Word Problems and Applied Math Practice Questions — GMAT

Question 1Question →

A fuel tank initially contains $60$ gallons of a fuel mixture that is $10\%$ ethanol and $90\%$ gasoline by volume. A mechanic removes $x$ gallons of this mixture and replaces it with an equal volume of pure ethanol to obtain a new mixture that is $25\%$ ethanol by volume. What is the value of $x$ ?

Show answer & explanation

Answer: 10

Answer

The volume of fuel mixture that must be removed and replaced with pure ethanol is

10

gallons.

Replacing

10

gallons of the

10\%

ethanol fuel with pure ethanol removes

1

gallon of ethanol and adds

10

gallons of pure ethanol. The total ethanol in the tank becomes

6 - 1 + 10 = 15

gallons, which represents exactly

25\%

of the total

60

-gallon volume.

Step-by-Step Solution

1

Calculate the initial volume of ethanol in the tank

Ethanol volume =

0.10 \times 60 = 6

gallons.

Establishes the starting quantity of the solute.

2

Express the amount of ethanol after removal and replacement in terms of

x

Final ethanol volume =

6 - 0.10x + x = 6 + 0.90x

gallons.

Removing

x

gallons of fuel removes

10\%

ethanol, while adding

x

gallons of pure ethanol adds

100\%

ethanol.

3

Set up an equation using the target ethanol concentration

6 + 0.90x = 0.25 \times 60 = 15

.

The final

60

-gallon mixture must contain

25\%

ethanol by volume.

4

Solve the equation for

x

0.90x = 9 \implies x = 10

.

Isolates

x

to find the required replacement volume.

Key Concept

Dilution and fluid replacement in mixture problems

Question 2Question →

An investor allocated a total principal of $\$25,000$ between two savings accounts, Account X and Account Y. Account X earns simple interest at an annual rate of $8\%$ , while Account Y earns interest at an annual rate of $10\%$ , compounded annually. If after $2$ years the total interest earned from Account Y exceeds the total interest earned from Account X by $\$1,550$ , what was the amount, in dollars, invested in Account Y?

Show answer & explanation

Answer: 15000

Answer

The amount invested in Account Y was $15,000 dollars.

Let

P_Y

be the principal invested in Account Y. Since the total investment is

\$25,000

, the principal invested in Account X is

25,000 - P_Y

. Account X earns simple interest over 2 years equal to

(25,000 - P_Y) \times 0.08 \times 2 = 4,000 - 0.16 P_Y

. Account Y earns compound interest over 2 years equal to

P_Y \times ((1 + 0.10)^2 - 1) = 0.21 P_Y

. The problem states that the interest from Account Y exceeds that from Account X by

\$1,550

, giving the equation

0.21 P_Y - (4,000 - 0.16 P_Y) = 1,550

. Combining like terms yields

0.37 P_Y = 5,550

, which simplifies to

P_Y = 15,000

.

Step-by-Step Solution

1

Express the principal of Account X in terms of Account Y

Principal of Account X =

25,000 - P_Y

The total sum invested across both accounts is

\$25,000

.

2

Calculate the interest earned from Account X over 2 years

I_X = 4,000 - 0.16 P_Y

Simple interest is calculated as

I = P \times r \times t = (25,000 - P_Y) \times 0.08 \times 2

.

3

Calculate the compound interest earned from Account Y over 2 years

I_Y = 0.21 P_Y

The compound multiplier for 2 years at

10\%

is

(1.10)^2 = 1.21

, giving a interest percentage of

1.21 - 1 = 0.21

or

21\%

.

4

Set up and simplify the equation for the difference in interest

0.37 P_Y = 5,550

Subtracting

I_X

from

I_Y

yields

0.21 P_Y - (4,000 - 0.16 P_Y) = 1,550

, which simplifies to

0.37 P_Y - 4,000 = 1,550

.

5

Solve for

P_Y

P_Y = 15,000

Dividing

5,550

by

0.37

gives

15,000

.

Key Concept

Combining Simple and Compound Interest Linear Equations

Question 3Question →

A commercial bakery produces artisan cakes at a base cost of $\$40$ per cake. The bakery marks up the base cost by $75\%$ to set the catalog price. To encourage bulk corporate orders, the bakery offers a $20\%$ discount off the catalog price. If the bakery incurs an additional packaging and delivery expense of $\$6$ per cake, what is the bakery's net profit, in dollars, on each corporate cake sold?

Show answer & explanation

Answer: 10

Answer

The bakery earns a net profit of $10 on each corporate cake sold.

The catalog price is

40 \times 1.75 =

70. Applying a 20% discount gives a discounted selling price of

70 \times 0.80 =

56. The total cost to produce and package each cake is

40 +

6 =

46. Subtracting total cost from selling price yields a net profit of

56 -

46 =

10.

Step-by-Step Solution

1

Calculate the catalog price using the base cost and markup percentage.

Catalog price =

40 \times (1 + 0.75) =

70

The catalog price is established by marking up the base production cost by 75%.

2

Calculate the selling price after applying the discount.

Selling price =

70 \times (1 - 0.20) =

56

The 20% promotional discount applies to the marked-up catalog price.

3

Calculate the total cost incurred by the bakery for each cake.

Total cost = Base cost + Overhead expense =

40 +

6 = $46

Net profit calculations must account for both production base cost and direct selling/packaging expenses.

4

Subtract total cost from selling price to determine net profit.

Net profit =

56 -

46 = $10

Net profit equals total revenue per unit minus total costs per unit.

Key Concept

Profit, Loss, and Markup with Successive Adjustments and Overhead Expenses

Question 4Question →

A delivery van leaves a warehouse traveling along a straight highway at a constant speed of $45$ miles per hour. Exactly $40$ minutes later, a courier motorcycle leaves the same warehouse along the exact same route, traveling at a constant speed of $60$ miles per hour. How many hours after the courier motorcycle departs will it overtake the delivery van?

Show answer & explanation

Answer: 2

Answer

The courier motorcycle will overtake the delivery van 2 hours after the motorcycle departs.

The delivery van travels for

\frac{2}{3}

hour before the motorcycle begins moving, establishing a initial lead of

45 \times \frac{2}{3} = 30

miles. Since both vehicles travel in the same direction, the motorcycle gains on the van at a rate equal to the difference of their speeds,

60 - 45 = 15

miles per hour. To cover the 30-mile gap, it takes

\frac{30}{15} = 2

hours after the motorcycle departs.

Step-by-Step Solution

1

Convert the van's head-start time from minutes to hours.

Head-start time is

\frac{40}{60} = \frac{2}{3}

hours.

Speed is given in miles per hour, so time units must be converted to hours for consistency.

2

Calculate the lead distance gained by the van before the motorcycle starts moving.

Lead distance =

45 \times \frac{2}{3} = 30

miles.

Distance equals rate multiplied by time (

d = r \cdot t

).

3

Calculate the relative speed at which the motorcycle gains on the van.

Relative speed =

60 - 45 = 15

miles per hour.

When two objects move in the same direction, the rate at which the distance between them decreases is the difference between their individual speeds.

4

Determine the time needed for the motorcycle to eliminate the 30-mile gap.

Time =

\frac{30}{15} = 2

hours.

Time equals distance divided by relative speed (

t = \frac{d}{r_{\text{rel}}}

).

Key Concept

Catch-up scenarios and relative speed in same-direction movement.

Alternative Method

Equate the distance expressions for both vehicles at the moment of overtaking. Let

t

represent the motorcycle's travel time in hours. The van's travel time is

t + \frac{2}{3}

hours. Since both travel equal distances from the warehouse:

60t = 45\left(t + \frac{2}{3}\right) \implies 60t = 45t + 30 \implies 15t = 30 \implies t = 2

hours.

Estimated Time:1m 30s

Question 5Question →

At the beginning of the year, a cloud storage service had a total of $S$ active subscribers. During the first quarter, the number of subscribers increased by $20\%$ . In the second quarter, the number of subscribers decreased by $10\%$ . In the third quarter, the number of subscribers increased by $25\%$ . If the total net increase in subscribers from the beginning of the year to the end of the third quarter was $7{}000$ , what was the value of $S$ ?

Show answer & explanation

Answer: 20000

Answer

The initial number of subscribers

S

was

20{}000

.

Each quarter's percent change applies to the balance at the start of that quarter. An increase of

20\%

multiplies the quantity by

1.20

. A decrease of

10\%

multiplies it by

0.90

. A subsequent increase of

25\%

multiplies it by

1.25

. The cumulative factor is

1.20 \times 0.90 \times 1.25 = 1.35

. Since the final subscriber total is

1.35S

, the net gain is

0.35S

. Given that

0.35S = 7{}000

, dividing

7{}000

by

0.35

yields

S = 20{}000

.

Step-by-Step Solution

1

Calculate the composite growth factor after three quarters of successive percentage changes.

Growth factor

= (1.20) \times (0.90) \times (1.25) = 1.35

Successive percentage changes multiply together, each using the previous period's value as its new base.

2

Express the net increase in terms of the initial amount

S

.

Net increase

= 1.35S - S = 0.35S

The net change is the final value minus the starting value.

3

Set the net increase equal to

7{}000

and solve for

S

.

S = \frac{7{}000}{0.35} = 20{}000

Dividing the absolute net increase by the net percentage increase rate yields the original base value.

Key Concept

Successive Percent Change and Base Value Calculation

Estimated Time:1m 30s

Question 6Question →

An artisanal coffee roaster creates a custom blend using Arabica, Robusta, and Liberica beans in the initial ratio of $5 : 3 : 2$ by weight, respectively. To adjust the flavor profile, the roaster adds $60$ kilograms of Arabica beans and $20$ kilograms of Robusta beans to the blend, while leaving the amount of Liberica beans unchanged. If the new ratio of Arabica to Robusta to Liberica beans is $4 : 2 : 1$ , respectively, what was the initial total weight, in kilograms, of the blend?

100

120

160

200

280

Show answer & explanation

Answer: 200

Answer

200 kilograms

Let the initial weights of Arabica, Robusta, and Liberica beans be

5x

,

3x

, and

2x

kilograms, respectively. The initial total weight is

5x + 3x + 2x = 10x

kg. After adding

60

kg of Arabica and

20

kg of Robusta, the new weights are

5x + 60

,

3x + 20

, and

2x

. Using the new ratio of Robusta to Liberica (

2 : 1

), we set up the equation

\frac{3x + 20}{2x} = 2

, which simplifies to

3x + 20 = 4x

, giving

x = 20

. Substituting

x = 20

into the initial total weight expression yields

10(20) = 200

kg.

Step-by-Step Solution

1

Define initial quantities using a common multiplier

x

.

Arabica

= 5x

, Robusta

= 3x

, Liberica

= 2x

. Initial total weight

= 5x + 3x + 2x = 10x

.

Expressing ratio terms in terms of a single multiplier allows algebraic modeling of changes to each component.

2

Express the new quantities after adding the specified weights.

New Arabica

= 5x + 60

, New Robusta

= 3x + 20

, New Liberica

= 2x

.

Arabica and Robusta amounts increase by 60 kg and 20 kg respectively, while Liberica remains constant.

3

Set up a proportion using the new ratio of Robusta to Liberica (

2 : 1

).

\frac{3x + 20}{2x} = \frac{2}{1} \implies 3x + 20 = 4x \implies x = 20

.

Equating the ratio of the updated Robusta and Liberica quantities to

2:1

yields the multiplier

x

.

4

Verify consistency with the Arabica to Liberica ratio (

4 : 1

).

\frac{5(20) + 60}{2(20)} = \frac{160}{40} = \frac{4}{1}

, which matches the given final ratio.

Ensures that

x = 20

satisfies the complete 3-part ratio of

4 : 2 : 1

.

5

Calculate the initial total weight.

Initial total

= 10x = 10 \times 20 = 200

kg.

Multiplying the initial sum of ratio parts (10) by

x = 20

gives the original total blend weight.

Key Concept

Ratio and Proportion Word Problems

Question 7Question →

A municipal water treatment facility utilizes three primary purification units—Unit P, Unit Q, and Unit R—to process large water batches. Operating alone at its constant rate, Unit P can process a full batch in $12$ hours, Unit Q can process a full batch in $18$ hours, and Unit R can process a full batch in $36$ hours. All three units begin processing a batch together. After $3$ hours, Unit P stops operating due to scheduled maintenance. Unit Q and Unit R continue operating together for an additional $2$ hours, after which Unit Q is shut down. Unit R then completes the remainder of the batch working alone. How many total hours does it take from start to finish to process the complete batch?

Show answer & explanation

Answer: 17

Answer

The total time required to process the complete batch from start to finish is 17 hours.

To find the total time from start to finish, analyze the work completed in each of the three stages:

1. In Stage 1 (3 hours), all three units operate together. Their combined rate is

\frac{1}{12} + \frac{1}{18} + \frac{1}{36} = \frac{3 + 2 + 1}{36} = \frac{6}{36} = \frac{1}{6}

batch per hour. In

3

hours, they complete

3 \times \frac{1}{6} = \frac{1}{2}

of the total batch.

2. In Stage 2 (2 hours), Unit P stops and Units Q and R continue together. Their combined rate is

\frac{1}{18} + \frac{1}{36} = \frac{2 + 1}{36} = \frac{3}{36} = \frac{1}{12}

batch per hour. In

2

hours, they complete

2 \times \frac{1}{12} = \frac{1}{6}

of the total batch.

3. After Stage 2, the total fraction of work completed is

\frac{1}{2} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}

of the batch. The remaining fraction of work is

1 - \frac{2}{3} = \frac{1}{3}

of the batch.

4. In Stage 3, Unit R completes the remaining

\frac{1}{3}

batch alone at its rate of

\frac{1}{36}

batch per hour. The time taken by Unit R is

\frac{1/3}{1/36} = 12

hours.

Adding the durations of all three stages gives

3 + 2 + 12 = 17

total hours.

Step-by-Step Solution

1

Calculate individual work rates

Rate of P =

\frac{1}{12}

batch/hr, Rate of Q =

\frac{1}{18}

batch/hr, Rate of R =

\frac{1}{36}

batch/hr

Work rate is the reciprocal of the total time required to complete one entire job.

2

Calculate work done during Stage 1 (first 3 hours)

Combined rate of P, Q, R =

\frac{1}{12} + \frac{1}{18} + \frac{1}{36} = \frac{6}{36} = \frac{1}{6}

batch/hr. Work done =

3 \times \frac{1}{6} = \frac{1}{2}

batch

All three units operate simultaneously for 3 hours.

3

Calculate work done during Stage 2 (next 2 hours)

Combined rate of Q and R =

\frac{1}{18} + \frac{1}{36} = \frac{3}{36} = \frac{1}{12}

batch/hr. Work done =

2 \times \frac{1}{12} = \frac{1}{6}

batch

Unit P stops working, leaving Q and R to operate together for 2 hours.

4

Determine remaining work and time required for Stage 3

Remaining work =

1 - \left(\frac{1}{2} + \frac{1}{6}\right) = \frac{1}{3}

batch. Time for R alone =

\frac{1/3}{1/36} = 12

hours

Unit R must complete the remaining fraction of the batch operating alone at its constant rate.

5

Calculate total elapsed time

3\text{ hours} + 2\text{ hours} + 12\text{ hours} = 17\text{ hours}

Sum the durations of all three distinct operational stages.

Key Concept

Combined Work Rate in Multi-Stage Work Scenarios

Question 8Question →

A home decor retailer imports hand-woven area rugs at a cost price of $500$ dollars each. The retailer marks up the cost price by $60\%$ to establish the regular list price. During a seasonal clearance sale, each rug is sold at a discount of $25\%$ off its regular list price. If the retailer incurs an additional non-refundable freight charge of $40$ dollars per rug, which is included in the total cost of each rug, what is the net profit percentage earned by the retailer relative to the total cost of each rug?

11\frac{1}{9}\%

10\%

20\%

25\%

7.5\%

Show answer & explanation

Answer: $11\frac{1}{9}\%$

Answer

The net profit percentage earned on the total cost of each rug is 11 1/9%.

The total cost of importing each rug includes both the initial cost price (

500) and the freight fee (

40), yielding a total cost base of

540. A 60% markup on the

500 cost price gives a list price of

500 * 1.60 =

800. Applying a 25% discount to the

800 list price yields a final selling price of

800 * 0.75 =

600. The net profit is

600 -

540 =

60. Expressed as a percentage of the total cost base of

540, the net profit percentage is (

60 / $540) * 100% = 1/9 * 100% = 11 1/9%.

Step-by-Step Solution

1

Calculate the total cost per rug

Total Cost =

500 +

40 =

540

Total cost must include both the initial purchase cost price and the additional freight charge.

2

Determine the regular list price after markup

List Price =

500 \times (1 + 0.60) =

500 \times 1.60 =

800

The markup of 60% is applied to the initial cost price of $500.

3

Calculate the discounted selling price

Selling Price =

800 \times (1 - 0.25) =

800 \times 0.75 =

600

The 25% discount is applied to the regular list price of $800.

4

Determine net profit per rug

Net Profit = Selling Price - Total Cost =

600 -

540 =

60

Net profit is the difference between final revenue (selling price) and total expenses (total cost).

5

Calculate the net profit percentage based on total cost

\text{Net Profit \%} = \left(\frac{60}{540}\right) \times 100\% = \frac{1}{9} \times 100\% = 11\frac{1}{9}\%

Percentage profit must be evaluated against the total cost base of $540.

Key Concept

Profit, Loss, and Markup Base Calculations

Estimated Time:1m 30s

Question 9Question →

An antique dealer purchased a vintage writing desk for $\$600$ . To prepare the desk for sale, the dealer spent $\$40$ on restoration. The dealer then marked up the initial purchase price of $\$600$ by $50\%$ to establish the list price. During a seasonal promotion, the desk was sold at a $20\%$ discount off the list price. What was the dealer's profit as a percentage of the total expenditure (purchase price plus restoration cost)?

Show answer & explanation

Answer: 12.5

Answer

The dealer's profit as a percentage of the total expenditure is

12.5\%

.

To find the net profit percentage on total expenditure, first compute total cost (

\$600 + \$40 = \$640

). Next, compute list price by applying a

50\%

markup to the initial purchase price (

\$600 \times 1.50 = \$900

). Then compute the selling price after a

20\%

discount on list price (

\$900 \times 0.80 = \$720

). Net profit is selling price minus total cost (

\$720 - \$640 = \$80

). Finally, express net profit as a percentage of total cost:

(\$80 / \$640) \times 100\% = 12.5\%

.

Step-by-Step Solution

1

Calculate the total cost (expenditure)

Total cost =

\$600 + \$40 = \$640

The total investment includes both the purchase price of the desk and the mandatory restoration fee incurred before sale.

2

Determine the list price

List price =

\$600 \times 1.50 = \$900

The markup of

50\%

is applied specifically to the initial purchase price of

\$600

.

3

Determine the selling price

Selling price =

\$900 \times 0.80 = \$720

The

20\%

discount reduces the list price of

\$900

.

4

Calculate the net dollar profit

Net profit =

\$720 - \$640 = \$80

Profit is calculated as total revenue (selling price) minus total cost.

5

Calculate the net profit percentage

Net profit percentage =

(\$80 / \$640) \times 100\% = 12.5\%

The percentage return must be evaluated using total expenditure (

\$640

) as the base.

Key Concept

Calculating successive percentage markups and discounts with secondary cost components

Estimated Time:1m 45s

Question 10Question →

A commuter travels to work along a straight route at a constant speed of $40$ miles per hour. On the return trip along the exact same route, traffic delays reduce the average speed to $24$ miles per hour. If the return trip takes $40$ minutes longer than the trip to work, what is the total distance, in miles, traveled by the commuter for the round trip?

Show answer & explanation

Answer: 80

Answer

The total distance traveled by the commuter for the round trip is 80 miles.

The total distance for the round trip is 80 miles. By expressing time as distance divided by speed, the outbound trip takes

\frac{D}{40}

hours and the return trip takes

\frac{D}{24}

hours. Since the return trip takes

40

minutes (

\frac{2}{3}

hours) longer, setting up the equation

\frac{D}{24} - \frac{D}{40} = \frac{2}{3}

yields

D = 40

miles for one way. The round-trip distance is

2 \times 40 = 80

miles.

Step-by-Step Solution

1

Define variables for the one-way distance and expressions for the travel time of each leg.

Let

D

represent the one-way distance in miles. Outbound time

t_1 = \frac{D}{40}

hours; return time

t_2 = \frac{D}{24}

hours.

Using the relation

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

expresses time in terms of a single unknown variable.

2

Convert the given time difference from minutes to hours.

40 \text{ minutes} = \frac{40}{60} \text{ hours} = \frac{2}{3} \text{ hours}

.

Speeds are given in miles per hour, so time units must be in hours for dimensional consistency.

3

Formulate the equation representing the time difference and solve for

D

.

\frac{D}{24} - \frac{D}{40} = \frac{2}{3} \implies \frac{2D}{120} = \frac{2}{3} \implies \frac{D}{60} = \frac{2}{3} \implies D = 40 \text{ miles}

.

Subtracting the faster outbound time from the slower return time equals the time difference.

4

Compute the total distance for the entire round trip.

\text{Total distance} = 2 \times D = 2 \times 40 = 80 \text{ miles}

.

The question asks for the round-trip distance, which includes both the outbound and return journeys (

2D

).

Key Concept

Rate, Time, and Distance Problems

Question 11Question →

An investor deposits $\$800$ into a savings account that pays a simple annual interest rate of $5\%$ . If no additional deposits or withdrawals are made, what is the total interest, in dollars, earned on the account after $3$ years?

Show answer & explanation

Answer: 120

Answer

The total interest earned after 3 years is 120 dollars.

The total interest earned is computed using the simple interest formula

I = P \times r \times t

. Substituting

P = 800

,

r = 0.05

, and

t = 3

yields

I = 800 \times 0.05 \times 3 = 120

dollars.

Step-by-Step Solution

1

Identify the principal, rate, and time from the problem stem.

Principal

P = \$800

, interest rate

r = 5\% = 0.05

, and time

t = 3

years.

These values are required to apply the simple interest formula.

2

Calculate simple interest using

I = P \times r \times t

.

I = 800 \times 0.05 \times 3 = 120

.

Simple interest accumulates linearly over time based on the initial principal.

Key Concept

Simple Interest Calculation

Estimated Time:45s

Question 12Question →

Technician A can assemble a solar panel frame in $6$ hours working alone at a constant rate. Technician B can assemble the exact same solar panel frame in $3$ hours working alone at a constant rate. Working together at their respective constant rates, how many hours will it take Technician A and Technician B to assemble one solar panel frame?

1.5

hours

2

hours

3

hours

4.5

hours

9

hours

Show answer & explanation

Answer: $2$ hours

Answer

Working together, Technician A and Technician B will complete the assembly in

2

hours.

The correct option is

2

hours because Technician A completes

\frac{1}{6}

of the work in one hour and Technician B completes

\frac{1}{3}

(or

\frac{2}{6}

) of the work in one hour. Combined, they complete

\frac{3}{6} = \frac{1}{2}

of the work each hour, so the entire job takes

2

hours.

Step-by-Step Solution

1

Determine individual work rates per hour.

Technician A's rate is

\frac{1}{6}

frame per hour, and Technician B's rate is

\frac{1}{3}

frame per hour.

Work rate is defined as the fraction of a job completed per unit of time, calculated as

\frac{1}{\text{time}}

.

2

Add the individual rates to find the combined rate.

Combined rate

= \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}

frame per hour.

When entities work simultaneously, their individual rates add up.

3

Calculate total time required for one full frame.

Total time

= \frac{1}{\text{Combined Rate}} = \frac{1}{1/2} = 2

hours.

Total time required is the reciprocal of the combined work rate.

Key Concept

Combined Work Rate Formula

Estimated Time:50s

Question 13Question →

Pump A can fill a water storage tank in $4$ hours when operating alone at a constant rate. Pump B can fill the same tank in $6$ hours when operating alone at a constant rate. If both pumps operate simultaneously at their respective constant rates, how many hours will it take to fill the empty tank completely?

2.4

hours

5.0

hours

10.0

hours

2.0

hours

\frac{5}{12}

hours

Show answer & explanation

Answer: $2.4$ hours

Answer

2.4

hours

To find the combined time, sum the hourly rates of both pumps:

\frac{1}{4} + \frac{1}{6} = \frac{5}{12}

tanks per hour. Inverting this combined rate gives the total hours required:

\frac{12}{5} = 2.4

hours.

Step-by-Step Solution

1

Determine individual work rates per hour.

Pump A's rate is

\frac{1}{4}

of the tank per hour, and Pump B's rate is

\frac{1}{6}

of the tank per hour.

Work rate is defined as

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

.

2

Add the individual rates to find the combined rate.

\text{Combined Rate} = \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}\text{ tank per hour}.

When entities work together, their work rates add linearly.

3

Calculate the total time required for the combined rate to complete

1

full tank.

\text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4\text{ hours}.

Time equals total work divided by the combined work rate.

Key Concept

Combined Work Rate Formula

Estimated Time:1m 0s

Question 14Question →

A delivery drone flies in a straight line from Hub A to Hub B against a constant headwind of $10$ miles per hour, taking $4$ hours to complete the flight. On the return flight from Hub B to Hub A along the exact same path, the wind direction reverses to become a tailwind, and its speed increases by $50$ percent. If the return flight takes $2$ hours and the drone maintains a constant airspeed in still air throughout both flights, what was the drone's average speed, in miles per hour, for the entire round trip?

16\frac{2}{3}

20

33\frac{1}{3}

35

37\frac{1}{2}

Show answer & explanation

Answer: $33\frac{1}{3}$

Answer

The drone's average speed for the entire round trip is

33\frac{1}{3}

miles per hour.

The drone's still-air speed

v

is determined by equating the outbound and return distance equations:

4(v - 10) = 2(v + 15)

, giving

v = 35

miles per hour. The outbound ground speed is

25

miles per hour over

100

miles, and the return ground speed is

50

miles per hour over

100

miles. Dividing the total round-trip distance of

200

miles by the total time of

6

hours yields an average speed of

\frac{200}{6} = 33\frac{1}{3}

miles per hour.

Step-by-Step Solution

1

Express the effective ground speeds and distance for both legs of the trip in terms of the drone's still-air speed

v

.

Outbound headwind speed =

10

mph, ground speed =

v - 10

mph, outbound distance

D = 4(v - 10)

. Return tailwind speed =

10 \times (1 + 0.50) = 15

mph, return ground speed =

v + 15

mph, return distance

D = 2(v + 15)

.

Distance equals speed multiplied by time, and wind speeds adjust the effective ground speed depending on direction.

2

Equate the distance expressions to solve for the still-air speed

v

.

4(v - 10) = 2(v + 15) \implies 4v - 40 = 2v + 30 \implies 2v = 70 \implies v = 35

mph.

The distance from Hub A to Hub B is identical in both directions.

3

Calculate the one-way distance

D

and total round-trip distance.

One-way distance

D = 4(35 - 10) = 100

miles. Total round-trip distance

= 100 + 100 = 200

miles.

Average speed requires total distance covered across both legs.

4

Calculate the average speed for the entire round trip using

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

.

Total time

= 4 + 2 = 6

hours. Average speed

= \frac{200}{6} = \frac{100}{3} = 33\frac{1}{3}

mph.

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

Key Concept

Average Speed for Multi-Leg Journeys with Wind Vector Effects

Estimated Time:2m 30s

Question 15Question →

A cosmetic chemist creates a skin care product by mixing two vitamin C solutions. Solution A consists of $200$ milliliters of a liquid that is $15\%$ vitamin C by volume. Solution B is a liquid that is $25\%$ vitamin C by volume. How many milliliters of Solution B must be added to Solution A so that the resulting mixture is $21\%$ vitamin C by volume?

Show answer & explanation

Answer: 300

Answer

300

milliliters of Solution B must be added.

Adding

300

mL of Solution B contributes

300 \times 0.25 = 75

mL of pure vitamin C. Combined with Solution A's

30

mL of vitamin C, the total amount of vitamin C is

105

mL in a total mixture volume of

200 + 300 = 500

mL. The resulting concentration is

\frac{105}{500} = 21\%

.

Step-by-Step Solution

1

Calculate the volume of pure solute (vitamin C) contained in Solution A.

200 \times 0.15 = 30

mL of pure vitamin C.

The solute volume is calculated by multiplying total volume by the percentage concentration.

2

Express the solute contribution of Solution B and set up the equation for the combined concentration.

\frac{30 + 0.25x}{200 + x} = 0.21

The final concentration is equal to total volume of pure solute divided by the total volume of the final mixture.

3

Solve the algebraic equation for

x

.

30 + 0.25x = 42 + 0.21x \implies 0.04x = 12 \implies x = 300

Cross-multiplying and isolating

x

yields the required volume of Solution B in milliliters.

Key Concept

Weighted average concentration equation for mixing two liquids.

Estimated Time:1m 30s

Question 16Question →

Tank P contains $80$ liters of a liquid fertilizer solution that is $30\%$ nitrogen by volume. Tank Q contains $120$ liters of a liquid fertilizer solution that is $15\%$ nitrogen by volume. If $x$ liters of solution are transferred from Tank P into Tank Q, causing the nitrogen concentration in Tank Q to become $20\%$ , what is the value of $x$ ?

20

30

40

60

80

Show answer & explanation

Answer: $60$

Answer

The volume transferred,

x

, is

60

liters.

The initial amount of nitrogen in Tank Q is

0.15 \times 120 = 18

liters. Transferring

x

liters from Tank P adds

0.30x

liters of nitrogen and increases Tank Q's total volume to

120 + x

liters. Setting the new concentration to

20\%

, we get

\frac{18 + 0.30x}{120 + x} = 0.20

. Solving for

x

gives

18 + 0.30x = 24 + 0.20x \implies 0.10x = 6 \implies x = 60

. Thus,

60

liters must be transferred.

Step-by-Step Solution

1

Calculate the initial volume of pure nitrogen in Tank Q.

Initial nitrogen in Tank Q =

15\% \text{ of } 120 = 0.15 \times 120 = 18

liters.

To determine the new concentration, we must know the starting amount of solute.

2

Express the added nitrogen and the new total volume of Tank Q in terms of

x

.

Nitrogen added from Tank P =

0.30x

liters; New total volume of Tank Q =

120 + x

liters.

The solution transferred carries

30\%

nitrogen per liter and increases both the nitrogen content and total volume of Tank Q.

3

Set up the concentration equation for Tank Q and solve for

x

.

\frac{18 + 0.30x}{120 + x} = 0.20 \implies 18 + 0.30x = 0.20(120 + x) \implies 18 + 0.30x = 24 + 0.20x \implies 0.10x = 6 \implies x = 60.

The concentration is defined as total solute divided by total solution volume.

Key Concept

Weighted Average and Mixture Equations

Question 17Question →

A university library surveyed $240$ graduate students regarding the research methodologies used in their dissertations: Quantitative Analysis ( $Q$ ), Qualitative Interviews ( $I$ ), and Archival Research ( $A$ ). Every student surveyed used at least one of these three methodologies. A total of $130$ students used Quantitative Analysis, $110$ used Qualitative Interviews, and $90$ used Archival Research. Additionally, $40$ students used both Quantitative Analysis and Qualitative Interviews, $30$ used both Qualitative Interviews and Archival Research, and $35$ used both Quantitative Analysis and Archival Research. How many students used all three research methodologies?

15

25

40

90

105

Show answer & explanation

Answer: $15$

Answer

15 students used all three research methodologies.

According to the Principle of Inclusion-Exclusion for three sets,

\text{Total} = |Q| + |I| + |A| - (|Q \cap I| + |I \cap A| + |Q \cap A|) + |Q \cap I \cap A| + \text{Neither}

. Substituting the given values gives

240 = 130 + 110 + 90 - (40 + 30 + 35) + x + 0

, which simplifies to

240 = 225 + x

, yielding

x = 15

. Thus, the option stating 15 is correct.

Step-by-Step Solution

1

State the three-set inclusion-exclusion principle formula for total population.

Total =

|Q| + |I| + |A| - (|Q \cap I| + |I \cap A| + |Q \cap A|) + |Q \cap I \cap A| + \text{Neither}

To set up an algebraic equation relating the total number of students to their set intersections.

2

Substitute the known numerical values into the formula.

240 = 130 + 110 + 90 - (40 + 30 + 35) + |Q \cap I \cap A| + 0

Every student used at least one methodology, so

\text{Neither} = 0

.

3

Simplify the equation and solve for the target variable

|Q \cap I \cap A|

.

240 = 330 - 105 + |Q \cap I \cap A| \implies 240 = 225 + |Q \cap I \cap A| \implies |Q \cap I \cap A| = 15

Subtracting 225 from 240 yields the number of students who used all three methodologies.

Key Concept

Three-Set Inclusion-Exclusion Principle

Estimated Time:1m 30s

Question 18Question →

An antique dealer purchased three items—an armchair, a desk, and a cabinet—for a total combined cost of $\$4,000$ . The cost of the desk was $20\%$ greater than the cost of the armchair, and the cost of the cabinet was $50\%$ greater than the cost of the desk. To determine the list price for each item, the dealer marked up the cost of the armchair by $40\%$ , the cost of the desk by $50\%$ , and the cost of the cabinet by $30\%$ . During a clearance event, the dealer sold the armchair at a $15\%$ discount off its list price, the desk at a $20\%$ discount off its list price, and the cabinet at its full list price with no discount. What was the dealer's overall profit as a percentage of the total combined purchase cost of the three items?

23.00\%

24.25\%

28.75\%

19.52\%

38.00\%

Show answer & explanation

Answer: $24.25\%$

Answer

The dealer's overall profit was

24.25\%

of the total combined purchase cost.

The correct response of

24.25\%

accurately calculates the individual item costs, applies sequential markup and discount multipliers (

1.40 \times 0.85 = 1.19

for armchair,

1.50 \times 0.80 = 1.20

for desk,

1.30

for cabinet), sums the final sales revenue to

\$4,970

, and evaluates net profit (

\$970

) over total cost (

\$4,000

).

Step-by-Step Solution

1

Determine the individual cost of each item

Armchair cost =

\$1,000

, Desk cost =

\$1,200

, Cabinet cost =

\$1,800

Let

A

be the cost of the armchair. The desk cost is

1.2A

, and the cabinet cost is

1.5 \times 1.2A = 1.8A

. The total cost is

A + 1.2A + 1.8A = 4.0A = 4,000 \implies A = 1,000

.

2

Calculate the list price and selling price for each item

Armchair selling price =

\$1,190

; Desk selling price =

\$1,440

; Cabinet selling price =

\$2,340

Armchair list price =

\$1,000 \times 1.40 = \$1,400

; selling price =

\$1,400 \times 0.85 = \$1,190

. Desk list price =

\$1,200 \times 1.50 = \$1,800

; selling price =

\$1,800 \times 0.80 = \$1,440

. Cabinet selling price = list price =

\$1,800 \times 1.30 = \$2,340

.

3

Find total revenue and total profit

Total revenue =

\$4,970

, Total profit =

\$970

Total revenue =

\$1,190 + \$1,440 + \$2,340 = \$4,970

. Total profit =

\$4,970 - \$4,000 = \$970

.

4

Compute overall profit percentage on total cost

24.25\%

Overall profit percentage =

(\frac{970}{4000}) \times 100\% = 24.25\%

.

Key Concept

Successive Percentage Changes and Weighted Profit Base Calculations

Estimated Time:2m 0s

Question 19Question →

An automotive testing center evaluated $350$ electric vehicle models for three advanced driver-assistance features: Lane Keeping Assist ( $L$ ), Automatic Emergency Braking ( $A$ ), and Blind Spot Detection ( $B$ ). The evaluation showed that $180$ models had feature $L$ , $150$ had feature $A$ , and $160$ had feature $B$ . Additionally, $65$ models had both $L$ and $A$ , $50$ had both $A$ and $B$ , $60$ had both $L$ and $B$ , and $25$ models had all three features. How many of the evaluated electric vehicle models had none of these three features?

Show answer & explanation

Answer: 10

Answer

The number of electric vehicle models that had none of the three features is 10.

Using the formula for three overlapping sets,

|L \cup A \cup B| = |L| + |A| + |B| - (|L \cap A| + |A \cap B| + |L \cap B|) + |L \cap A \cap B|

, we get

180 + 150 + 160 - (65 + 50 + 60) + 25 = 340

vehicles with at least one feature. Subtracting this from the total evaluated group of

350

yields

350 - 340 = 10

vehicles with none of the features.

Step-by-Step Solution

1

Use the Principle of Inclusion-Exclusion for three overlapping sets to find the total number of models with at least one feature.

|L \cup A \cup B| = 180 + 150 + 160 - (65 + 50 + 60) + 25 = 340

models.

Adding individual set counts overcounts pairwise overlap regions twice and the central triple overlap three times. Subtracting pairwise intersections corrects for double counting, and adding back the triple intersection accounts for its over-subtraction.

2

Subtract the number of models having at least one feature from the total number of tested models.

\text{None} = 350 - 340 = 10

models.

The entire group consists of models with at least one feature plus models with none of the features.

Key Concept

Three-Set Inclusion-Exclusion Principle and Venn Diagram Region Partitioning

Question 20Question →

A boutique jewelry designer crafts customized gold necklaces. The total cost to produce each necklace is the sum of a fixed material cost of $\$800$ and a variable design cost. To determine the list price, the designer marks up the total production cost by $60\%$ . During a seasonal promotion, the designer offers a $25\%$ discount off the list price. If the designer earns a net profit of $\$360$ on each necklace sold during the promotion, what is the variable design cost, in dollars, per necklace?

Show answer & explanation

Answer: 1000

Answer

The variable design cost per necklace is $1,000.

The total production cost per necklace is

C = 800 + L

. Marking up by

60\%

yields a list price of

1.60C

. A

25\%

discount reduces the price to

0.75 \times 1.60C = 1.20C

. The net profit per necklace is

1.20C - C = 0.20C

. Given that the net profit is

\$360

, we set

0.20C = 360

, which gives

C = 1,800

. Subtracting the fixed material cost of

\$800

yields the variable design cost of

\$1,000

.

Step-by-Step Solution

1

Define total production cost in terms of variable cost

C = 800 + L

, where

C

is total production cost and

L

is variable design cost

Total cost is the sum of fixed material cost and variable design cost.

2

Determine the selling price after markup and discount

Selling Price

S = 0.75 \times (1.60 C) = 1.20 C

A

60\%

markup increases total cost by a factor of

1.60

, and a

25\%

discount reduces that marked price to

75\%

(

0.75

).

3

Formulate the profit equation and solve for total production cost

C

Profit

= S - C = 1.20 C - C = 0.20 C = 360 \implies C = 1,800

Net profit is the selling price minus the total production cost.

4

Calculate variable design cost

L

L = 1,800 - 800 = 1,000

Subtract fixed material cost from total production cost.

Key Concept

Profit, Loss, and Markup with Successive Percentage Adjustments

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