Word Problems and Applied Math Practice Questions — GMAT — Page 2

Question 21Question →

A laboratory technician has $40$ liters of a saline solution that is $15\%$ salt by volume. How many liters of pure water must be evaporated from the solution so that the remaining solution is $25\%$ salt by volume?

Show answer & explanation

Answer: 16

Answer

The correct answer is

16

liters.

Because evaporation removes only pure water, the quantity of salt remains constant at

6

liters (

40 \times 0.15

). For

6

liters of salt to constitute

25\%

of the final mixture volume

V

, we set up

0.25V = 6

, which yields

V = 24

liters. Subtracting the final volume of

24

liters from the initial volume of

40

liters gives

16

liters of evaporated water.

Step-by-Step Solution

1

Calculate the volume of pure salt in the initial solution.

The initial salt volume is

40 \times 0.15 = 6

liters.

The solute amount is determined by multiplying total volume by concentration.

2

Determine the required total solution volume after evaporation to achieve a 25% concentration.

The required total final volume is

\frac{6}{0.25} = 24

liters.

Since evaporation removes only water, the volume of salt remains 6 liters, which must equal 25% of the new total volume.

3

Calculate the amount of water evaporated by taking the difference between the initial and final total volumes.

The volume of water evaporated is

40 - 24 = 16

liters.

The decrease in total solution volume equals the volume of pure water removed by evaporation.

Key Concept

Concentration change via evaporation (solute mass conservation)

Question 22Question →

A venture capital firm allocated its initial investment fund between two startups, Enterprise X and Enterprise Y. Over a two-year period, the investment in Enterprise X earned simple interest at an annual rate of $15\%$ . During the same period, the value of the investment in Enterprise Y increased by $25\%$ in the first year and then decreased by $20\%$ of its new value in the second year. If the total combined value of the two investments at the end of the two years was $12\%$ greater than the initial investment fund, what percentage of the initial fund was invested in Enterprise X?

28\%

40\%

50\%

60\%

80\%

Show answer & explanation

Answer: $40\%$

Answer

40\%

of the initial fund was invested in Enterprise X.

The correct answer is

40\%

. Enterprise X earns simple interest of

15\%

per year for 2 years, yielding a growth factor of

1 + 2(0.15) = 1.30

. Enterprise Y increases by

25\%

and then decreases by

20\%

, yielding a net growth factor of

1.25 \times 0.80 = 1.00

. Setting the overall weighted multiplier equal to

1.12

gives

1.30x + 1.00(1-x) = 1.12

, which yields

x = 0.40

, or

40\%

.

Step-by-Step Solution

1

Define variables and determine the growth factor for Enterprise X.

Let

x

be the fraction of the fund invested in Enterprise X, so

(1-x)

is the fraction invested in Enterprise Y. Over 2 years at an annual simple interest rate of

15\%

, the growth multiplier for Enterprise X is

1 + 2 \times 0.15 = 1.30

.

Simple interest accumulates linearly over time.

2

Calculate the net successive percentage multiplier for Enterprise Y.

A

25\%

increase followed by a

20\%

decrease gives a net multiplier of

(1 + 0.25) \times (1 - 0.20) = 1.25 \times 0.80 = 1.00

.

Successive percentage changes must be multiplied sequentially based on the intermediate values.

3

Set up the weighted average equation for the total investment.

1.30x + 1.00(1-x) = 1.12

.

The total combined value at the end of 2 years is

12\%

greater than the initial fund, corresponding to a total multiplier of

1.12

.

4

Solve for

x

.

1.30x + 1.00 - 1.00x = 1.12 \Rightarrow 0.30x = 0.12 \Rightarrow x = 0.40 = 40\%

.

Isolating

x

gives the proportion of the fund allocated to Enterprise X.

Key Concept

Combining simple interest and successive percentage change in weighted portfolio problems.

Question 23Question →

An educational foundation set an annual fundraising goal for a given year. During the first quarter, the foundation raised $30\%$ of its annual goal. In the second quarter, the amount raised was $25\%$ greater than the amount raised in the first quarter. In the third quarter, the amount raised was $40\%$ less than the total amount raised in the first two quarters combined. If the total amount raised across the first three quarters combined was $\$129,600$ , what was the foundation's annual fundraising goal, in dollars?

Show answer & explanation

Answer: 120000

Answer

The foundation's annual fundraising goal was $120,000.

To determine the annual goal

G

, calculate the raised amount in each quarter relative to

G

. The first quarter yields

Q_1 = 0.30G

. The second quarter amount is

25\%

greater than

Q_1

, which equals

1.25 \times 0.30G = 0.375G

. The sum of the first two quarters is

0.30G + 0.375G = 0.675G

. The third quarter amount is

40\%

less than this combined sum, giving

0.60 \times 0.675G = 0.405G

. Summing the three quarters gives a total of

0.30G + 0.375G + 0.405G = 1.08G

. Equating

1.08G = 129,600

and dividing by

1.08

gives

G = 120,000

.

Step-by-Step Solution

1

Define the variable for the unknown quantity

Let

G

represent the foundation's annual fundraising goal in dollars.

Establishing a variable allows all quarterly contributions to be expressed as a linear algebraic function of the goal.

2

Express the amount raised in the first quarter in terms of

G

Q_1 = 0.30G

The foundation raised

30\%

of its annual goal during the first quarter.

3

Express the amount raised in the second quarter in terms of

G

Q_2 = 1.25 \times 0.30G = 0.375G

An increase of

25\%

over

Q_1

means

Q_2 = (1 + 0.25) Q_1 = 1.25 \times 0.30G

.

4

Calculate the combined total raised in the first two quarters

Q_1 + Q_2 = 0.30G + 0.375G = 0.675G

This combined sum serves as the base value for calculating the third quarter's contribution.

5

Express the amount raised in the third quarter in terms of

G

Q_3 = (1 - 0.40) \times 0.675G = 0.60 \times 0.675G = 0.405G

The third quarter raised

40\%

less than the combined amount of the first two quarters, meaning it equaled

60\%

of

(Q_1 + Q_2)

.

6

Sum the contributions of all three quarters and solve for

G

Total

= 0.30G + 0.375G + 0.405G = 1.08G = 129,600 \implies G = 120,000

Setting the sum equal to the total dollar amount raised (

\$129,600

) yields

1.08G = 129,600

, so

G = \frac{129,600}{1.08} = 120,000

.

Key Concept

Successive Percent Change and Base Identification

Question 24Question →

A store owner purchased a shipment of coats. She marked up the cost price of each coat by $40\%$ to set the regular selling price. During a clearance sale, she offered a discount of $15\%$ off the regular selling price on all remaining coats. If a coat was sold at the clearance price for $\$238$ , what was the cost price of the coat?

\$168.00

\$170.00

\$190.40

\$200.00

\$212.50

Show answer & explanation

Answer: $\$200.00$

Answer

The cost price of the coat was

\$200.00

.

The correct answer is

\$200.00

. The regular price is

140\%

of the cost price (

1.40C

). A

15\%

discount on the regular price means the clearance price is

85\%

of the regular price, which translates to

0.85 \times 1.40C = 1.19C

. Setting

1.19C = \$238

yields

C = \$200.00

.

Step-by-Step Solution

1

Define variables and express the regular selling price in terms of the cost price.

Let

C

be the cost price of a coat. The regular selling price is

R = C + 0.40C = 1.40C

.

Markup is calculated as a percentage of the cost price.

2

Express the clearance price in terms of the regular selling price and cost price.

The clearance price is

R \times (1 - 0.15) = 1.40C \times 0.85 = 1.19C

.

The discount of

15\%

is applied to the regular selling price, resulting in

85\%

of the regular selling price.

3

Solve for the cost price

C

using the given clearance price.

1.19C = 238 \implies C = \frac{238}{1.19} = 200

.

Equating the algebraic expression for the clearance price to the given numerical value gives the original cost price.

Key Concept

Successive Percentage Changes in Profit and Loss

Question 25Question →

An investor allocated a total initial capital of $\$60,000$ between two funds, Account X and Account Y. During the first year, Account X increased in value by $40\%$ , while Account Y increased in value by $20\%$ . During the second year, Account X decreased in value by $15\%$ relative to its value at the end of the first year, while Account Y increased in value by $10\%$ relative to its value at the end of the first year. If the total combined value of both accounts at the end of the second year was $\$74,000$ , what was the initial amount invested in Account X?

\$20,000

\$30,000

\$40,000

\$45,000

\$50,000

Show answer & explanation

Answer: $\$40,000$

Answer

The initial amount invested in Account X was

\$40,000

.

The correct answer is

\$40,000

. Account X increases by

40\%

in Year 1 to

1.40X

and decreases by

15\%

in Year 2, yielding

1.40X \times 0.85 = 1.19X

. Account Y increases by

20\%

in Year 1 and

10\%

in Year 2, yielding

(60,000 - X) \times 1.20 \times 1.10 = 1.32(60,000 - X)

. Setting

1.19X + 1.32(60,000 - X) = 74,000

gives

-0.13X = -5,200

, so

X = 40,000

.

Step-by-Step Solution

1

Calculate the net multiplier for Account X over the two-year period.

Multiplier for Account X =

(1 + 0.40) \times (1 - 0.15) = 1.40 \times 0.85 = 1.19

Successive percentage changes must be applied sequentially to updated base values.

2

Calculate the net multiplier for Account Y over the two-year period.

Multiplier for Account Y =

(1 + 0.20) \times (1 + 0.10) = 1.20 \times 1.10 = 1.32

Successive percentage increases compound on the value at the end of the first year.

3

Set up an algebraic equation for the total combined final value.

Let

X

be the initial capital in Account X. Then

60,000 - X

is the initial capital in Account Y.

1.19X + 1.32(60,000 - X) = 74,000

The sum of the final values of both accounts equals the total combined value of

\$74,000

.

4

Solve the linear equation for

X

.

1.19X + 79,200 - 1.32X = 74,000 \implies -0.13X = -5,200 \implies X = 40,000

Dividing

-5,200

by

-0.13

yields the initial amount invested in Account X.

Key Concept

Successive Percent Change and Base Value Determination

Question 26Question →

A high-speed passenger train travels from Station X to Station Y at a constant speed of $60$ miles per hour. On the return trip along the exact same track, due to maintenance, the train travels at a constant speed of $40$ miles per hour. What is the average speed of the train, in miles per hour, for the entire round trip?

45

48

50

52

54

Show answer & explanation

Answer: $48$

Answer

48

miles per hour

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

d

, the outbound travel time is

\frac{d}{60}

and the return time is

\frac{d}{40}

. The total distance is

2d

and total time is

\frac{5d}{120} = \frac{d}{24}

. Dividing total distance

2d

by total time

\frac{d}{24}

gives

48

miles per hour.

Step-by-Step Solution

1

Define variables for distance and time for each leg of the journey.

Let the distance between Station X and Station Y be

d

miles. Outbound time

t_1 = \frac{d}{60}

hours, and return time

t_2 = \frac{d}{40}

hours.

Average speed is defined as total distance divided by total time, not the simple average of speeds.

2

Calculate total distance and total time.

Total distance

= d + d = 2d

. Total time

= t_1 + t_2 = \frac{d}{60} + \frac{d}{40} = \frac{2d + 3d}{120} = \frac{5d}{120} = \frac{d}{24}

hours.

Summing individual distances and times provides the components needed for the average speed formula.

3

Divide total distance by total time to find the average speed.

Average speed

= \frac{2d}{\frac{d}{24}} = 2d \times \frac{24}{d} = 48

miles per hour.

Executing the division yields the overall rate for the trip.

Key Concept

Average Speed for Multi-Leg Trips

Question 27Question →

An executive drives from Office A to Office B, a distance of $120$ miles, at a constant speed of $40$ miles per hour. After spending exactly $1$ hour at Office B for a meeting, she returns to Office A along the exact same route at a constant speed of $60$ miles per hour. What is her average speed, in miles per hour, for the entire round trip including the time spent at Office B?

35

40

45

48

50

Show answer & explanation

Answer: $40$

Answer

The average speed for the entire round trip, including the meeting time, is

40

miles per hour.

To find the average speed for the entire trip, divide total distance covered by total time elapsed. The total distance is

120 + 120 = 240

miles. The travel time outbound is

120 / 40 = 3

hours, and the travel time return is

120 / 60 = 2

hours. Including the

1

-hour meeting at Office B, the total elapsed time is

3 + 1 + 2 = 6

hours. Thus, the overall average speed is

240 / 6 = 40

miles per hour.

Step-by-Step Solution

1

Calculate the travel time for the outbound leg from Office A to Office B.

\text{Time}_{\text{outbound}} = \frac{120\text{ miles}}{40\text{ mph}} = 3\text{ hours}

.

Time equals distance divided by speed.

2

Calculate the travel time for the return leg from Office B to Office A.

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

.

Time equals distance divided by speed.

3

Determine the total distance and total elapsed time for the entire trip.

\text{Total Distance} = 120 + 120 = 240\text{ miles}

;

\text{Total Time} = 3\text{ hours (outbound)} + 1\text{ hour (meeting)} + 2\text{ hours (return)} = 6\text{ hours}

.

Average speed requires total distance divided by total time elapsed, which must include non-moving meeting time.

4

Calculate the overall average speed.

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

.

Divide total distance by total time.

Key Concept

Average Speed for Multi-Leg Journeys

Estimated Time:2m 0s

Question 28Question →

A bakery bakes vanilla cakes and chocolate cakes in a ratio of $3:4$ . If the bakery bakes a total of $35$ cakes in one morning, how many chocolate cakes were baked?

15

20

21

28

45

Show answer & explanation

Answer: $20$

Answer

20 chocolate cakes were baked.

The ratio of vanilla to chocolate cakes is

3:4

, meaning there are

3 + 4 = 7

total equal parts. The fraction of total cakes that are chocolate is

\frac{4}{7}

. Multiplying this fraction by the total of

35

cakes yields

\frac{4}{7} \times 35 = 20

chocolate cakes.

Step-by-Step Solution

1

Determine the total number of ratio parts

Sum of ratio terms:

3 + 4 = 7

total parts.

The given ratio of vanilla to chocolate cakes is

3:4

, which means out of every

7

cakes,

3

are vanilla and

4

are chocolate.

2

Calculate the value of one ratio part

Value per part:

\frac{35}{7} = 5

cakes per part.

Dividing the total count of

35

cakes by the total number of ratio parts gives the multiplier for each part.

3

Compute the total number of chocolate cakes

Chocolate cakes:

4 \times 5 = 20

.

Multiply the chocolate ratio term (

4

) by the value of one part (

5

).

Key Concept

Part-to-Whole Ratio Scaling

Question 29Question →

A community library maintains fiction and non-fiction books in a ratio of $5 : 2$ . If the library has $350$ fiction books, how many non-fiction books are in the library?

100

140

250

700

875

Show answer & explanation

Answer: 140

Answer

There are 140 non-fiction books in the library.

The ratio of fiction to non-fiction books is

5 : 2

. Since there are

350

fiction books, each unit of the ratio represents

\frac{350}{5} = 70

books. Multiplying the non-fiction ratio part (

2

) by

70

gives

2 \times 70 = 140

non-fiction books.

Step-by-Step Solution

1

Set up the ratio of fiction books to non-fiction books.

\frac{\text{Fiction}}{\text{Non-fiction}} = \frac{5}{2}

The problem states that the ratio of fiction to non-fiction is 5 to 2.

2

Substitute the known number of fiction books (350) and let x represent the number of non-fiction books.

\frac{350}{x} = \frac{5}{2}

This establishes a direct proportion between the actual counts and the ratio values.

3

Cross-multiply and solve for x.

5x = 350 \times 2 \implies 5x = 700 \implies x = 140

Solving for x yields the total number of non-fiction books.

Key Concept

Solving ratio word problems using direct proportion

Estimated Time:45s

Question 30Question →

An architectural restoration project allocates its workforce across three specialized teams: Masonry, Carpentry, and Glasswork. Initially, the ratio of Masonry workers to Carpentry workers is $5 : 6$ , and the ratio of Carpentry workers to Glasswork workers is $4 : 5$ .

During phase two of the project:
• The number of Masonry workers is increased by $50\%$ .
• The number of Carpentry workers is decreased by $25\%$ .
• The number of Glasswork workers is increased by $X\%$ .

If the new ratio of Masonry workers to the total number of workers across all three teams is $1 : 3$ , what is the value of $X$ ?

Show answer & explanation

Answer: 40

Answer

The value of

X

is 40.

To find

X

, first combine the initial ratios by equating the common Carpentry term: Masonry : Carpentry = 10 : 12 and Carpentry : Glasswork = 12 : 15, yielding a joint ratio of 10 : 12 : 15. Representing the initial teams as

10k

,

12k

, and

15k

, the updated Masonry team is

10k \times 1.5 = 15k

and Carpentry is

12k \times 0.75 = 9k

. Using the part-to-total ratio

\frac{\text{New Masonry}}{\text{New Total}} = \frac{1}{3}

, we set up

\frac{15k}{15k + 9k + \text{New Glasswork}} = \frac{1}{3}

, which yields

\text{New Glasswork} = 21k

. The percentage increase from

15k

to

21k

is

\frac{21k - 15k}{15k} \times 100\% = 40\%

. Thus,

X = 40

.

Step-by-Step Solution

1

Unify the two given ratios into a single three-part ratio.

The ratio Masonry : Carpentry : Glasswork is

10 : 12 : 15

.

Carpentry is common to both given ratios (

5 : 6

and

4 : 5

). The least common multiple of 6 and 4 is 12. Scaling

5 : 6

by 2 yields

10 : 12

, and scaling

4 : 5

by 3 yields

12 : 15

.

2

Define initial workforce quantities using a multiplier

k

.

Initial Masonry =

10k

, Carpentry =

12k

, Glasswork =

15k

.

Using an algebraic variable preserves the proportional relationships across all teams.

3

Calculate the updated worker counts for Masonry and Carpentry.

New Masonry =

15k

, New Carpentry =

9k

.

Masonry increases by

50\%

(

10k \times 1.50 = 15k

). Carpentry decreases by

25\%

(

12k \times 0.75 = 9k

).

4

Set up the part-to-total ratio equation to solve for the new Glasswork team size.

New Glasswork worker count =

21k

.

The total workforce is

15k + 9k + \text{New Glasswork} = 24k + \text{New Glasswork}

. Given

\frac{\text{New Masonry}}{\text{New Total}} = \frac{1}{3}

, we have

\frac{15k}{24k + \text{New Glasswork}} = \frac{1}{3} \implies 45k = 24k + \text{New Glasswork} \implies \text{New Glasswork} = 21k

.

5

Compute the percentage increase

X

for the Glasswork team.

X = 40

.

The initial Glasswork count was

15k

and the new count is

21k

. The percentage increase is

\frac{21k - 15k}{15k} \times 100 = \frac{6}{15} \times 100 = 40\%

.

Key Concept

Combining compound ratios and solving multi-step part-to-whole proportion equations.

Estimated Time:2m 30s

Question 31Question →

A container holds $20$ liters of a solution that is $15\%$ acid by volume. If $5$ liters of pure water are added to the solution, what is the percentage concentration of acid in the resulting mixture?

Show answer & explanation

Answer: 12

Answer

The concentration of acid in the resulting mixture is 12%.

The correct concentration is found by taking the invariant amount of pure acid (

3

liters) and dividing it by the total updated volume of the liquid (

25

liters), which yields

\frac{3}{25} = 0.12

, or

12\%

.

Step-by-Step Solution

1

Determine the amount of solute (pure acid) in the original mixture.

Amount of acid =

0.15 \times 20 = 3

liters.

The solution initially consists of

15\%

acid out of a total volume of

20

liters.

2

Determine the total volume of the new solution after dilution.

New total volume =

20 + 5 = 25

liters.

Adding

5

liters of pure water increases the total liquid volume without adding any additional acid.

3

Compute the new percentage concentration.

New concentration =

\frac{3}{25} \times 100\% = 12\%

.

The concentration is the ratio of pure solute to total solution volume expressed as a percentage.

Key Concept

Dilution of a mixture by adding pure solvent

Question 32Question →

A jeweler melts two gold alloys together. Alloy X is $80\%$ gold by weight, and Alloy Y is $50\%$ gold by weight. How many grams of Alloy X must be mixed with Alloy Y to create a $60$ -gram alloy that is $70\%$ gold by weight?

20

24

30

40

50

Show answer & explanation

Answer: $40$

Answer

The mass of Alloy X required is

40

grams.

By writing the gold balance equation

0.80x + 0.50(60 - x) = 0.70(60)

, we solve to find

0.30x = 12

, which gives

x = 40

grams of Alloy X.

Step-by-Step Solution

1

Define variables for the mass of each alloy.

Let

x

be the mass of Alloy X in grams. Since the total mixture is

60

grams, the mass of Alloy Y is

(60 - x)

grams.

The sum of the component masses equals the total mass of the resulting mixture.

2

Set up the equation for total gold content.

0.80x + 0.50(60 - x) = 0.70(60)

The amount of pure gold from Alloy X plus the amount of pure gold from Alloy Y must equal the amount of pure gold in the final

60

-gram mixture.

3

Solve for

x

.

0.80x + 30 - 0.50x = 42 \Rightarrow 0.30x = 12 \Rightarrow x = 40

Simplifying the algebraic equation yields the exact amount of Alloy X required.

Key Concept

Weighted Average in Mixture Problems

Estimated Time:1m 0s

Question 33Question →

An archival digitization laboratory uses two high-resolution scanner units, Unit $A$ and Unit $B$ , to process historic document collections. Working alone at its constant rate, Unit $A$ can digitize a standard collection in $24$ hours. Working alone at its constant rate, Unit $B$ can digitize the same collection in $40$ hours. Unit $A$ begins digitizing a standard collection alone. After $8$ hours of operation, Unit $B$ is turned on, and both units work together at their respective constant rates until the entire collection is digitized. How many total hours does it take to digitize the collection from start to finish?

Show answer & explanation

Answer: 18

Answer

The total time required to digitize the collection from start to finish is 18 hours.

Unit A completes

\frac{1}{3}

of the total job during its 8 solo hours. The remaining

\frac{2}{3}

of the job is completed by Units A and B working together at a combined rate of

\frac{1}{15}

per hour, taking 10 hours. Adding the 8 solo hours to the 10 combined hours yields a total time of 18 hours.

Step-by-Step Solution

1

Determine the individual work rates per hour for each unit.

Unit A's rate is

\frac{1}{24}

of the collection per hour, and Unit B's rate is

\frac{1}{40}

of the collection per hour.

Work rate is defined as the fraction of the job completed per unit of time (

R = \frac{1}{T}

).

2

Calculate the fraction of the work completed by Unit A alone during the first 8 hours.

Work completed

= 8 \times \frac{1}{24} = \frac{8}{24} = \frac{1}{3}

of the collection.

Work done equals rate multiplied by time (

W = R \times t

).

3

Calculate the remaining fraction of the collection to be digitized.

Remaining work

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

of the collection.

The total job represents 1 whole unit.

4

Calculate the combined work rate when both units operate together.

Combined rate

= \frac{1}{24} + \frac{1}{40} = \frac{5}{120} + \frac{3}{120} = \frac{8}{120} = \frac{1}{15}

of the collection per hour.

When working simultaneously, individual rates add together.

5

Find the time required for both units working together to complete the remaining work.

Combined time

= \frac{2/3}{1/15} = \frac{2}{3} \times 15 = 10

hours.

Time equals remaining work divided by the combined rate (

t = \frac{W_{\text{remaining}}}{R_{\text{combined}}}

).

6

Calculate the total time required from start to finish.

Total time

= 8 \text{ hours} + 10 \text{ hours} = 18

hours.

Total time is the sum of the time Unit A worked alone and the time both units worked together.

Key Concept

Work Rate and Combined Work

Question 34Question →

A commercial bakery uses two automated dough mixing machines, Machine $A$ and Machine $B$ , to process large batches of dough. Working alone at its constant rate, Machine $A$ can process a full batch of dough in $12$ hours. Machine $B$ , working alone at its constant rate, can process a full batch of dough in $6$ hours. Machine $A$ starts processing a batch alone. After working for $3$ hours, Machine $B$ joins Machine $A$ , and both machines work together at their respective constant rates to finish the batch. What is the total number of hours required to process the entire batch of dough from start to finish?

Show answer & explanation

Answer: 6

Answer

The total number of hours required to process the entire batch of dough from start to finish is 6 hours.

Machine A processes 1/12 of the batch per hour. In the first 3 hours, Machine A finishes 3/12 = 1/4 of the batch, leaving 3/4 of the batch uncompleted. When Machine B joins, the combined processing rate is 1/12 + 1/6 = 1/4 batch per hour. Dividing the remaining 3/4 batch by the combined rate of 1/4 batch per hour yields 3 hours. Adding the initial 3 hours worked by Machine A alone gives a total elapsed time of 6 hours.

Step-by-Step Solution

1

Determine the individual work rates for Machine

A

and Machine

B

.

Machine

A

's rate is

\frac{1}{12}

batch/hour, and Machine

B

's rate is

\frac{1}{6}

batch/hour.

Work rate is the fraction of the job completed per hour (1 divided by total hours required alone).

2

Calculate the portion of the batch completed by Machine

A

operating alone during the first 3 hours.

Machine

A

completes

3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4}

of the batch.

Work completed equals work rate multiplied by time worked.

3

Determine the remaining fraction of the batch to be completed.

Remaining work is

1 - \frac{1}{4} = \frac{3}{4}

of the batch.

The entire job corresponds to 1 whole batch.

4

Calculate the combined work rate of Machine

A

and Machine

B

.

Combined rate

= \frac{1}{12} + \frac{1}{6} = \frac{1}{12} + \frac{2}{12} = \frac{3}{12} = \frac{1}{4}

batch per hour.

When machines work together, their individual rates add up.

5

Calculate the time required for both machines working together to complete the remaining work.

Time together

= \frac{\text{Remaining Work}}{\text{Combined Rate}} = \frac{3/4}{1/4} = 3

hours.

Time equals work divided by rate.

6

Calculate the total time required from start to finish.

Total time

= 3 \text{ hours (Machine A alone)} + 3 \text{ hours (both together)} = 6

hours.

The total time is the sum of time spent in each phase of the process.

Key Concept

Work Rate and Combined Work

Question 35Question →

An architectural firm initially allocates its annual project hours among three divisions—Residential, Commercial, and Urban Planning—in the ratio of $7 : 5 : 3$ , respectively. Mid-year, to meet new project demands, $300$ hours from Residential and $100$ hours from Commercial are transferred to Urban Planning. Following this transfer, the ratio of Commercial hours to Urban Planning hours becomes $1 : 1$ . What was the total number of project hours initially allocated across all three divisions?

Show answer & explanation

Answer: 3750

Answer

The total number of project hours initially allocated across all three divisions was 3,750 hours.

Representing the initial hours as

7x

,

5x

, and

3x

gives a total of

15x

hours. After the transfers, Commercial hours equal

5x - 100

and Urban Planning hours equal

3x + 300 + 100 = 3x + 400

. Setting these equal gives

5x - 100 = 3x + 400

, which simplifies to

2x = 500

, or

x = 250

. The initial total hours were

15 \times 250 = 3,750

.

Step-by-Step Solution

1

Define initial quantities using a multiplier variable

Let

x

represent the multiplier. Initial hours are Residential =

7x

, Commercial =

5x

, and Urban Planning =

3x

. Total initial hours =

15x

.

Ratios define the proportional relationship among the three divisions.

2

Determine updated hours after the transfers

Commercial hours become

5x - 100

. Urban Planning receives

300

hours from Residential and

100

hours from Commercial, becoming

3x + 300 + 100 = 3x + 400

.

Quantities added to Urban Planning must equal the sum of hours subtracted from the other two divisions.

3

Set up and solve the equation for the new ratio

Since Commercial and Urban Planning hours are now in a

1 : 1

ratio,

5x - 100 = 3x + 400 \implies 2x = 500 \implies x = 250

.

A

1 : 1

ratio means both quantities are equal.

4

Calculate the initial total project hours

Total initial hours =

15 \times 250 = 3,750

.

Substitute

x = 250

into the expression for total initial hours (

15x

).

Key Concept

Solving multi-part ratio word problems involving internal transfers by setting up algebraic equations with a single unknown multiplier.

Question 36Question →

The price of a jacket was increased by $20\%$ . During a clearance sale, the new price was reduced by $10\%$ . What was the net percentage increase in the price of the jacket from its original value?

18\%

10\%

8\%

12\%

30\%

Show answer & explanation

Answer: $8\%$

Answer

An

8\%

net increase

Applying successive percentage changes requires multiplying the growth/decay factors. A

20\%

increase corresponds to a multiplier of

1.20

, and a

10\%

decrease corresponds to a multiplier of

0.90

. The net multiplier is

1.20 \times 0.90 = 1.08

, which represents a net increase of

8\%

over the original price.

Step-by-Step Solution

1

Assume a convenient original price to track percentage changes easily.

Let the original price of the jacket be

\$100

.

Choosing

\$100

as a base makes calculating percentages straightforward.

2

Calculate the price after the

20\%

increase.

\$\$100 + 0.20 \times \$100 = \$120

.

The price increases by

20\%

of the original price.

3

Calculate the price after the

10\%

clearance discount.

\$\$120 - 0.10 \times \$120 = \$108

.

The discount of

10\%

applies to the intermediate price of

\$120

, not the original

\$100

.

4

Determine the net percentage change from the original price.

\frac{\$108 - \$100}{\$100} \times 100\% = 8\%

.

Comparing the final price of

\$108

to the initial price of

\$100

gives a net increase of

8\%

.

Key Concept

Successive Percent Change

Estimated Time:45s

Question 37Question →

An investment fund initially holds shares of Technology, Healthcare, and Energy stocks in the ratio $5 : 3 : 2$ , respectively. After the fund sells $30$ shares of Technology stock and purchases $30$ shares of Energy stock, the ratio of Technology, Healthcare, and Energy shares becomes $4 : 3 : 3$ . What was the total number of shares in the investment fund initially?

Show answer & explanation

Answer: 300

Answer

The initial total number of shares in the investment fund was 300.

The initial shares can be represented as

5x

,

3x

, and

2x

for a total of

10x

shares. After selling

30

Technology shares and buying

30

Energy shares, the Healthcare shares remain unchanged at

3x

. Since Healthcare represents

3

parts in both the initial and final ratios (

5:3:2

and

4:3:3

), the value of one ratio part remains constant at

x

. Therefore, the decrease in Technology shares from

5x

to

4x

corresponds directly to

1x = 30

shares. Multiplying

x = 30

by the initial total of

10

parts gives

300

shares.

Step-by-Step Solution

1

Define initial quantities using a common ratio multiplier

x

Technology =

5x

, Healthcare =

3x

, Energy =

2x

. Total initial shares =

10x

.

Representing ratio terms algebraically preserves their proportional relationships.

2

Account for changes in share counts

New Technology shares =

5x - 30

, Healthcare shares =

3x

, new Energy shares =

2x + 30

.

30 shares of Technology were sold and 30 shares of Energy were added.

3

Set up an equation comparing the new ratio of Technology to Healthcare

\frac{5x - 30}{3x} = \frac{4}{3}

The new ratio of Technology to Healthcare is given as

4 : 3

.

4

Solve for multiplier

x

3(5x - 30) = 12x \implies 15x - 90 = 12x \implies 3x = 90 \implies x = 30

Cross-multiplying eliminates fractions to solve for the unknown multiplier.

5

Calculate the initial total number of shares

10x = 10 \times 30 = 300

The initial total is the sum of the initial parts (

5x + 3x + 2x = 10x

).

Key Concept

Ratio Modification and Algebraic Scaling

Estimated Time:1m 30s

Question 38Question →

A manufacturer produces custom electric scooters at a manufacturing cost of $250 per unit. To establish the list price, the manufacturer marks up the cost price by 60%. During a promotional campaign, 70% of the manufactured scooters are sold at a 15% discount off the list price, and the remaining 30% are sold at a clearance discount of 35% off the list price. If the manufacturer incurs a fixed promotional overhead expense of$ 14,400, and the overall net profit across all scooters sold (after deducting the fixed overhead expense) is equal to 12% of the total manufacturing cost of all the scooters, how many scooters were manufactured and sold in total?

240

400

720

800

3,600

Show answer & explanation

Answer: 400

Answer

400 scooters

To find the total number of scooters

N

, first determine the list price:

250 \times 1.60 = \$400

. Next, calculate the selling prices for the two tiers:

400 \times 0.85 = \$340

for 70% of units, and

400 \times 0.65 = \$260

for 30% of units. The weighted average revenue per scooter is

0.70(340) + 0.30(260) = \$316

. The profit before fixed overhead per scooter is

316 - 250 = \$66

, yielding a total net profit of

66N - 14,400

. Setting this equal to the target net profit of 12% of total manufacturing costs (

0.12 \times 250N = 30N

) gives

66N - 14,400 = 30N

, which simplifies to

36N = 14,400

, or

N = 400

.

Step-by-Step Solution

1

Calculate the list price of a single scooter.

List price =

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

.

The list price is established by marking up the cost price ($250) by 60%.

2

Calculate the discounted selling prices for both sales categories.

Promotional price (70% of inventory) =

400 \times (1 - 0.15) = \$340

. Clearance price (30% of inventory) =

400 \times (1 - 0.35) = \$260

.

Discounts are applied directly to the list price.

3

Determine the weighted average selling price per unit and profit per unit before overhead.

Weighted average selling price =

0.70(340) + 0.30(260) = 238 + 78 = \$316

. Gross profit per unit =

316 - 250 = \$66

.

Combining the two sales segments gives the effective revenue generated per unit sold.

4

Set up and solve the net profit equation for total units

N

.

Total net profit =

66N - 14,400

. Target profit =

0.12 \times 250N = 30N

. Equating gives

66N - 14,400 = 30N \implies 36N = 14,400 \implies N = 400

.

Net profit after deducting fixed overhead must equal 12% of total manufacturing cost.

Key Concept

Weighted average selling price, markup vs. discount percentage bases, and multi-step net profit equation modeling.

Alternative Method

Work per unit on a percentage basis: List price is 160% of cost. Revenue from 70% of inventory is

0.70 \times 1.60 \times 0.85 = 0.952

of cost. Revenue from 30% of inventory is

0.30 \times 1.60 \times 0.65 = 0.312

of cost. Total revenue is

(0.952 + 0.312) = 1.264

of cost. Gross profit before overhead is

1.264 - 1.000 = 0.264

of cost. Net profit is

0.264 \times \text{Total Cost} - 14,400

. Setting this equal to

0.120 \times \text{Total Cost}

gives

0.144 \times \text{Total Cost} = 14,400

, so

\text{Total Cost} = \$100,000

. Since total cost is

250 \times N = 100,000

, we get

N = 400

.

Estimated Time:2m 0s

Question 39Question →

A corporate training agency surveyed $180$ professionals to determine which of three skill workshops they attended: Leadership ( $L$ ), Negotiation ( $N$ ), and Communication ( $C$ ). The survey revealed that $90$ professionals attended Leadership, $80$ attended Negotiation, and $70$ attended Communication. Additionally, $35$ attended both Leadership and Negotiation, $25$ attended both Negotiation and Communication, $40$ attended both Leadership and Communication, and $20$ attended none of the three workshops. How many of the surveyed professionals attended exactly one workshop?

Show answer & explanation

Answer: 100

Answer

The number of professionals who attended exactly one workshop is 100.

By applying the principle of inclusion-exclusion for three overlapping sets, the number of professionals attending at least one workshop is

180 - 20 = 160

. Setting up the equation

160 = 90 + 80 + 70 - (35 + 25 + 40) + x

reveals that

x = 20

professionals attended all three workshops. Subtracting

20

from each pairwise intersection yields the counts for those attending exactly two workshops (

15

,

5

, and

20

, totaling

40

). Subtracting the exactly-two count (

40

) and the all-three count (

20

) from the total attending at least one (

160

) gives

100

professionals who attended exactly one workshop.

Step-by-Step Solution

1

Determine the number of professionals in the union of all three sets

|L \cup N \cup C| = 180 - 20 = 160

The total number of professionals who attended at least one workshop is equal to the total surveyed minus those who attended none.

2

Apply the principle of inclusion-exclusion for three sets to find the intersection of all three workshops

160 = 90 + 80 + 70 - (35 + 25 + 40) + |L \cap N \cap C| \implies |L \cap N \cap C| = 20

Summing individual set counts double-counts pairwise intersections and triple-counts the triple intersection, so we adjust using the standard three-set formula.

3

Calculate the number of professionals who attended exactly two workshops

(35 - 20) + (25 - 20) + (40 - 20) = 15 + 5 + 20 = 40

Each pairwise intersection includes those who attended all three workshops; subtracting the triple intersection leaves those in exactly two sets.

4

Calculate the number of professionals who attended exactly one workshop

\text{Exactly one} = 160 - 40 - 20 = 100

Subtracting the number of professionals who attended exactly two workshops and all three workshops from the total attending at least one leaves those attending exactly one workshop.

Key Concept

Overlapping Sets (Three-Set Inclusion-Exclusion Principle)

Estimated Time:2m 0s

Question 40Question →

A commercial distributor purchased a shipment of 50 identical medical imaging monitors for a total cost of $20,000. To set the original list price of each monitor, the distributor marked up the cost price per unit by 75 percent. The distributor sold 30 monitors at the original list price. To clear the remaining inventory, the distributor sold the rest of the monitors at a discount of$ d $percent off the original list price. If the distributor realized an overall net profit of 33 percent on the entire shipment of 50 monitors, what is the value of$ d$?

24

30

40

60

84

Show answer & explanation

Answer: 60

Answer

The value of

d

is 60.

The correct response accurately determines that the distributor needs

5,600 from the remaining 20 monitors to achieve an overall 33 percent profit on the

20,000 investment. This requires selling each remaining monitor for

280, which reflects a

420 price reduction from the

700 list price. Expressed as a percentage of the

700 list price, a $420 reduction is equal to 60 percent.

Step-by-Step Solution

1

Calculate the unit cost price and original list price per monitor

Unit cost =

\frac{\$20,000}{50} = \$400

. List price =

\$400 \times (1 + 0.75) = \$700

.

Establishing individual unit costs and list prices is necessary to determine total revenues.

2

Determine total revenue required for an overall net profit of 33 percent

Total Target Revenue =

\$20,000 \times (1 + 0.33) = \$26,600

.

An overall 33 percent profit means total revenue must equal 133 percent of the total shipment cost.

3

Calculate revenue from the first 30 monitors and the remaining required revenue

Revenue from 30 units =

30 \times \$700 = \$21,000

. Remaining revenue needed from 20 units =

\$26,600 - \$21,000 = \$5,600

.

Subtracting the revenue generated by full-price sales isolates the revenue needed from the discounted items.

4

Determine the selling price per discounted monitor and calculate the discount percentage

d

Discounted selling price =

\frac{\$5,600}{20} = \$280

. Discount amount =

\$700 - \$280 = \$420

. Discount percentage

d = \left(\frac{\$420}{\$700}\right) \times 100 = 60

.

The discount percentage is the ratio of the dollar discount to the original list price.

Key Concept

Profit, Loss, and Markup (Successive percentage changes with mixed pricing)

Word Problems and Applied Math

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

Answer

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