Word Problems and Applied Math Practice Questions — GMAT — Page 9

Question 161Question →

An autonomous electric shuttle completed a 240-mile test run from Point A to Point B. For the first 120 miles, the shuttle traveled at a constant speed of $v$ miles per hour. For the remaining 120 miles, due to battery efficiency controls, the shuttle traveled at a reduced speed equal to $\frac{2}{3}v$ miles per hour. On the return trip along the exact same 240-mile route from Point B to Point A, the shuttle maintained a uniform speed of $v + 10$ miles per hour. If the total time for the outbound trip from Point A to Point B was 2 hours longer than the total time for the return trip from Point B to Point A, what was the shuttle's average speed, in miles per hour, for the outbound trip from Point A to Point B?

32

40

41\frac{2}{3}

45

50

Show answer & explanation

Answer: 40

Answer

40 miles per hour

To find the average speed for the outbound trip, calculate the total time taken for both 120-mile legs. The first half takes

\frac{120}{v}

hours, and the second half at

\frac{2}{3}v

speed takes

\frac{120}{\frac{2}{3}v} = \frac{180}{v}

hours, giving a total outbound time of

\frac{300}{v}

hours. Setting up the time difference equation against the return trip gives

\frac{300}{v} - \frac{240}{v + 10} = 2

. Solving this quadratic equation yields

v = 50

mph. Substituting

v = 50

into the total outbound time formula gives

\frac{300}{50} = 6

hours. Finally, dividing the total distance of 240 miles by 6 hours gives an average speed of 40 miles per hour.

Step-by-Step Solution

1

Express the total outbound travel time in terms of

v

.

T_{\text{outbound}} = \frac{120}{v} + \frac{120}{\frac{2}{3}v} = \frac{120}{v} + \frac{180}{v} = \frac{300}{v}

hours.

Time equals distance divided by speed for each 120-mile segment.

2

Express the return travel time and set up the time-difference equation.

\frac{300}{v} - \frac{240}{v + 10} = 2

.

The return trip covers 240 miles at speed

v + 10

mph and takes 2 hours less than the outbound trip.

3

Solve the algebraic equation for

v

.

Simplifying

\frac{150}{v} - \frac{120}{v + 10} = 1

leads to

v^2 - 20v - 1500 = 0

, giving positive root

v = 50

.

Factoring

(v - 50)(v + 30) = 0

yields

v = 50

mph since speed must be positive.

4

Calculate total outbound time and outbound average speed.

Total outbound time =

\frac{300}{50} = 6

hours; Average speed =

\frac{240}{6} = 40

mph.

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

Key Concept

Average Speed across Multi-Leg Trips (Total Distance / Total Time)

Estimated Time:2m 30s

Question 162Question →

At the beginning of 2021, a biotechnology firm allocated a fixed annual budget to its principal research laboratory. In 2022, the laboratory's budget was increased by $20\%$ relative to its 2021 budget. In 2023, the budget was reduced by $15\%$ from its 2022 level. In 2024, the budget was increased by $25\%$ over its 2023 level. If the budget in 2024 exceeded the budget in 2021 by $\$55,000$ , what was the laboratory's budget in 2021, in dollars?

Show answer & explanation

Answer: 200000

Answer

200,000 dollars

The initial 2021 budget

B

undergoes successive percentage changes over three years. A

20\%

increase in 2022 results in

1.20B

. A

15\%

decrease in 2023 yields

1.20B \times 0.85 = 1.02B

. A

25\%

increase in 2024 yields

1.02B \times 1.25 = 1.275B

. The net increase over the initial budget is

1.275B - B = 0.275B

. Setting

0.275B = 55,000

gives

B = \frac{55,000}{0.275} = 200,000

dollars.

Step-by-Step Solution

1

Represent the annual budgets sequentially in terms of the initial 2021 budget

B

2022 budget =

1.20B

, 2023 budget =

1.20B \times 0.85 = 1.02B

, 2024 budget =

1.02B \times 1.25 = 1.275B

Calculate successive percentage changes sequentially by multiplying the respective multipliers for each period

2

Calculate the net change in budget from 2021 to 2024

1.275B - B = 0.275B

Determine how much the final year's budget exceeds the initial base budget

3

Set up and solve the linear equation for

B

0.275B = 55,000 \implies B = \frac{55,000}{0.275} = 200,000

Equate the net algebraic difference to the given dollar amount to solve for the initial 2021 budget

Key Concept

Successive Percent Change and Base Value Tracking

Question 163Question →

An agricultural enterprise harvests wheat across three distinct parcels of land: Parcel X, Parcel Y, and Parcel Z. Parcel X covers $120$ acres and yields an average of $45$ bushels per acre. Parcel Y covers $80$ acres and yields an average of $60$ bushels per acre. Parcel Z covers $50$ acres and yields an average of $70$ bushels per acre. What is the overall average yield, in bushels per acre, for the entire enterprise across all three parcels?

54.8

58.3

61.3

45.7

61.8

Show answer & explanation

Answer: $54.8$

Answer

The overall average yield for the entire enterprise is

54.8

bushels per acre.

To find the overall average yield across all land, compute total production divided by total acreage. Parcel X yields

5,400

bushels (

120 \times 45

), Parcel Y yields

4,800

bushels (

80 \times 60

), and Parcel Z yields

3,500

bushels (

50 \times 70

). Dividing the combined production of

13,700

bushels by the combined area of

250

acres gives

54.8

bushels per acre.

Step-by-Step Solution

1

Calculate total bushels produced on Parcel X

120 \text{ acres} \times 45 \text{ bushels/acre} = 5,400 \text{ bushels}

Total production for a parcel equals its area multiplied by its yield per acre.

2

Calculate total bushels produced on Parcel Y

80 \text{ acres} \times 60 \text{ bushels/acre} = 4,800 \text{ bushels}

Determine total wheat harvested from Parcel Y.

3

Calculate total bushels produced on Parcel Z

50 \text{ acres} \times 70 \text{ bushels/acre} = 3,500 \text{ bushels}

Determine total wheat harvested from Parcel Z.

4

Sum total bushels and total acreage

Total bushels =

5,400 + 4,800 + 3,500 = 13,700 \text{ bushels}

; Total acres =

120 + 80 + 50 = 250 \text{ acres}

Weighted average requires total quantity produced divided by total weight (area).

5

Compute the weighted average yield

\frac{13,700}{250} = 54.8 \text{ bushels per acre}

Divide aggregate bushels by aggregate acreage.

Key Concept

Weighted Averages in Applied Contexts

Estimated Time:2m 0s

Question 164Question →

A motorboat travels $60$ miles downstream in a river with a constant current and then returns upstream along the same route to its starting point. If the motorboat's speed in still water is $20$ miles per hour and the total round-trip journey takes $8$ hours, what is the speed of the river current, in miles per hour?

5

8

10

12

15

Show answer & explanation

Answer: $10$

Answer

The speed of the river current is

10

miles per hour.

The correct answer is

10

miles per hour. Substituting

c = 10

into the rates gives a downstream rate of

30

mph (taking

60 / 30 = 2

hours) and an upstream rate of

10

mph (taking

60 / 10 = 6

hours). The total trip time is

2 + 6 = 8

hours, which matches the problem description.

Step-by-Step Solution

1

Define variables and expressions for downstream and upstream rates.

Let

c

be the speed of the river current in mph. Downstream rate is

20 + c

mph and upstream rate is

20 - c

mph.

When traveling downstream, the current adds to the boat's still-water speed; when traveling upstream, the current subtracts from it.

2

Set up the total time equation using Time = Distance / Rate.

\frac{60}{20 + c} + \frac{60}{20 - c} = 8

The sum of the time spent traveling downstream and the time spent traveling upstream equals the total trip duration of

8

hours.

3

Find a common denominator and solve the algebraic equation for

c

.

60(20 - c) + 60(20 + c) = 8(20 + c)(20 - c) \implies 1200 - 60c + 1200 + 60c = 8(400 - c^2) \implies 2400 = 3200 - 8c^2

Combining the fractions over the common denominator

(20+c)(20-c) = 400 - c^2

allows clear quadratic isolation.

4

Isolate

c^2

and solve for

c

.

8c^2 = 800 \implies c^2 = 100 \implies c = 10

Since speed must be a positive quantity,

c = 10

miles per hour.

Key Concept

Rate, Time, and Distance Problems in Currents (Upstream and Downstream)

Question 165Question →

An electric utility provider generates energy from three distinct sources: Solar, Wind, and Natural Gas. Solar power accounts for $20\%$ of total production at a generation cost of $\$40$ per megawatt-hour (MWh). Wind power accounts for $40\%$ of total production at a generation cost of $\$30$ per MWh. Natural Gas accounts for the remaining production at a generation cost of $\$80$ per MWh. What is the weighted average cost, in dollars per MWh, of the electricity generated by the utility provider?

\$44

\$48

\$50

\$52

\$56

Show answer & explanation

Answer: $\$52$

Answer

The weighted average cost of electricity is

\$52

per MWh.

The correct answer reflects the sum of each energy source's unit cost multiplied by its proportion of total output: Solar (

0.20 \times \$40 = \$8

), Wind (

0.40 \times \$30 = \$12

), and Natural Gas (

0.40 \times \$80 = \$32

). Adding these components yields

\$8 + \$12 + \$32 = \$52

per MWh.

Step-by-Step Solution

1

Determine the percentage weight of each energy source.

Solar

= 20\% = 0.20

, Wind

= 40\% = 0.40

, Natural Gas

= 100\% - (20\% + 40\%) = 40\% = 0.40

.

The total proportion of all energy sources must sum to

100\%

(

1.00

).

2

Calculate the weighted cost contribution for each energy source.

Solar:

0.20 \times \$40 = \$8

. Wind:

0.40 \times \$30 = \$12

. Natural Gas:

0.40 \times \$80 = \$32

.

Multiplying individual unit costs by their corresponding weights yields their contribution to the total average cost per unit.

3

Sum the weighted contributions to find the overall weighted average cost.

Weighted Average Cost

= \$8 + \$12 + \$32 = \$52

per MWh.

Adding the individual component contributions yields the overall weighted average.

Key Concept

Weighted Average Formula:

\text{Weighted Average} = \sum (w_i \times x_i)

where

w_i

represents the relative weight (fraction or percentage) of each component

x_i

, and

\sum w_i = 1

.

Estimated Time:1m 30s

Question 166Question →

An express delivery drone flies from Station Alpha to Station Beta at a constant speed of $60$ miles per hour against a headwind. On the return flight along the exact same path from Station Beta to Station Alpha, with the wind acting as a tailwind of identical strength, the drone travels at a constant speed of $90$ miles per hour. If the entire round-trip flight took a total of $5$ hours, what is the distance, in miles, between Station Alpha and Station Beta?

Show answer & explanation

Answer: 180

Answer

The distance between Station Alpha and Station Beta is 180 miles.

To find the distance

D

, set up the total time equation:

\frac{D}{60} + \frac{D}{90} = 5

. Finding the common denominator yields

\frac{5D}{180} = 5

, leading directly to

D = 180

miles.

Step-by-Step Solution

1

Define the unknown variable and write time expressions for each leg.

Let

D

be the distance between Station Alpha and Station Beta. Outbound time is

\frac{D}{60}

hours and return time is

\frac{D}{90}

hours.

Using the rate-time-distance formula

t = \frac{d}{r}

allows expressing unknown time components using distance.

2

Set up an equation for total round-trip time.

\frac{D}{60} + \frac{D}{90} = 5

The sum of the time spent on the outbound leg and the return leg equals the total given flight time of

5

hours.

3

Solve the algebraic equation for

D

.

Finding a common denominator of

180

gives

\frac{3D + 2D}{180} = 5 \implies \frac{5D}{180} = 5 \implies 5D = 900 \implies D = 180

.

Multiplying both sides by

180

clears denominators and isolates

D

.

Key Concept

Rate, Time, and Distance Relationship

Question 167Question →

An investor deposits $\$5,000$ into Account B, which earns interest at a rate of $20\%$ per year compounded annually for $2$ years. The same investor deposits another $\$5,000$ into Account A, which earns simple annual interest at a rate of $r\%$ per year for $3$ years. If the total interest earned from Account B exceeds the total interest earned from Account A by $\$400$ , what is the value of $r$ ?

Show answer & explanation

Answer: 12

Answer

The annual simple interest rate r is 12.

To find

r

, calculate the interest from each account. Account B grows compounded annually to

\$5,000 \times (1.20)^2 = \$7,200

, producing

\$2,200

in interest. Account A produces

5,000 \times \frac{r}{100} \times 3 = 150r

in simple interest. The problem states that

2,200 - 150r = 400

. Solving for

r

gives

150r = 1,800

, so

r = 12

.

Step-by-Step Solution

1

Calculate the compound interest earned from Account B

Interest from Account B = $2,200

Using the compound interest formula

A = P(1 + i)^n

, the balance after 2 years is

\$5,000 \times (1.20)^2 = \$7,200

. Subtracting the principal gives

\$7,200 - \$5,000 = \$2,200

.

2

Express the simple interest earned from Account A in terms of r

Interest from Account A = 150r

Simple interest is calculated as

I = P \times \frac{r}{100} \times t = 5,000 \times \frac{r}{100} \times 3 = 150r

.

3

Formulate and solve the linear equation relating the two interest amounts

r = 12

Subtracting the simple interest from the compound interest gives

2,200 - 150r = 400

. Solving yields

150r = 1,800

, which gives

r = 12

.

Key Concept

Comparing simple interest and compound interest expressions to solve for an unknown rate

Question 168Question →

A commuter drives a total distance of $60$ miles from home to work. Due to heavy morning traffic, the commuter travels the first $30$ miles at a constant speed of $30$ miles per hour. At what constant speed, in miles per hour, must the commuter travel the remaining $30$ miles to achieve an average speed of $40$ miles per hour for the entire $60$ -mile trip?

45

48

50

60

75

Show answer & explanation

Answer: $60$

Answer

The commuter must travel the remaining

30

miles at a speed of

60

miles per hour.

The total trip distance is

60

miles and the desired overall average speed is

40

miles per hour, requiring a total time of

1.5

hours. Since traveling the first

30

miles at

30

miles per hour takes

1

hour, only

0.5

hours remain to complete the remaining

30

miles. Traveling

30

miles in

0.5

hours requires an average speed of

60

miles per hour.

Step-by-Step Solution

1

Calculate the total time required for the entire trip using

\text{Total Time} = \frac{\text{Total Distance}}{\text{Average Speed}}

.

\text{Total Time} = \frac{60 \text{ miles}}{40 \text{ mph}} = 1.5 \text{ hours}

.

Average speed for a multi-leg trip is defined strictly as total distance divided by total time.

2

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

\text{Time for Leg 1} = \frac{30 \text{ miles}}{30 \text{ mph}} = 1.0 \text{ hour}

.

Using

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

for the first

30

miles.

3

Determine the remaining time available for the second leg of the trip.

\text{Time for Leg 2} = 1.5 \text{ hours} - 1.0 \text{ hour} = 0.5 \text{ hours}

.

The sum of the times for both legs must equal the total required trip time.

4

Calculate the required constant speed for the remaining

30

miles.

\text{Speed for Leg 2} = \frac{30 \text{ miles}}{0.5 \text{ hours}} = 60 \text{ mph}

.

Dividing the remaining distance by the remaining time gives the required speed for the second leg.

Key Concept

Average Speed Formula (

V_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}}

)

Estimated Time:2m 0s

Question 169Question →

A wholesaler purchased $80$ identical smartphones for a total cost of $\$16,000$ . The wholesaler marked up the cost price of each smartphone by $50\%$ to establish the regular retail price. During a promotional sale, $20\%$ of the smartphones were sold at a $20\%$ discount off the regular retail price, and the remaining smartphones were sold at the full regular retail price. What was the total profit earned by the wholesaler on all $80$ smartphones?

\$5,760

\$6,400

\$7,040

\$7,360

\$8,000

Show answer & explanation

Answer: $\$7,040$

Answer

The total profit earned by the wholesaler on all

80

smartphones was

\$7,040

.

The cost per smartphone is

\$200

. Marking up by

50\%

gives a regular retail price of

\$300

.

16

smartphones (

20\%

of

80

) are sold at a

20\%

discount for

\$240

each, generating

\$3,840

in revenue. The remaining

64

smartphones are sold at

\$300

each, generating

\$19,200

in revenue. The total revenue is

\$23,040

, and subtracting the initial total cost of

\$16,000

yields a net profit of

\$7,040

.

Step-by-Step Solution

1

Calculate the cost price per unit and the regular retail price per unit.

Cost per unit =

\frac{\$16,000}{80} = \$200

. Regular retail price =

\$200 \times (1 + 0.50) = \$300

.

Markup is applied to the cost price per unit.

2

Calculate the number of units sold at a discount and at regular price, along with their respective selling prices.

Discounted units =

80 \times 0.20 = 16

units. Regular units =

80 - 16 = 64

units. Discounted price =

\$300 \times (1 - 0.20) = \$240

.

The

20\%

discount is taken off the regular retail price of

\$300

.

3

Calculate total revenue and total profit.

Total revenue =

(16 \times \$240) + (64 \times \$300) = \$3,840 + \$19,200 = \$23,040

. Total profit =

\$23,040 - \$16,000 = \$7,040

.

Profit is total revenue minus total cost.

Key Concept

Successive percentage markup and discount calculations with split inventory

Estimated Time:2m 0s

Question 170Question →

A cyclist travels from City A to City B at a constant speed of $20$ miles per hour. On the return trip along the exact same route, the cyclist increases their speed by $25\%$ . If the total time taken for the round trip is $9$ hours, what is the distance, in miles, between City A and City B?

Show answer & explanation

Answer: 100

Answer

The distance between City A and City B is 100 miles.

To find the one-way distance

d

, determine the return speed as

20 \times 1.25 = 25

miles per hour. The outbound time is

\frac{d}{20}

hours and the return time is

\frac{d}{25}

hours. Adding these gives

\frac{d}{20} + \frac{d}{25} = 9

, which simplifies to

\frac{9d}{100} = 9

, so

d = 100

miles.

Step-by-Step Solution

1

Calculate the speed on the return trip.

The return speed is

20 \times 1.25 = 25

miles per hour.

The return speed increases by

25\%

over the outbound speed of

20

miles per hour.

2

Express outbound and inbound travel times using distance

d

.

Outbound time

t_1 = \frac{d}{20}

hours, Inbound time

t_2 = \frac{d}{25}

hours.

Using the rate-time-distance formula

t = \frac{d}{r}

.

3

Set up and solve the total time equation.

\frac{d}{20} + \frac{d}{25} = 9 \implies \frac{9d}{100} = 9 \implies d = 100

miles.

The total duration of the round trip is given as

9

hours.

Key Concept

Rate, Time, and Distance Relationship in Round Trips

Question 171Question →

A specialty beverage distributor purchased a crate of rare botanical spirits at a total cost price of $\$400$ . The distributor marked up the cost price by $50\%$ to establish the regular list price. During a promotional event, the list price was discounted by $x\%$ . If the distributor earned a net profit of $20\%$ on the original cost price of the crate, what is the value of $x$ ?

15

20

25

30

33\frac{1}{3}

Show answer & explanation

Answer: $20$

Answer

The value of

x

is

20

.

The regular list price is calculated as

1.50 \times \$400 = \$600

. Since the distributor earns a

20\%

profit on the cost price, the discounted selling price must be

1.20 \times \$400 = \$480

. The discount amount is

\$600 - \$480 = \$120

. Expressed as a percentage of the list price,

x = \frac{120}{600} \times 100 = 20

.

Step-by-Step Solution

1

Calculate the regular list price based on the

50\%

markup.

List Price =

\$400 \times (1 + 0.50) = \$600

.

Markup is applied directly to the original cost price.

2

Determine the actual selling price after the discount using the

20\%

net profit target.

Selling Price =

\$400 \times (1 + 0.20) = \$480

.

Net profit percentage is calculated relative to the cost price base.

3

Calculate the dollar discount and express it as a percentage of the list price.

Discount Amount =

\$600 - \$480 = \$120

. Discount Percentage

x\% = \frac{\$120}{\$600} \times 100\% = 20\%

. Thus,

x = 20

.

Percentage discount is always measured relative to the original list price.

Key Concept

Markup, Profit, and Successive Percentage Discounts

Question 172Question →

A jogger runs from point A to point B at a constant speed of $8$ miles per hour and then immediately returns from point B to point A along the exact same route at a constant speed of $12$ miles per hour. If the total time for the entire round trip is $5$ hours, what is the distance, in miles, from point A to point B?

Show answer & explanation

Answer: 24

Answer

The distance from point A to point B is 24 miles.

Let

d

be the distance in miles between point A and point B. The time taken for the outbound journey is

\frac{d}{8}

hours, and the time taken for the return journey is

\frac{d}{12}

hours. Since the total time for the round trip is

5

hours, we set up the equation

\frac{d}{8} + \frac{d}{12} = 5

. Finding a common denominator of

24

, the equation becomes

\frac{3d + 2d}{24} = 5

, which simplifies to

\frac{5d}{24} = 5

. Multiplying both sides by

24

gives

5d = 120

, and dividing by

5

yields

d = 24

miles.

Step-by-Step Solution

1

Define the variable

d

as the distance in miles between point A and point B, and express the time for each leg of the trip.

Outbound time =

\frac{d}{8}

hours, Return time =

\frac{d}{12}

hours.

Using the fundamental rate formula

t = \frac{d}{r}

, time equals distance divided by speed.

2

Sum the time spent on the outbound leg and the return leg to match the total round-trip time.

\frac{d}{8} + \frac{d}{12} = 5

.

The total time for the trip is given as

5

hours.

3

Find a common denominator to add the algebraic fractions and solve for

d

.

\frac{3d + 2d}{24} = 5 \implies \frac{5d}{24} = 5 \implies 5d = 120 \implies d = 24

.

The common denominator of

8

and

12

is

24

. Simplifying the resulting linear equation yields the one-way distance.

Key Concept

Rate, Time, and Distance Problems:

d = r \cdot t

. Total trip time is the sum of individual leg times (

t_{\text{total}} = t_1 + t_2

).

Estimated Time:1m 30s

Question 173Question →

A specialty audio equipment distributor purchased a batch of $40$ high-fidelity headphones for a total cost of $\$10,000$ . The distributor marked up the wholesale cost price of each unit by $60\%$ to establish its list price. During a seasonal promotion, $30$ of the headphones were sold at a $20\%$ discount off the list price. The remaining $10$ headphones were sold during an end-of-season clearance at a $40\%$ discount off the list price. What was the distributor's net profit percentage on the total purchase of all $40$ headphones?

12\%

16\frac{2}{3}\%

20\%

28\%

35\%

Show answer & explanation

Answer: $20\%$

Answer

The distributor's net profit percentage on the total purchase is

20\%

.

To find the overall net profit percentage, calculate total revenue across all sales and compare it to total cost. The cost per unit is

\frac{\$10,000}{40} = \$250

. Marking up by

60\%

yields a list price of

\$250 \times 1.60 = \$400

. The

30

units sold at a

20\%

discount generated

30 \times (\$400 \times 0.80) = \$9,600

. The remaining

10

units sold at a

40\%

discount generated

10 \times (\$400 \times 0.60) = \$2,400

. Total revenue was

\$9,600 + \$2,400 = \$12,000

. Net profit is

\$12,000 - \$10,000 = \$2,000

. Expressed as a percentage of total cost,

\frac{\$2,000}{\$10,000} \times 100\% = 20\%

.

Step-by-Step Solution

1

Calculate the wholesale cost price per unit and the list price per unit.

Wholesale cost per unit =

\frac{\$10,000}{40} = \$250

. List price per unit =

\$250 \times (1 + 0.60) = \$400

.

Establishing the individual cost and marked-up retail base price is necessary to calculate discount prices.

2

Calculate the selling price and revenue for the two sales batches.

Promotional price =

\$400 \times (1 - 0.20) = \$320

. Revenue from first batch =

30 \times \$320 = \$9,600

. Clearance price =

\$400 \times (1 - 0.40) = \$240

. Revenue from second batch =

10 \times \$240 = \$2,400

.

Determining total revenue requires evaluating each portion of inventory sold at its respective discounted price.

3

Sum total revenue and compute total net profit.

Total Revenue =

\$9,600 + \$2,400 = \$12,000

. Total Net Profit =

\$12,000 - \$10,000 = \$2,000

.

Net profit is the absolute difference between total revenue generated and total wholesale cost incurred.

4

Compute the net profit percentage.

Net Profit Percentage =

\left(\frac{\$2,000}{\$10,000}\right) \times 100\% = 20\%

.

Profit percentage must always use the total original cost price as its base.

Key Concept

Calculating overall profit percentage across multi-tiered pricing by finding total revenue relative to original cost base.

Question 174Question →

In January, a logistics company's total monthly operational expenditure for its electric delivery fleet was $\$C$ . In February, the number of vehicles in the fleet increased by $25\%$ , while the monthly operational cost per vehicle decreased by $10\%$ due to optimized charging schedules. If the fleet's total operational expenditure in February was $\$33,750$ , what was the total operational expenditure, in dollars, in January?

Show answer & explanation

Answer: 30000

Answer

The total operational expenditure in January was $30,000.

The total expenditure is the product of the number of vehicles and the cost per vehicle. An increase of

25\%

in fleet size introduces a multiplier of

1.25

, while a

10\%

decrease in unit cost introduces a multiplier of

0.90

. The net multiplier for the total expenditure is

1.25 \times 0.90 = 1.125

(or

\frac{9}{8}

). Dividing the February total of

\$33,750

by

1.125

yields the January expenditure of

\$30,000

.

Step-by-Step Solution

1

Define variables for January's components and express January's total cost

Let

N

be the number of vehicles in January and

V

be the operational cost per vehicle in January. Total January expenditure

C = N \times V

.

Total cost is the product of quantity and unit cost.

2

Determine the updated fleet size and unit cost for February

February fleet size

= N \times (1 + 0.25) = 1.25 N = \frac{5}{4} N

. February unit cost

= V \times (1 - 0.10) = 0.90 V = \frac{9}{10} V

.

A

25\%

increase corresponds to a factor of

1.25

, and a

10\%

decrease corresponds to a factor of

0.90

.

3

Calculate February's total expenditure in terms of

C

February total cost

= \left(\frac{5}{4} N\right) \times \left(\frac{9}{10} V\right) = \frac{45}{40} NV = 1.125 C = \frac{9}{8} C

.

The total cost in February is the product of the updated fleet size and the updated unit cost.

4

Solve for January's total cost

C

\frac{9}{8} C = 33,750 \implies C = 33,750 \times \frac{8}{9} = 3,750 \times 8 = 30,000

.

Divide February's known cost by the net multiplier

1.125

(or

\frac{9}{8}

) to reverse the percent change.

Key Concept

Net percentage change on a product of variables and reverse calculation

Question 175Question →

An express cargo train leaves Station A heading toward Station B at a constant speed of $50$ miles per hour. At the exact same time, a local freight train leaves Station B heading toward Station A along a parallel track at a constant speed of $30$ miles per hour. If the total distance between Station A and Station B is $360$ miles, how many miles from Station A will the two trains meet?

Show answer & explanation

Answer: 225

Answer

The two trains will meet 225 miles from Station A.

When two objects move directly toward each other, their rate of approach is the sum of their speeds (

50 + 30 = 80

mph). Covering the total distance of

360

miles at a rate of

80

mph takes

4.5

hours. In

4.5

hours, the train traveling from Station A at

50

mph covers a distance of

50 \times 4.5 = 225

miles.

Step-by-Step Solution

1

Calculate the combined relative speed of both trains.

Relative speed =

50 + 30 = 80

miles per hour.

Since the two trains move toward each other, their relative rate of convergence is the sum of their individual speeds.

2

Calculate the time required for the two trains to meet.

Time

t = \frac{360 \text{ miles}}{80 \text{ mph}} = 4.5

hours.

Time equals total distance divided by combined relative speed.

3

Calculate the distance covered by the express train from Station A.

Distance =

50 \text{ mph} \times 4.5 \text{ hours} = 225

miles.

Distance equals the rate of the train departing Station A multiplied by the elapsed time.

Key Concept

Relative speed of converging objects moving toward each other

Estimated Time:1m 30s

Question 176Question →

A boutique clothing retailer purchases a line of jackets for a wholesale price of $x$ dollars. The retailer marks up the wholesale price by $50\%$ to set the original retail price. During a seasonal clearance sale, the original retail price is discounted by $30\%$ . If a customer purchases a jacket during the clearance sale for $\$126$ , what was the original wholesale price of the jacket, $x$ ?

\$90

\$105

\$120

\$150

\$180

Show answer & explanation

Answer: $\$120$

Answer

The original wholesale price of the jacket is

\$120

.

The wholesale price

x

is increased by

50\%

, giving a retail price of

1.5x

. A

30\%

discount reduces this retail price to

70\%

of its value:

1.5x \times 0.7 = 1.05x

. Given that the final price is

\$126

, solving

1.05x = 126

gives

x = \$120

.

Step-by-Step Solution

1

Express the original retail price in terms of the wholesale price

x

.

Original Retail Price

= x \times (1 + 0.50) = 1.50x

A

50\%

markup increases the initial value by a factor of

1.50

.

2

Express the clearance sale price in terms of the original retail price.

Clearance Sale Price

= 1.50x \times (1 - 0.30) = 1.50x \times 0.70 = 1.05x

A

30\%

discount means paying

70\%

of the marked-up retail price.

3

Set the sale price expression equal to

\$126

and solve for

x

.

1.05x = 126 \implies x = \frac{126}{1.05} = \frac{12600}{105} = 120

Dividing the clearance sale price by the net multiplier yields the original wholesale price.

Key Concept

Successive Percent Change and Multiplier Bases

Question 177Question →

A service technician travels by boat from Pier 1 to an offshore platform at a constant speed of $30$ miles per hour. Immediately upon arrival, the technician returns along the exact same route to Pier 1 at a constant speed of $60$ miles per hour. If the entire round-trip journey takes $3$ hours, what is the total distance, in miles, traveled by the technician?

60

90

120

135

180

Show answer & explanation

Answer: $120$

Answer

120 miles

To find the total distance, we must account for the time spent on each leg. Letting

d

be the one-way distance, the outbound time is

\frac{d}{30}

and the return time is

\frac{d}{60}

. Summing these gives

\frac{d}{30} + \frac{d}{60} = 3

hours, which simplifies to

\frac{3d}{60} = 3

, leading to

d = 60

miles. Since the technician completes a round trip, the total distance traveled is

2 \times 60 = 120

miles.

Step-by-Step Solution

1

Define variables for distance and express travel times for each leg of the trip.

Let

d

be the one-way distance between Pier 1 and the offshore platform. Time to reach platform:

t_1 = \frac{d}{30}

hours. Time to return:

t_2 = \frac{d}{60}

hours.

Time equals distance divided by speed (

t = \frac{d}{v}

).

2

Set up an equation using total time.

t_1 + t_2 = \frac{d}{30} + \frac{d}{60} = 3

The total duration for both legs combined is given as

3

hours.

3

Solve for the one-way distance

d

.

Find a common denominator:

\frac{2d}{60} + \frac{d}{60} = \frac{3d}{60} = \frac{d}{20} = 3 \implies d = 60

miles.

Combining fractional terms allows isolated calculation of the one-way distance.

4

Calculate the total round-trip distance.

Total distance

= 2d = 2 \times 60 = 120

miles.

The question specifically asks for the total distance traveled during the entire round trip.

Key Concept

Average Speed and Total Distance in Round-Trip Scenarios

Estimated Time:2m 0s

Question 178Question →

A motorist travels from Town A to Town B, a distance of $120$ miles, at a constant speed of $60$ miles per hour. On the return trip from Town B to Town A along the exact same route, heavy traffic slows the vehicle's speed to $20$ miles per hour for the first half of the distance. At what constant speed, in miles per hour, must the motorist travel for the remaining half of the return trip so that the average speed for the entire $240$ -mile round trip is $32$ miles per hour?

16

24

28

36

44

Show answer & explanation

Answer: 24

Answer

24 miles per hour

The correct answer is 24 miles per hour. To achieve an overall average speed of 32 mph over the total 240-mile trip, the entire journey must take exactly 7.5 hours (

240 / 32 = 7.5

). The outbound 120-mile trip at 60 mph takes 2 hours (

120 / 60 = 2

). The first 60 miles of the return trip at 20 mph takes 3 hours (

60 / 20 = 3

). This leaves

7.5 - 5 = 2.5

hours to complete the final 60 miles. Dividing 60 miles by 2.5 hours yields a required speed of 24 mph.

Step-by-Step Solution

1

Calculate the total time required for the entire round trip.

\text{Total Time} = \frac{\text{Total Distance}}{\text{Average Speed}} = \frac{240\text{ miles}}{32\text{ mph}} = 7.5\text{ hours}

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

2

Calculate the time taken for the outbound trip and the first half of the return trip.

\text{Outbound Time} = \frac{120}{60} = 2\text{ hours}; \quad \text{Return Leg 1 Time} = \frac{60}{20} = 3\text{ hours}

Time equals distance divided by rate for each specific segment.

3

Determine the remaining time and required speed for the final segment of

60

miles.

\text{Remaining Time} = 7.5 - (2 + 3) = 2.5\text{ hours}; \quad \text{Required Speed} = \frac{60\text{ miles}}{2.5\text{ hours}} = 24\text{ mph}

Dividing the remaining

60

miles by the remaining

2.5

hours gives the required speed.

Key Concept

Average speed for multi-leg journeys must always be calculated as Total Distance divided by Total Time, not by taking the arithmetic mean of individual speeds.

Estimated Time:2m 0s

Question 179Question →

At the beginning of 2021, a technology startup had an initial valuation of $V$ dollars. Over the next three years, its valuation experienced the following successive annual percentage changes:
- In 2021, the valuation increased by $20\%$ .
- In 2022, the valuation decreased by $10\%$ relative to its valuation at the end of 2021.
- In 2023, the valuation increased by $50\%$ relative to its valuation at the end of 2022.

If the startup's valuation at the end of 2023 was $\$648,000$ , what was its initial valuation $V$ at the beginning of 2021?

\$360,000

\$400,000

\$405,000

\$432,000

\$450,000

Show answer & explanation

Answer: $\$400,000$

Answer

The initial valuation of the startup at the beginning of 2021 was

\$400,000

.

To find the initial valuation

V

, we must apply successive growth multipliers rather than adding simple percentages. A

20\%

increase corresponds to a multiplier of

1.20

, a

10\%

decrease corresponds to

0.90

, and a

50\%

increase corresponds to

1.50

. Multiplying these factors yields

1.20 \times 0.90 \times 1.50 = 1.62

. Setting up the equation

1.62 V = \$648,000

yields

V = \$400,000

.

Step-by-Step Solution

1

Express each year's percentage change as a multiplier of the value at the start of that year.

Year 2021 multiplier =

1 + 0.20 = 1.20

; Year 2022 multiplier =

1 - 0.10 = 0.90

; Year 2023 multiplier =

1 + 0.50 = 1.50

.

Percentage increases and decreases compound sequentially on the updated base value of each preceding period.

2

Calculate the combined successive percentage multiplier for the three-year period.

Combined Multiplier =

1.20 \times 0.90 \times 1.50 = 1.08 \times 1.50 = 1.62

.

Successive changes multiply together to determine the final overall factor relative to the initial value

V

.

3

Set up the linear equation relating the final valuation to the initial valuation

V

and solve for

V

.

1.62 V = 648,000 \implies V = \frac{648,000}{1.62} = \frac{648,000}{\frac{162}{100}} = 4,000 \times 100 = 400,000

.

Dividing the final amount by the combined multiplier gives the initial amount

V

.

Key Concept

Successive Percent Change

Estimated Time:2m 0s

Question 180Question →

A commercial cargo drone flies from Base Alpha to Outpost Bravo against a headwind at a constant speed of $30$ miles per hour. On the return flight along the exact same route, traveling with a tailwind, the drone maintains a constant speed of $60$ miles per hour. If the drone spent a total of $3$ hours in flight for the entire round trip, what was its average speed, in miles per hour, for the complete trip?

36

40

45

50

60

Show answer & explanation

Answer: $40$

Answer

The average speed of the drone for the entire round trip is

40

miles per hour.

The average speed is calculated as total distance divided by total time. Since the one-way distance is

60

miles, the total round-trip distance is

120

miles. Dividing

120

miles by the total time of

3

hours yields

40

miles per hour.

Step-by-Step Solution

1

Set up distance and time relationships for each leg of the trip.

Let

d

be the distance in miles between Base Alpha and Outpost Bravo. Time taken for the outbound leg is

t_1 = \frac{d}{30}

hours, and time taken for the return leg is

t_2 = \frac{d}{60}

hours.

Time equals distance divided by rate (

t = \frac{d}{r}

).

2

Express total time in terms of

d

and solve for

d

.

Total time

T = t_1 + t_2 = \frac{d}{30} + \frac{d}{60} = \frac{3d}{60} = \frac{d}{20}

. Given total time is

3

hours,

\frac{d}{20} = 3 \implies d = 60

miles.

Summing the times for both legs equals the total flight time given in the problem.

3

Calculate the total distance and average speed.

Total distance

= 2d = 2 \times 60 = 120

miles. Average speed

= \frac{\text{Total Distance}}{\text{Total Time}} = \frac{120}{3} = 40

miles per hour.

Average speed for any multi-leg trip is always defined as total distance divided by total time.

Key Concept

Average Speed for a Round Trip

Estimated Time:1m 30s

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