

0.16 P_X + 0.21(20,000 - P_X) = 3,600

.

The sum of interest from both accounts equals the total interest given.

4

Solve for

P_X

.

0.16 P_X + 4,200 - 0.21 P_X = 3,600 \implies -0.05 P_X = 3,600 - 4,200 \implies -0.05 P_X = -600 \implies P_X = 12,000

.

Isolating the variable

P_X

gives the initial principal invested in Account X.

Key Concept

Simple vs. Compound Interest and Algebraic Mixture Equations

Estimated Time:1m 30s

Question 186Question →

An initial investment of $\$8,000$ was deposited into an account earning an annual interest rate of $r\%$ , compounded annually. If no additional deposits or withdrawals were made and the total balance in the account after $2$ years was $\$11,520$ , what is the value of $r$ ?

Show answer & explanation

Answer: 20

Answer

The value of

r

is 20.

The compound interest formula for annual compounding is

A = P\left(1 + \frac{r}{100}\right)^t

. Substituting

P = 8,000

,

A = 11,520

, and

t = 2

gives

11,520 = 8,000\left(1 + \frac{r}{100}\right)^2

. Dividing both sides by

8,000

yields

\left(1 + \frac{r}{100}\right)^2 = 1.44

. Taking the square root gives

1 + \frac{r}{100} = 1.2

, so

\frac{r}{100} = 0.20

, which corresponds to

r = 20

.

Step-by-Step Solution

1

Set up the compound interest equation using given principal and accumulated balance.

11,520 = 8,000 \left(1 + \frac{r}{100}\right)^2

Interest compounded annually over

2

years scales the initial principal by a factor of

\left(1 + \frac{r}{100}\right)^2

.

2

Divide both sides by

8,000

to isolate the annual growth factor squared.

\left(1 + \frac{r}{100}\right)^2 = \frac{11,520}{8,000} = 1.44

Finding the ratio of final balance to initial principal gives the overall multiplier over the two-year period.

3

Take the square root of both sides to obtain the single-year multiplier.

1 + \frac{r}{100} = \sqrt{1.44} = 1.2

Since the interest rate

r > 0

, taking the positive square root gives the multiplier for one year.

4

Solve for

r

.

r = (1.2 - 1) \times 100 = 20

Subtracting

1

yields the annual decimal interest rate of

0.20

, which corresponds to

20\%

.

Key Concept

Annual Compound Interest Rate Calculation

Question 187Question →

A high-end audio equipment manufacturer evaluated the market value of a specialized manufacturing unit over a three-year period. In the first year, the unit's value increased by $25\%$ . In the second year, its value increased by $20\%$ relative to its value at the end of the first year. In the third year, its value decreased by $20\%$ relative to its value at the end of the second year. If the value of the unit at the end of the third year was $\$144,000$ , what was its original value at the beginning of the first year?

\$80,000

\$96,000

\$103,680

\$115,200

\$120,000

Show answer & explanation

Answer: $\$120,000$

Answer

The original value at the beginning of the first year was

\$120,000

.

To determine the original value, find the overall multiplier for successive changes: an increase of

25\%

(factor

1.25

), an increase of

20\%

(factor

1.20

), and a decrease of

20\%

(factor

0.80

). The overall cumulative factor is

1.25 \times 1.20 \times 0.80 = 1.20

. Dividing the final value of

\$144,000

by

1.20

yields the correct initial value of

\$120,000

.

Step-by-Step Solution

1

Express each year's percentage change as a growth/decay multiplier.

Year 1 factor

= 1 + 0.25 = 1.25 = \frac{5}{4}

. Year 2 factor

= 1 + 0.20 = 1.20 = \frac{6}{5}

. Year 3 factor

= 1 - 0.20 = 0.80 = \frac{4}{5}

.

Successive percentage changes must be calculated relative to each new intermediate value.

2

Calculate the cumulative multiplier over the three years.

Cumulative Multiplier

= \frac{5}{4} \times \frac{6}{5} \times \frac{4}{5} = \frac{6}{5} = 1.20

.

Multiplying individual period factors yields the net relationship between initial and final values.

3

Solve for the initial value

V_0

.

V_0 \times 1.20 = 144,000 \implies V_0 = \frac{144,000}{1.20} = 120,000

.

Dividing the final value by the overall cumulative factor yields the original price.

Key Concept

Successive Percent Change and Base Multipliers

Estimated Time:1m 30s

Question 188Question →

A boutique dairy farm produced $4,000$ liters of organic milk in 2021. The annual milk production increased by $15\%$ from 2021 to 2022, decreased by $10\%$ from 2022 to 2023, and then increased by $20\%$ from 2023 to 2024. How many liters of organic milk did the farm produce in 2024?

Show answer & explanation

Answer: 4968

Answer

The farm produced 4,968 liters of organic milk in 2024.

To calculate the final quantity after sequential percent changes, multiply the starting amount by the multiplier for each period. The net expression is

4,000 \times 1.15 \times 0.90 \times 1.20

. Stepwise,

4,000 \times 1.15 = 4,600

, then

4,600 \times 0.90 = 4,140

, and finally

4,140 \times 1.20 = 4,968

liters.

Step-by-Step Solution

1

Calculate milk production in 2022 after a 15% increase.

4,000 \times 1.15 = 4,600

liters

An increase of 15% corresponds to multiplying the initial base value by

1 + 0.15 = 1.15

.

2

Calculate milk production in 2023 after a 10% decrease.

4,600 \times 0.90 = 4,140

liters

A decrease of 10% corresponds to multiplying the new base value of 4,600 by

1 - 0.10 = 0.90

.

3

Calculate milk production in 2024 after a 20% increase.

4,140 \times 1.20 = 4,968

liters

An increase of 20% corresponds to multiplying the current base value of 4,140 by

1 + 0.20 = 1.20

.

Key Concept

Successive Percent Change

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