Word Problems and Applied Math Practice Questions — GMAT — Page 4

Question 61Question →

A freight logistics warehouse initially stores Standard, Refrigerated, and Oversized cargo containers in the ratio $4 : 3 : 2$ , respectively. During a morning shift, $12$ Standard containers are shipped out and $6$ Refrigerated containers arrive at the warehouse, while no Oversized containers are moved. Following these changes, the ratio of Standard containers to Refrigerated containers in the warehouse becomes $1 : 1$ . What was the total number of containers (Standard, Refrigerated, and Oversized) initially stored in the warehouse?

18

72

126

156

162

Show answer & explanation

Answer: 162

Answer

162

The correct answer 162 is obtained by representing the initial numbers of containers as

4x

,

3x

, and

2x

. Equating the modified quantities of Standard (

4x - 12

) and Refrigerated (

3x + 6

) yields

x = 18

. Summing all three parts gives

9x = 9 \times 18 = 162

.

Step-by-Step Solution

1

Define initial quantities using a common variable based on the given three-part ratio.

Let the initial numbers of Standard, Refrigerated, and Oversized containers be

4x

,

3x

, and

2x

, respectively. The initial total count of containers is

4x + 3x + 2x = 9x

.

Ratios represent relative parts, so multiplying each term by a constant multiplier

x

yields the actual quantities.

2

Set up an equation reflecting the change in container quantities.

After shipping

12

Standard containers and adding

6

Refrigerated containers, the new quantities are

(4x - 12)

Standard containers and

(3x + 6)

Refrigerated containers. Since their new ratio is

1 : 1

, we write

4x - 12 = 3x + 6

.

A

1 : 1

ratio means the two quantities are equal.

3

Solve for the multiplier

x

.

4x - 3x = 6 + 12 \implies x = 18

.

Isolating

x

gives the ratio scaling factor.

4

Calculate the initial total number of containers.

Initial total

= 9x = 9 \times 18 = 162

.

The question asks for the total initial count across all three container categories (

4x + 3x + 2x = 9x

).

Key Concept

Altering linear ratios by solving for a common multiplier across multi-part ratio components.

Question 62Question →

At a corporate law firm, the ratio of Partners to Associates was initially $3 : 8$ , and the ratio of Associates to Paralegals was initially $4 : 5$ . During an internal restructuring, $15$ Associates were promoted to Partners, and $2$ Paralegals resigned. As a result of these two changes, the ratio of Partners to Paralegals became $5 : 8$ . Shortly thereafter, the firm conducted a hiring drive, recruiting additional Associates until the ratio of Associates to Paralegals became $7 : 6$ , while the numbers of Partners and Paralegals remained unchanged. What was the total number of staff members (Partners, Associates, and Paralegals) at the firm after the hiring drive?

Show answer & explanation

Answer: 134

Answer

134

The solution unifies the initial ratios into 3 : 8 : 10, sets up the algebraic equations reflecting both promotions and resignations to find the multiplier k = 5, updates the staff counts, and applies the final ratio to find 56 Associates, yielding a final total of 134 staff members.

Step-by-Step Solution

1

Unify initial two-variable ratios into a continuous three-part ratio.

Partners : Associates : Paralegals = 3 : 8 : 10

Associates is the common term. Scaling 4 : 5 by 2 gives 8 : 10, matching the 8 parts in the Partners-to-Associates ratio.

2

Formulate algebraic expressions for post-restructuring staff counts.

Partners = 3k + 15, Associates = 8k - 15, Paralegals = 10k - 2

Promotions transfer 15 from Associates to Partners, and 2 Paralegals leave.

3

Solve for the ratio multiplier k using the new Partner-to-Paralegal ratio.

k = 5

Setting (3k + 15) / (10k - 2) = 5 / 8 yields 24k + 120 = 50k - 10, so 26k = 130.

4

Calculate exact staff counts after restructuring.

Partners = 30, Associates = 25, Paralegals = 48

Substitute k = 5 into the expressions from Step 2.

5

Determine final Associate count and total staff after hiring drive.

Final Associates = 56; Total Staff = 134

Associates = (7/6) * 48 = 56. Summing all roles gives 30 + 56 + 48 = 134.

Key Concept

Multi-part ratio unification, internal transfer ratio alteration, and sequential proportion scaling

Question 63Question →

A research laboratory mixes Solution A and Solution B in a ratio of $3 : 5$ by volume to create a standard compound. If a technician needs to prepare $240$ milliliters of the compound, how many milliliters of Solution A are required?

48

milliliters

80

milliliters

90

milliliters

144

milliliters

150

milliliters

Show answer & explanation

Answer: $90$ milliliters

Answer

Solution A requires

90

milliliters of liquid.

The total compound consists of

3 + 5 = 8

parts. The fraction of Solution A in the total mixture is

\frac{3}{8}

. Multiplying this fraction by the total volume of

240

milliliters yields

\frac{3}{8} \times 240 = 90

milliliters.

Step-by-Step Solution

1

Calculate the total number of ratio parts

The ratio

3 : 5

yields

3 + 5 = 8

equal parts in total.

To find the value of one part, the total volume must be divided by the total sum of the ratio terms.

2

Determine the volume of a single ratio part

\frac{240 \text{ mL}}{8 \text{ parts}} = 30 \text{ mL per part}

.

Dividing the total volume by the total parts gives the multiplier per ratio unit.

3

Multiply the single part volume by Solution A's ratio portion

3 \times 30 \text{ mL} = 90 \text{ mL}

.

Solution A accounts for

3

of the

8

total parts.

Key Concept

Part-to-Whole Ratio Scaling

Estimated Time:1m 0s

Question 64Question →

Scanner Unit Alpha and Scanner Unit Beta, operating independently at their respective constant rates, can complete a document digitization project together in $12$ hours. Scanner Unit Alpha operates at a rate that is $50\%$ faster than that of Scanner Unit Beta. The project is carried out in three consecutive stages: first, Scanner Unit Alpha operates alone for $4$ hours; second, Scanner Unit Beta joins Alpha and both operate together until $70\%$ of the entire project is completed; finally, Scanner Unit Beta finishes the remaining portion of the project alone. How many total hours does it take to complete the entire digitization project?

Show answer & explanation

Answer: 19

Answer

The total time required to complete the entire digitization project is 19 hours.

To find the total time needed, calculate the individual rates first. With a combined rate of

r_A + r_B = \frac{1}{12}

and

r_A = 1.5 r_B

, solving

2.5 r_B = \frac{1}{12}

yields

r_B = \frac{1}{30}

and

r_A = \frac{1}{20}

. During Stage 1 (

4

hours), Alpha completes

4 \times \frac{1}{20} = 0.20

of the job. In Stage 2, both units work together to bring completion from

20\%

to

70\%

(

0.50

work), taking

\frac{0.50}{1/12} = 6

hours. In Stage 3, Beta completes the remaining

0.30

work alone, taking

\frac{0.30}{1/30} = 9

hours. Summing all stage durations yields

4 + 6 + 9 = 19

hours.

Step-by-Step Solution

1

Determine the individual work rates of Scanner Unit Alpha (

r_A

) and Scanner Unit Beta (

r_B

).

r_B = \frac{1}{30}

project per hour, and

r_A = \frac{1}{20}

project per hour.

Since their combined rate is

\frac{1}{12}

project per hour and

r_A = 1.5 r_B

, we solve

2.5 r_B = \frac{1}{12}

to find

r_B = \frac{1}{30}

and

r_A = \frac{1}{20}

.

2

Calculate the fraction of work completed during Stage 1.

Scanner Unit Alpha completes

0.20

(

20\%

) of the project in

4

hours.

Alpha works alone for

4

hours at a rate of

\frac{1}{20}

project per hour:

4 \times \frac{1}{20} = 0.20

.

3

Calculate the duration of Stage 2 where both units work together.

Stage 2 takes

6

hours.

The combined units must complete the portion from

20\%

to

70\%

, which represents

0.70 - 0.20 = 0.50

of the project. At a combined rate of

\frac{1}{12}

project per hour, the time required is

\frac{0.50}{1/12} = 6

hours.

4

Calculate the duration of Stage 3 where Scanner Unit Beta works alone.

Stage 3 takes

9

hours.

Beta must complete the remaining

30\%

(

0.30

) of the project alone. At a rate of

\frac{1}{30}

project per hour, the time required is

\frac{0.30}{1/30} = 9

hours.

5

Sum the durations of all three stages to determine total project time.

Total time =

19

hours.

Adding the duration of each stage:

4 \text{ hours} + 6 \text{ hours} + 9 \text{ hours} = 19 \text{ hours}

.

Key Concept

Work Rate and Combined Work

Question 65Question →

At a research institute, the ratio of senior scientists to associate scientists to junior researchers was initially $2 : 3 : 5$ , respectively. During an organizational expansion, 8 new senior scientists were hired, and 4 junior researchers were promoted to associate scientists, while no other personnel changes occurred. If the new ratio of senior scientists to associate scientists became $4 : 5$ , what was the initial total number of researchers at the institute?

40

60

100

120

128

Show answer & explanation

Answer: 120

Answer

120

By assigning a multiplier

x

to the initial ratio

2 : 3 : 5

, the counts are

2x

senior scientists,

3x

associate scientists, and

5x

junior researchers, making the initial total

10x

. Adding 8 senior scientists gives

2x + 8

, and adding 4 promoted associate scientists gives

3x + 4

. Equating their ratio to

\frac{4}{5}

yields

\frac{2x + 8}{3x + 4} = \frac{4}{5}

, which simplifies to

2x = 24

, or

x = 12

. Multiplying

x = 12

by the total 10 parts gives the correct initial total of 120.

Step-by-Step Solution

1

Define initial quantities using a common ratio multiplier

x

.

Senior scientists =

2x

, Associate scientists =

3x

, Junior researchers =

5x

. Initial total researchers =

2x + 3x + 5x = 10x

.

Expressing quantities in terms of

x

allows setting up algebraic equations based on personnel updates.

2

Apply the specified personnel changes to find the new counts of senior and associate scientists.

New Senior scientists =

2x + 8

. New Associate scientists =

3x + 4

(since 4 junior researchers were promoted to associate scientists).

Promoting 4 junior researchers increases the associate scientist count by 4.

3

Set up the proportion equation for the new ratio of senior scientists to associate scientists.

\frac{2x + 8}{3x + 4} = \frac{4}{5}

The problem states the new ratio between senior and associate scientists is

4 : 5

.

4

Cross-multiply and solve for

x

.

5(2x + 8) = 4(3x + 4) \implies 10x + 40 = 12x + 16 \implies 2x = 24 \implies x = 12.

Solving for

x

provides the multiplier required to compute the initial total.

5

Calculate the initial total number of researchers.

Initial total =

10x = 10 \times 12 = 120

.

The total number of initial researchers corresponds to

10x

parts.

Key Concept

Setting up and solving algebraic proportions involving multi-part ratios after internal transfers and external additions

Estimated Time:2m 0s

Question 66Question →

Three specialized synthesis columns—Alpha, Beta, and Gamma—are used in a pharmaceutical facility to purify batches of a chemical compound. Operating simultaneously at their respective constant rates, Column Alpha and Column Beta can process a standard batch in $8$ hours, while Column Beta and Column Gamma working together can process the exact same batch in $12$ hours.

Column Alpha begins processing a standard batch alone. After $4$ hours of operation, Column Alpha experiences a mechanical restriction that reduces its processing rate by $33\frac{1}{3}\%$ . At that exact moment, Column Gamma is brought online to assist Column Alpha. Working together, Column Alpha (at its reduced rate) and Column Gamma complete the remaining portion of the batch in $8.4$ hours.

How many hours would it take Column Beta, operating alone at its normal constant rate, to process an entire standard batch of the chemical compound?

15 hours

18 hours

20 hours

24 hours

30 hours

Show answer & explanation

Answer: 20 hours

Answer

Column Beta would take 20 hours to complete an entire batch working alone.

The correct response of 20 hours is determined by establishing the rates of the three columns. With

a + b = 1/8

and

b + c = 1/12

, subtracting these equations gives

c = a - 1/24

. During the first 4 hours, Column Alpha completes

4a

of the batch. In the second stage, Column Alpha works at

2/3 a

while Column Gamma works at

c

, giving a joint rate of

5/3 a - 1/24

. Multiplying this combined rate by

8.4

hours and setting it equal to the remaining work

1 - 4a

yields

a = 3/40

. Substituting this back into

a + b = 1/8

yields

b = 1/20

, meaning Column Beta requires 20 hours to complete a batch alone.

Step-by-Step Solution

1

Define individual work rates in batches per hour for Column Alpha (

a

), Column Beta (

b

), and Column Gamma (

c

).

From the given combined rates:

a + b = \frac{1}{8}

and

b + c = \frac{1}{12}

.

Combined work rates equal the sum of individual work rates.

2

Express Column Gamma's rate (

c

) in terms of Column Alpha's rate (

a

).

Subtracting the second equation from the first gives

(a + b) - (b + c) = \frac{1}{8} - \frac{1}{12} \implies a - c = \frac{1}{24} \implies c = a - \frac{1}{24}

.

Isolating one rate variable simplifies the multi-stage work equation.

3

Calculate the work completed during Stage 1 and express the remaining work.

Work completed in Stage 1 (

4

hours at rate

a

) is

4a

. The remaining work to be done is

1 - 4a

.

Total work equals

1

batch.

4

Formulate the combined rate for Stage 2 and set up the equation for the remaining work.

Alpha's reduced rate is

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

. The combined rate with Gamma is

\frac{2}{3}a + c = \frac{2}{3}a + \left(a - \frac{1}{24}\right) = \frac{5}{3}a - \frac{1}{24}

. Stage 2 takes

8.4 = \frac{42}{5}

hours, so

\frac{42}{5}\left(\frac{5}{3}a - \frac{1}{24}\right) = 1 - 4a

.

Work completed in Stage 2 equals combined rate multiplied by time spent in Stage 2.

5

Solve the algebraic equation for

a

, then find

b

.

Expanding the equation:

14a - \frac{7}{20} = 1 - 4a \implies 18a = \frac{27}{20} \implies a = \frac{3}{40}

. Then

b = \frac{1}{8} - a = \frac{1}{8} - \frac{3}{40} = \frac{2}{40} = \frac{1}{20}

.

Since Beta's rate is

\frac{1}{20}

batch/hour, the time taken by Beta alone is

\frac{1}{1/20} = 20

hours.

Key Concept

Work Rate Equations and Multi-Stage Combined Work

Question 67Question →

Machine X and Machine Y, operating independently at their respective constant rates, can complete a bottling order together in $6$ hours. If Machine X works alone for $2$ hours and is then joined by Machine Y, both machines work together for an additional $4.8$ hours to finish the remaining part of the order. How many hours would it take Machine Y, operating alone at its constant rate, to complete the entire bottling order?

10

12

15

16

18

Show answer & explanation

Answer: $15$

Answer

15

hours

The correct answer is

15

hours. Using the work equation

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

, the combined rate of Machine X and Machine Y is

\frac{1}{6}

. The work completed by both machines in the

4.8

-hour period is

4.8 \times \frac{1}{6} = 0.8

of the job. Since the remaining

0.2

of the job was completed by Machine X in

2

hours, Machine X's rate is

\frac{0.2}{2} = 0.1 = \frac{1}{10}

per hour. Subtracting Machine X's rate from the combined rate yields Machine Y's rate:

\frac{1}{6} - \frac{1}{10} = \frac{1}{15}

per hour. Therefore, Machine Y takes

15

hours operating alone to finish the entire bottling order.

Step-by-Step Solution

1

Define rates for Machine X and Machine Y

Let Machine X's rate be

r_x

orders per hour and Machine Y's rate be

r_y

orders per hour. Their combined rate is

r_x + r_y = \frac{1}{6}

orders per hour.

Combined work rate is the reciprocal of the combined completion time of

6

hours.

2

Express the total work completed in two stages

Machine X works alone for

2

hours completing

2 r_x

of the order. Then both work together for

4.8

hours completing

4.8(r_x + r_y)

of the order. Thus,

2 r_x + 4.8(r_x + r_y) = 1

.

The sum of work completed in the two stages equals

1

full job.

3

Substitute the combined rate into the equation to find

r_x

Since

r_x + r_y = \frac{1}{6}

, we substitute:

2 r_x + 4.8\left(\frac{1}{6}\right) = 1 \implies 2 r_x + 0.8 = 1 \implies 2 r_x = 0.2 \implies r_x = 0.1 = \frac{1}{10}

.

Substituting the combined rate simplifies the equation to a single variable,

r_x

.

4

Calculate

r_y

and the time needed for Machine Y working alone

r_y = \frac{1}{6} - \frac{1}{10} = \frac{5 - 3}{30} = \frac{2}{30} = \frac{1}{15}

orders per hour. Time taken by Machine Y alone

= \frac{1}{r_y} = 15

hours.

Subtracting Machine X's rate from the combined rate yields Machine Y's rate, whose reciprocal gives the time to complete the job alone.

Key Concept

Work Rate and Combined Work

Estimated Time:2m 0s

Question 68Question →

A specialized mapping drone, Model X, can complete an aerial survey of a solar farm in $20$ hours operating alone at a constant rate. A second drone, Model Y, can complete the exact same survey in $30$ hours operating alone at a constant rate. If both drones operate simultaneously at their respective constant rates, how many hours will it take them to complete the aerial survey together?

Show answer & explanation

Answer: 12

Answer

It will take 12 hours for both drones operating together to complete the survey.

Model X completes

\frac{1}{20}

of the job per hour and Model Y completes

\frac{1}{30}

of the job per hour. Combined, their rate is

\frac{1}{20} + \frac{1}{30} = \frac{5}{60} = \frac{1}{12}

of the job per hour. Taking the reciprocal yields a total time of 12 hours.

Step-by-Step Solution

1

Calculate individual work rates

Model X rate =

\frac{1}{20}

job/hr, Model Y rate =

\frac{1}{30}

job/hr

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

2

Calculate the combined rate of work

Combined rate =

\frac{1}{20} + \frac{1}{30} = \frac{5}{60} = \frac{1}{12}

job/hr

When working simultaneously, individual rates add together.

3

Compute total time required for combined work

Total time =

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

hours

Time is equal to total work (1 job) divided by the combined work rate.

Key Concept

Combined Work Rate formula:

\text{Rate}_{\text{total}} = \text{Rate}_1 + \text{Rate}_2

, and

\text{Time}_{\text{total}} = \frac{1}{\text{Rate}_{\text{total}}}

.

Question 69Question →

A graphic design department has senior designers and junior designers in a ratio of $3 : 4$ . If there are $24$ junior designers in the department, what is the total number of senior and junior designers in the department?

Show answer & explanation

Answer: 42

Answer

The total number of senior and junior designers in the department is 42.

The ratio of senior to junior designers is

3 : 4

. Given that there are

24

junior designers,

4

ratio parts equal

24

, making

1

ratio part equal to

6

. The total department consists of

3 + 4 = 7

ratio parts. Therefore, the total number of designers is

7 \times 6 = 42

.

Step-by-Step Solution

1

Determine the value of one ratio unit using the given number of junior designers.

Since

4

ratio parts correspond to

24

junior designers,

1

ratio part

= 24 / 4 = 6

.

The given count of

24

aligns with the

4

parts of junior designers in the

3 : 4

ratio.

2

Calculate the total ratio parts and find the total count of designers.

Total ratio parts

= 3 + 4 = 7

. Total designers

= 7 \times 6 = 42

.

The total department size is represented by the sum of the ratio components.

Key Concept

Ratio and Proportion Word Problems

Question 70Question →

At a textile weaving mill, high-grade cotton, synthetic polyester, and natural silk threads are blended to produce a specialized fabric. Initially, the weights of cotton, polyester, and silk in the blending vat are in the ratio of $5 : 3 : 2$ , respectively. To meet a custom order requirement, $12$ kilograms of silk are added to the vat, and $4$ kilograms of polyester are removed, while the amount of cotton remains unchanged. If the new ratio of cotton weight to silk weight is $5 : 4$ , how many kilograms of synthetic polyester are in the blending vat after these adjustments?

Show answer & explanation

Answer: 14

Answer

14 kg

Representing the initial weights as

5x

,

3x

, and

2x

, the addition of

12

kg of silk updates the silk quantity to

2x + 12

. Setting the unchanged cotton quantity (

5x

) over the new silk quantity equal to

\frac{5}{4}

yields

x = 6

. The initial weight of polyester was

3(6) = 18

kg, and removing

4

kg gives a final polyester weight of

14

kg.

Step-by-Step Solution

1

Express initial component weights using a common ratio multiplier

x

Cotton =

5x

, Polyester =

3x

, Silk =

2x

The given initial ratio of cotton to polyester to silk is

5 : 3 : 2

.

2

Set up a proportion for the updated Cotton-to-Silk ratio

\frac{5x}{2x + 12} = \frac{5}{4}

Adding

12

kg of silk increases its weight to

(2x + 12)

kg, while cotton remains at

5x

kg.

3

Solve the algebraic proportion for

x

x = 6

Cross-multiplying gives

20x = 5(2x + 12) = 10x + 60

, which simplifies to

10x = 60

, so

x = 6

.

4

Calculate the final weight of synthetic polyester

14 kg

Initial polyester was

3(6) = 18

kg. Subtracting the

4

kg removed leaves

18 - 4 = 14

kg.

Key Concept

Multi-part ratios and algebraic setting of updated proportional quantities

Question 71Question →

An investment consultancy allocates capital across three asset classes: Equities, Fixed Income, and Real Estate. Initially, the ratio of Equities to Fixed Income is $3 : 4$ , and the ratio of Fixed Income to Real Estate is $2 : 3$ . Following a market expansion, the total value of the portfolio increases by $50\%$ . To comply with updated risk guidelines, the firm reallocates the expanded capital such that the amount in Fixed Income decreases by $25\%$ , and the remaining portfolio value is divided between Equities and Real Estate in the ratio $3 : 2$ , respectively. If the new allocation in Real Estate is $\$18,000$ greater than its initial allocation, what was the initial total value of the investment portfolio?

\$130,000

\$300,000

\$390,000

\$520,000

\$585,000

Show answer & explanation

Answer: $\$390,000$

Answer

The initial total value of the investment portfolio was

\$

390,000.

Combining the initial ratios gives an Equities to Fixed Income to Real Estate ratio of

3 : 4 : 6

, making the initial total

13x

. After a

50\%

increase in total value (

19.5x

) and a

25\%

decrease in Fixed Income (

3x

), the remaining

16.5x

yields a new Real Estate value of

\frac{2}{5} \times 16.5x = 6.6x

. The difference

6.6x - 6x = 0.6x = 18,000

gives

x = 30,000

, resulting in an initial total portfolio of

13 \times 30,000 = \$390,000

.

Step-by-Step Solution

1

Combine the initial two-part ratios into a single three-part ratio for Equities (

E_1

), Fixed Income (

F_1

), and Real Estate (

R_1

).

E_1 : F_1 = 3 : 4

and

F_1 : R_1 = 2 : 3 = 4 : 6

, giving

E_1 : F_1 : R_1 = 3 : 4 : 6

.

A common term for Fixed Income (

4

) is needed to express all three asset classes in terms of a single variable

x

.

2

Express initial amounts and initial total portfolio value in terms of

x

.

E_1 = 3x

,

F_1 = 4x

,

R_1 = 6x

, and Total Initial Portfolio

T_1 = 3x + 4x + 6x = 13x

.

Summing the ratio units gives the total initial quantity representation.

3

Calculate the updated total portfolio value (

T_2

) and updated Fixed Income amount (

F_2

).

T_2 = 1.50 \times 13x = 19.5x

, and

F_2 = (1 - 0.25) \times 4x = 3x

.

The overall portfolio increases by

50\%

while Fixed Income decreases by

25\%

.

4

Determine the remaining portfolio value for Equities and Real Estate, and calculate the new Real Estate value (

R_2

).

Remaining Value

= 19.5x - 3x = 16.5x

.

R_2 = \frac{2}{3 + 2} \times 16.5x = \frac{2}{5} \times 16.5x = 6.6x

.

The remaining capital is divided between Equities and Real Estate in a

3 : 2

ratio.

5

Set up the change equation for Real Estate to solve for

x

and determine

T_1

.

R_2 - R_1 = 6.6x - 6x = 0.6x = 18,000 \implies x = 30,000

. Therefore,

T_1 = 13 \times 30,000 = 390,000

.

The problem states the final Real Estate amount is

\$

18,000 greater than its initial value.

Key Concept

Combining compound ratios and applying sequential percentage modifications to part-to-part and part-to-whole relationships.

Question 72Question →

At an architectural firm, the ratio of senior architects to junior architects to structural engineers was initially $3 : 7 : 4$ , respectively. After the firm hired $6$ additional senior architects and $4$ structural engineers left the firm, the ratio of senior architects to structural engineers became $3 : 2$ , while the number of junior architects remained unchanged. What was the total number of senior architects, junior architects, and structural engineers combined initially at the firm?

Show answer & explanation

Answer: 56

Answer

The total number of senior architects, junior architects, and structural engineers combined initially at the firm was 56.

Let the initial numbers of senior architects, junior architects, and structural engineers be

3x

,

7x

, and

4x

, respectively. After adding

6

senior architects and subtracting

4

structural engineers, the updated counts are

3x + 6

and

4x - 4

. Setting their ratio to

3 : 2

yields

\frac{3x + 6}{4x - 4} = \frac{3}{2}

. Cross-multiplying gives

6x + 12 = 12x - 12

, which simplifies to

6x = 24

, so

x = 4

. The total number of initial employees is

3x + 7x + 4x = 14x = 14(4) = 56

.

Step-by-Step Solution

1

Define initial quantities using ratio multiplier x

Senior Architects

= 3x

, Junior Architects

= 7x

, Structural Engineers

= 4x

Ratios specify relative proportions, so multiplying each term by

x

gives actual counts.

2

Formulate new quantities after staff changes

Senior Architects

= 3x + 6

, Structural Engineers

= 4x - 4

6 senior architects joined and 4 structural engineers left.

3

Set up ratio equation and solve for x

\frac{3x + 6}{4x - 4} = \frac{3}{2} \implies 2(3x + 6) = 3(4x - 4) \implies 6x + 12 = 12x - 12 \implies 6x = 24 \implies x = 4

Cross-multiplication converts the proportion into a linear equation.

4

Compute total initial staff count

Initial Total

= 3x + 7x + 4x = 14x = 14 \times 4 = 56

Summing all 3 initial ratio parts and substituting

x = 4

yields the total initial count.

Key Concept

Multi-part ratio formulation and linear equation setup based on additions and subtractions

Estimated Time:2m 0s

Question 73Question →

A pharmaceutical production facility utilizes three automated synthesis modules—Module A, Module B, and Module C—to produce a specific batch of medication. Working alone at its constant rate, Module A can complete a full batch in $15$ hours. Modules B and C, working together at their respective constant rates, can complete a full batch in $20$ hours. A production run begins with Module A and Module B working together for $4$ hours. At that point, Module A is turned off for recalibration, and Module C immediately joins Module B to complete the remaining portion of the batch. If Modules B and C work together for $12$ hours to finish the batch, how many hours would it take Module B to complete a full batch working alone?

20

hours

25

hours

30

hours

40

hours

60

hours

Show answer & explanation

Answer: $30$ hours

Answer

Module B would take

30

hours to complete a full batch working alone.

The correct answer is

30

hours. In Stage 2, Modules B and C work together for

12

hours at their combined rate of

\frac{1}{20}

batch per hour, completing

\frac{12}{20} = \frac{3}{5}

of the entire batch. This leaves

1 - \frac{3}{5} = \frac{2}{5}

of the batch that was completed during Stage 1 by Modules A and B working together for

4

hours. Setting up the Stage 1 equation:

4(r_A + r_B) = \frac{2}{5}

. Substituting Module A's rate

r_A = \frac{1}{15}

, we get

4\left(\frac{1}{15} + r_B\right) = \frac{2}{5} \implies \frac{4}{15} + 4r_B = \frac{6}{15} \implies 4r_B = \frac{2}{15} \implies r_B = \frac{1}{30}

. Therefore, Module B takes

30

hours operating alone.

Step-by-Step Solution

1

Express the individual and combined rates of the modules

Rate of Module A (

r_A

) =

\frac{1}{15}

batch/hr; Combined rate of Modules B and C (

r_B + r_C

) =

\frac{1}{20}

batch/hr.

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

2

Calculate the work completed during Stage 2

Work in Stage 2 =

12 \times (r_B + r_C) = 12 \times \frac{1}{20} = \frac{3}{5}

of the batch.

Modules B and C worked together for

12

hours at their known combined rate of

\frac{1}{20}

batch/hr.

3

Calculate the work done in Stage 1 and solve for Module B's rate (

r_B

)

Total Work = Stage 1 Work + Stage 2 Work =

1

. Thus,

4(r_A + r_B) + \frac{3}{5} = 1 \implies 4\left(\frac{1}{15} + r_B\right) = \frac{2}{5} \implies \frac{4}{15} + 4r_B = \frac{6}{15} \implies 4r_B = \frac{2}{15} \implies r_B = \frac{1}{30}

batch/hr.

The sum of the work performed across both stages must equal

1

full batch.

4

Convert Module B's rate into total time required working alone

Time for Module B alone =

\frac{1}{r_B} = 30

hours.

Total time working alone is the reciprocal of the individual work rate.

Key Concept

Work Rate and Combined Work

Question 74Question →

A software engineering team uses three automated refactoring scripts—Script A, Script B, and Script C—to update a large codebase. Working alone at their respective constant rates, Script A can finish the entire codebase in $10$ hours, Script B in $15$ hours, and Script C in $30$ hours. All three scripts begin running simultaneously. After $2$ hours, Script A encounters an error and stops running. Script B and Script C continue running together without interruption until the codebase is completely refactored. How many additional hours will it take for Script B and Script C to complete the remaining work?

Show answer & explanation

Answer: 6

Answer

6 hours

The rate of Script A is

\frac{1}{10}

of the codebase per hour, Script B is

\frac{1}{15}

per hour, and Script C is

\frac{1}{30}

per hour. Working together for

2

hours, all three scripts complete

2 \times \left(\frac{1}{10} + \frac{1}{15} + \frac{1}{30}\right) = 2 \times \frac{1}{5} = \frac{2}{5}

of the codebase, leaving

1 - \frac{2}{5} = \frac{3}{5}

of the task remaining. With Script A offline, the combined rate of Script B and Script C is

\frac{1}{15} + \frac{1}{30} = \frac{1}{10}

per hour. The additional time required to complete the remaining work is

\frac{3/5}{1/10} = 6

hours.

Step-by-Step Solution

1

Determine the individual work rates of Script A, Script B, and Script C.

Rate of Script A =

\frac{1}{10}

codebase/hour, Rate of Script B =

\frac{1}{15}

codebase/hour, Rate of Script C =

\frac{1}{30}

codebase/hour.

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

2

Calculate the combined work rate of all three scripts during the initial 2-hour period.

Combined Rate(A+B+C) =

\frac{1}{10} + \frac{1}{15} + \frac{1}{30} = \frac{3 + 2 + 1}{30} = \frac{6}{30} = \frac{1}{5}

codebase/hour.

When multiple agents work together simultaneously, their individual work rates are added.

3

Determine the fraction of the codebase refactored during the first 2 hours.

Work Completed =

2 \text{ hours} \times \frac{1}{5} \text{ codebase/hour} = \frac{2}{5}

of the total codebase.

Total work done equals rate multiplied by time.

4

Calculate the remaining fraction of the codebase left to be processed.

Remaining Work =

1 - \frac{2}{5} = \frac{3}{5}

of the total codebase.

Subtracting the completed fraction from 1 gives the uncompleted portion of the job.

5

Calculate the combined rate of Script B and Script C after Script A stops.

Rate(B+C) =

\frac{1}{15} + \frac{1}{30} = \frac{2 + 1}{30} = \frac{3}{30} = \frac{1}{10}

codebase/hour.

Only Script B and Script C contribute to the remaining work.

6

Calculate the additional time required for Script B and Script C to finish the remaining work.

Additional Time =

\frac{3/5}{1/10} = \frac{3}{5} \times 10 = 6

hours.

Time equals remaining work divided by the active combined rate.

Key Concept

Work Rate and Combined Work

Estimated Time:1m 30s

Question 75Question →

An agricultural processing facility uses three independent conveyor systems—Conveyor $X$ , Conveyor $Y$ , and Conveyor $Z$ —to fill a grain storage silo. Working together at their respective constant rates, Conveyors $X$ and $Y$ can fill the empty silo in $12$ hours, while Conveyors $Y$ and $Z$ working together can fill the empty silo in $20$ hours. Initially, Conveyors $X$ and $Z$ work together for $5$ hours, completing exactly $\frac{1}{3}$ of the silo. How many hours would it take Conveyor $Y$ operating alone to fill the remaining $\frac{2}{3}$ of the silo?

Show answer & explanation

Answer: 20

Answer

It would take Conveyor

Y

operating alone

20

hours to fill the remaining

\frac{2}{3}

of the silo.

By representing the rate of each pair of conveyors as a fraction of the total job per hour, we find

R_X + R_Y = 1/12

,

R_Y + R_Z = 1/20

, and

R_X + R_Z = 1/15

. Summing these three equations yields

2(R_X + R_Y + R_Z) = 1/5

, so

R_X + R_Y + R_Z = 1/10

. Subtracting

R_X + R_Z = 1/15

gives

R_Y = 1/30

silo per hour. To fill the remaining

2/3

of the silo, Conveyor

Y

requires

(2/3) / (1/30) = 20

hours.

Step-by-Step Solution

1

Express the combined rate of each pair of conveyors as a fraction of the silo filled per hour.

R_X + R_Y = \frac{1}{12}

,

R_Y + R_Z = \frac{1}{20}

, and

R_X + R_Z = \frac{1/3}{5} = \frac{1}{15}

.

Work rate equals work done divided by time taken.

2

Sum the three pairwise rates to determine the combined rate of all three conveyors working together.

2(R_X + R_Y + R_Z) = \frac{1}{12} + \frac{1}{20} + \frac{1}{15} = \frac{12}{60} = \frac{1}{5} \implies R_X + R_Y + R_Z = \frac{1}{10}

.

Adding pairwise rates counts each conveyor's individual rate twice.

3

Subtract the combined rate of Conveyors

X

and

Z

from the total rate of all three conveyors to isolate the rate of Conveyor

Y

.

R_Y = \frac{1}{10} - \frac{1}{15} = \frac{3 - 2}{30} = \frac{1}{30}

of the silo per hour.

Subtracting

(R_X + R_Z)

from

(R_X + R_Y + R_Z)

yields

R_Y

directly.

4

Divide the remaining fraction of work by Conveyor

Y

's individual rate to find the required time.

\text{Time} = \frac{2/3}{1/30} = 20

hours.

Time required equals remaining work divided by the individual work rate.

Key Concept

Solving systems of simultaneous work rate equations by summing pairwise rates.

Question 76Question →

An artisan chocolate factory produces a signature dark chocolate blend using cocoa mass, cocoa butter, and cane sugar in the initial ratio of $2 : 3 : 4$ by weight, respectively. To modify the flavor profile for a new batch, the master chocolatier adds $10$ kilograms of cocoa mass and $20$ kilograms of cocoa butter to the mix, while removing $5$ kilograms of cane sugar. As a result, the ratio of cocoa mass to cocoa butter in the modified mixture becomes $4 : 7$ . What was the total weight, in kilograms, of the initial dark chocolate blend?

25

35

45

60

70

Show answer & explanation

Answer: $45$

Answer

The total weight of the initial dark chocolate blend was

45

kilograms.

By setting the initial weights of cocoa mass, cocoa butter, and cane sugar as

2x

,

3x

, and

4x

, the initial total weight is

9x

. The modified weights for cocoa mass and cocoa butter are

2x + 10

and

3x + 20

, respectively. Equating their ratio to

\frac{4}{7}

gives

\frac{2x + 10}{3x + 20} = \frac{4}{7}

, which simplifies to

14x + 70 = 12x + 80

, so

2x = 10

and

x = 5

. Substituting

x = 5

into the total weight expression

9x

yields

9(5) = 45

kilograms.

Step-by-Step Solution

1

Define initial ingredient weights using a multiplier variable.

Cocoa mass =

2x

, Cocoa butter =

3x

, Cane sugar =

4x

. Total initial weight =

2x + 3x + 4x = 9x

.

Expressing quantities in terms of a common ratio multiplier

x

enables setting up algebraic equations after alterations.

2

Express the modified weights of cocoa mass and cocoa butter.

New cocoa mass =

2x + 10

; New cocoa butter =

3x + 20

.

The problem states

10

kg of cocoa mass and

20

kg of cocoa butter were added.

3

Set up the proportion equation using the new ratio of cocoa mass to cocoa butter.

\frac{2x + 10}{3x + 20} = \frac{4}{7}

The modified ratio of cocoa mass to cocoa butter is given as

4 : 7

.

4

Cross-multiply and solve for

x

.

7(2x + 10) = 4(3x + 20) \implies 14x + 70 = 12x + 80 \implies 2x = 10 \implies x = 5

.

Solving the linear equation determines the common ratio multiplier value.

5

Calculate the initial total weight of the blend.

Initial total weight =

9x = 9(5) = 45

kg.

Multiplying the sum of all initial ratio parts by the multiplier gives the requested initial total weight.

Key Concept

Ratio setup and algebraic scaling in multi-part word problems with changing quantities

Estimated Time:2m 0s

Question 77Question →

A specialized express freight facility uses three automated sorting systems—System 1, System 2, and System 3—to process incoming shipments. Operating continuously at their respective constant rates, System 1 and System 2 working together can process 1 full shipment in 6 hours; System 2 and System 3 working together can process the same shipment in 10 hours; and System 1 and System 3 working together can process the shipment in 7.5 hours. Processing begins with System 1 and System 2 working together. After 3 hours, System 1 breaks down and stops operating, at which point System 3 is immediately activated to work alongside System 2 until the entire shipment is completed. How many total hours does it take, from the start of processing, to complete the entire shipment?

Show answer & explanation

Answer: 8

Answer

The total time required from start to completion is 8 hours.

Converting completion times to work rates per hour gives paired rates of 1/6, 1/10, and 2/15. Summing these and dividing by 2 yields a combined three-system rate of 1/5 shipment per hour. In the first 3 hours, System 1 and System 2 complete 3 * (1/6) = 1/2 of the shipment. System 2 and System 3 then finish the remaining 1/2 at their combined rate of 1/10 per hour, requiring (1/2) / (1/10) = 5 hours. Total elapsed time is 3 + 5 = 8 hours.

Step-by-Step Solution

1

Set up equations for the rate of work done per hour by each pair of systems

Let

r_1, r_2, r_3

be the individual work rates in shipments per hour. Then

r_1 + r_2 = \frac{1}{6}

,

r_2 + r_3 = \frac{1}{10}

, and

r_1 + r_3 = \frac{1}{7.5} = \frac{2}{15}

.

Work rate is inversely proportional to completion time (

Rate = \frac{Work}{Time}

).

2

Calculate the combined processing rate of all three systems

2(r_1 + r_2 + r_3) = \frac{1}{6} + \frac{1}{10} + \frac{2}{15} = \frac{5 + 3 + 4}{30} = \frac{12}{30} = \frac{2}{5}

, which simplifies to

r_1 + r_2 + r_3 = \frac{1}{5}

shipment per hour.

Adding the three paired rates accounts for each individual system's rate exactly twice.

3

Determine the amount of work completed during the initial 3-hour period

Work completed =

3 \text{ hours} \times (r_1 + r_2) = 3 \times \frac{1}{6} = \frac{1}{2}

of the total shipment.

System 1 and System 2 operate together at a combined rate of

\frac{1}{6}

shipment per hour for 3 hours.

4

Calculate the time needed for System 2 and System 3 to complete the remaining shipment

Remaining work =

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

. Time required =

\frac{1/2}{r_2 + r_3} = \frac{1/2}{1/10} = 5

hours.

System 2 and System 3 work together at a combined rate of

\frac{1}{10}

shipment per hour to finish the remaining half of the job.

5

Calculate the total time elapsed from start to finish

Total time =

3 \text{ hours} + 5 \text{ hours} = 8 \text{ hours}

.

The question asks for the total duration of the process from the beginning.

Key Concept

Solving systems of simultaneous work rate equations and analyzing multi-stage combined work.

Question 78Question →

An industrial semiconductor fabrication facility operates three specialized chemical vapor deposition (CVD) chambers—Chamber 1, Chamber 2, and Chamber 3—to process a large batch of silicon wafers.

When Chamber 1 and Chamber 2 operate together at their respective constant rates, they can complete the entire batch in $12$ hours. When Chamber 2 and Chamber 3 operate together at their respective constant rates, they can complete the entire batch in $20$ hours.

To process a new batch, Chamber 1 operates alone for $4$ hours. Next, Chamber 2 is turned on to assist Chamber 1, and both chambers operate together for an additional $6$ hours. Finally, Chambers 1 and 2 are shut down, and Chamber 3 operates alone for $18$ hours to finish the remaining portion of the batch.

How many hours would it take Chamber 3 to process the entire batch working alone at its constant rate?

40 hours

48 hours

54 hours

60 hours

72 hours

Show answer & explanation

Answer: 60 hours

Answer

60 hours

The correct answer is 60 hours. By subtracting the combined rate of Chambers 2 and 3 (

\frac{1}{20}

) from the combined rate of Chambers 1 and 2 (

\frac{1}{12}

), we find that Chamber 1's rate exceeds Chamber 3's rate by

\frac{1}{30}

batch per hour. Substituting

r_1 = r_3 + \frac{1}{30}

into the total work equation for the three stages (

4 r_1 + 6(r_1 + r_2) + 18 r_3 = 1

) yields

22 r_3 + \frac{2}{15} = \frac{1}{2}

, giving

r_3 = \frac{1}{60}

. Therefore, Chamber 3 requires 60 hours alone.

Step-by-Step Solution

1

Define rate variables and establish equations for the combined pairs.

Let

r_1, r_2, r_3

represent the work rates of Chambers 1, 2, and 3 in fraction of batch per hour. We are given

r_1 + r_2 = \frac{1}{12}

and

r_2 + r_3 = \frac{1}{20}

.

Work rate is the reciprocal of completion time for a full job.

2

Find the difference between the rates of Chamber 1 and Chamber 3.

(r_1 + r_2) - (r_2 + r_3) = \frac{1}{12} - \frac{1}{20} \implies r_1 - r_3 = \frac{5}{60} - \frac{3}{60} = \frac{2}{60} = \frac{1}{30}

. Thus,

r_1 = r_3 + \frac{1}{30}

.

Eliminating

r_2

allows us to express

r_1

strictly in terms of

r_3

.

3

Calculate the work completed in each stage of the multi-stage schedule.

Stage 1 work =

4 r_1

. Stage 2 work =

6(r_1 + r_2) = 6 \left(\frac{1}{12}\right) = \frac{1}{2}

. Stage 3 work =

18 r_3

.

Work completed equals rate multiplied by time operating.

4

Set up the total work equation and solve for Chamber 3's rate

r_3

.

4 r_1 + \frac{1}{2} + 18 r_3 = 1 \implies 4 \left(r_3 + \frac{1}{30}\right) + 18 r_3 = \frac{1}{2} \implies 22 r_3 + \frac{2}{15} = \frac{1}{2} \implies 22 r_3 = \frac{15 - 4}{30} = \frac{11}{30} \implies r_3 = \frac{11}{30 \times 22} = \frac{1}{60}

.

Sum of work across all stages equals

1

complete batch.

5

Determine the time required for Chamber 3 to complete the entire batch alone.

Time

= \frac{1}{r_3} = 60

hours.

Total time for a single entity is the reciprocal of its individual work rate.

Key Concept

Multi-stage work rate modeling with system of linear equations

Estimated Time:2m 30s

Question 79Question →

In a market research study of $100$ consumers, $65$ consumers stated that they regularly purchase Brand A, and $45$ consumers stated that they regularly purchase Brand B. If $20$ consumers purchase neither Brand A nor Brand B, how many consumers regularly purchase both Brand A and Brand B?

Show answer & explanation

Answer: 30

Answer

The number of consumers who regularly purchase both Brand A and Brand B is

30

.

According to the principle of inclusion-exclusion for two sets,

\text{Total} = \text{Brand A} + \text{Brand B} - \text{Both} + \text{Neither}

. Substituting the given numbers into this relationship gives

100 = 65 + 45 - \text{Both} + 20

, which simplifies to

100 = 130 - \text{Both}

. Subtracting

100

from

130

yields

\text{Both} = 30

.

Step-by-Step Solution

1

Identify the given values from the problem statement.

\text{Total} = 100

,

\text{Brand A} = 65

,

\text{Brand B} = 45

,

\text{Neither} = 20

.

Extracting all known set quantities to set up the overlapping sets equation.

2

Apply the inclusion-exclusion formula for two overlapping sets.

\text{Total} = \text{Brand A} + \text{Brand B} - \text{Both} + \text{Neither}

.

Subtracting the overlap prevents double-counting elements belonging to both groups.

3

Substitute the values and solve for the unknown overlap.

100 = 65 + 45 - \text{Both} + 20 \implies 100 = 130 - \text{Both} \implies \text{Both} = 30

.

Algebraically determining the number of consumers in the intersection.

Key Concept

Two-Set Inclusion-Exclusion Principle

Question 80Question →

A commercial coffee roasting facility operates two industrial roasters, Roaster $X$ and Roaster $Y$ . Working alone at its constant rate, Roaster $X$ can process a standard batch of coffee beans in $12$ hours. Working alone at its constant rate, Roaster $Y$ can process the exact same batch in $8$ hours.

Roaster $X$ begins processing a standard batch alone. After $3$ hours, Roaster $Y$ is turned on, and both roasters work together at their respective constant rates until the batch is completely processed. What is the total time, in hours, required to process the entire batch of coffee beans?

3.6

hours

6.6

hours

7.8

hours

9.0

hours

10.5

hours

Show answer & explanation

Answer: $6.6$ hours

Answer

6.6 hours

In the first

3

hours, Roaster

X

completes

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

of the total batch, leaving

\frac{3}{4}

of the batch remaining. When both roasters work together, their combined rate is

\frac{1}{12} + \frac{1}{8} = \frac{5}{24}

batch per hour. The time required for both roasters to finish the remaining

\frac{3}{4}

batch is

\frac{3/4}{5/24} = 3.6

hours. Adding the initial

3

hours yields a total elapsed time of

6.6

hours.

Step-by-Step Solution

1

Calculate individual work rates per hour.

Roaster

X

rate =

\frac{1}{12}

batch/hr; Roaster

Y

rate =

\frac{1}{8}

batch/hr.

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

2

Determine the amount of work completed by Roaster

X

in the first

3

hours and the remaining work.

Work done =

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

batch. Remaining work =

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

batch.

Roaster

X

worked alone for the first

3

hours.

3

Calculate the combined work rate of both roasters working together.

Combined rate =

\frac{1}{12} + \frac{1}{8} = \frac{2}{24} + \frac{3}{24} = \frac{5}{24}

batch/hr.

Simultaneous work rates are additive.

4

Find the time taken to complete the remaining work together.

Combined time =

\frac{3/4}{5/24} = \frac{3}{4} \times \frac{24}{5} = \frac{18}{5} = 3.6

hours.

Time equals remaining work divided by the combined work rate.

5

Add the initial single-roaster phase to the combined work phase to find total time.

Total time =

3 + 3.6 = 6.6

hours.

The question asks for the total time to process the entire batch.

Key Concept

Work Rate and Combined Work

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