Word Problems and Applied Math Practice Questions — GMAT — Page 5

Question 81Question →

A merchant combines $10$ kilograms of Brand A tea costing $\$20$ per kilogram with $30$ kilograms of Brand B tea costing $\$40$ per kilogram. What is the average cost per kilogram of the combined mixture?

\$30

\$35

\$25

\$28

\$60

Show answer & explanation

Answer: $\$35$

Answer

The average cost per kilogram of the combined tea mixture is

\$35

.

The correct option of

\$35

properly weights each unit price by its respective quantity. Brand A contributes

10 \times 20 = \$200

and Brand B contributes

30 \times 40 = \$1200

, giving a combined total cost of

\$1400

for

40

kg. Dividing

\$1400

by

40

yields

\$35

per kilogram.

Step-by-Step Solution

1

Calculate the total cost of Brand A tea

10 \text{ kg} \times \$20/\text{kg} = \$200

To find the overall weighted cost, we first determine the total dollar amount spent on Brand A.

2

Calculate the total cost of Brand B tea

30 \text{ kg} \times \$40/\text{kg} = \$1200

Next, we calculate the total dollar amount spent on Brand B.

3

Find total combined cost and total combined weight

\text{Total cost} = \$200 + \$1200 = \$1400; \text{Total weight} = 10 + 30 = 40 \text{ kg}

Sum the total cost of both components and their combined weights.

4

Divide total combined cost by total combined weight

\text{Weighted Average Cost} = \frac{\$1400}{40 \text{ kg}} = \$35/\text{kg}

The weighted average formula is Total Cost divided by Total Quantity.

Key Concept

Weighted Average in Applied Contexts

Estimated Time:1m 0s

Question 82Question →

A commercial bakery prepares a specialty grain blend using Organic Wheat, Spelt, and Rye flour. Initially, the ratio of Organic Wheat to Spelt to Rye flour in the batch is $4 : 3 : 2$ by weight. To meet specific recipe requirements, $30$ kilograms of Organic Wheat and $15$ kilograms of Spelt are added to the batch, while $10$ kilograms of Rye are removed. After these adjustments, the ratio of Organic Wheat to Rye flour in the batch becomes $3 : 1$ . What was the total weight, in kilograms, of the initial batch of flour before any adjustments were made?

120

135

210

270

305

Show answer & explanation

Answer: 270

Answer

270 kilograms

By representing the initial amounts of Organic Wheat, Spelt, and Rye as

4x

,

3x

, and

2x

kilograms respectively, the initial total weight is

9x

kilograms. Following the adjustments, the new amount of Wheat is

4x + 30

kg and the new amount of Rye is

2x - 10

kg. Equating their ratio to

3 : 1

gives

\frac{4x + 30}{2x - 10} = \frac{3}{1}

. Solving for

x

gives

4x + 30 = 6x - 30

, leading to

2x = 60

or

x = 30

. Therefore, the initial total weight of the flour batch was

9 \times 30 = 270

kilograms.

Step-by-Step Solution

1

Define initial quantities using a common multiplier

x

.

Organic Wheat

= 4x

, Spelt

= 3x

, Rye

= 2x

. Initial total weight

= 4x + 3x + 2x = 9x

.

The initial ratio

4 : 3 : 2

specifies the relative proportions of the three components in the batch.

2

Express the updated amounts after additions and removals.

New Organic Wheat

= 4x + 30

, New Spelt

= 3x + 15

, New Rye

= 2x - 10

.

Incorporating the specified additions (

+30

kg Wheat,

+15

kg Spelt) and removal (

-10

kg Rye).

3

Set up an equation using the new ratio of Organic Wheat to Rye (

3 : 1

).

\frac{4x + 30}{2x - 10} = \frac{3}{1}

The problem states that the updated ratio of Wheat to Rye is

3 : 1

.

4

Solve for the multiplier

x

.

4x + 30 = 3(2x - 10) \implies 4x + 30 = 6x - 30 \implies 2x = 60 \implies x = 30

.

Cross-multiplying and solving the linear equation yields

x = 30

.

5

Calculate the total weight of the initial batch.

Initial total

= 9x = 9 \times 30 = 270

kilograms.

Substituting

x = 30

back into the initial total expression

9x

gives the required initial batch mass.

Key Concept

Ratio Modification and Multi-Part Proportions

Estimated Time:2m 0s

Question 83Question →

At a technology company, a team of $60$ software engineers works on product development. Of these engineers, $35$ work on the mobile application, $28$ work on the web platform, and $10$ work on neither project. How many engineers work on both the mobile application and the web platform?

3

13

18

25

63

Show answer & explanation

Answer: $13$

Answer

13 engineers work on both the mobile application and the web platform.

According to the fundamental principle of overlapping sets for two groups, the total number of individuals in the universe is equal to the number of individuals in Group A plus the number in Group B, minus those counted in both groups (to prevent double counting), plus those in neither group:

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

. Substituting the given values gives

60 = 35 + 28 - \text{Both} + 10

. Simplifying the right side yields

60 = 73 - \text{Both}

, so

\text{Both} = 13

. Thus, the value

13

is correct.

Step-by-Step Solution

1

Identify the given set values and formula

\text{Total} = 60

,

\text{Mobile} = 35

,

\text{Web} = 28

,

\text{Neither} = 10

Define all components of the standard two-set overlapping formula:

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

.

2

Substitute the values into the overlapping sets formula

60 = 35 + 28 - \text{Both} + 10

Set up the linear equation to solve for the unknown intersection (Both).

3

Simplify and solve for Both

60 = 73 - \text{Both} \implies \text{Both} = 73 - 60 = 13

Combine like terms (

35 + 28 + 10 = 73

) and isolate the variable to find the number of engineers in both groups.

Key Concept

Two-Set Overlapping Venn Diagram Principle

Estimated Time:1m 0s

Question 84Question →

A boutique perfumery blends three essential oils—Jasmine, Sandalwood, and Bergamot—in an initial volume ratio of $4 : 3 : 2$ , respectively. To modify the fragrance profile, the perfumer adds $20\text{ milliliters}$ of Sandalwood and $40\text{ milliliters}$ of Bergamot to the mixture, leaving the amount of Jasmine unchanged. If the resulting volume ratio of Sandalwood to Bergamot is $5 : 6$ , what was the total initial volume, in milliliters, of the fragrance batch?

Show answer & explanation

Answer: 90

Answer

The total initial volume of the fragrance batch was 90 milliliters.

By representing the initial amounts of Jasmine, Sandalwood, and Bergamot as

4x

,

3x

, and

2x

respectively, the total initial volume is

9x

. Setting up the proportion

\frac{3x + 20}{2x + 40} = \frac{5}{6}

yields

x = 10

. Substituting

x = 10

into

9x

gives the initial volume of 90 milliliters.

Step-by-Step Solution

1

Define variables for the initial quantities using the given ratio.

Jasmine volume =

4x

, Sandalwood volume =

3x

, Bergamot volume =

2x

, where

x

is a positive multiplier. Total initial volume =

4x + 3x + 2x = 9x

.

Representing ratio parts algebraically allows setting up equations for the modified amounts.

2

Formulate an equation based on the new ratio after adding essential oils.

New Sandalwood volume =

3x + 20

, New Bergamot volume =

2x + 40

. Equating their ratio to

5 : 6

gives

\frac{3x + 20}{2x + 40} = \frac{5}{6}

.

The problem specifies the new relationship between Sandalwood and Bergamot.

3

Solve the proportion for

x

.

Cross-multiplying gives

6(3x + 20) = 5(2x + 40) \implies 18x + 120 = 10x + 200 \implies 8x = 80 \implies x = 10

.

Finding the multiplier

x

allows us to calculate the exact initial quantities.

4

Compute the total initial volume.

Total initial volume =

9x = 9 \times 10 = 90

milliliters.

The question asks for the total initial volume of the batch.

Key Concept

Solving multi-step ratio problems involving additions to individual components by establishing an algebraic multiplier.

Question 85Question →

A pharmaceutical laboratory prepares a liquid vaccine solution by blending three ingredients: Active Compound X, Active Compound Y, and Distilled Water in the volume ratio of $3 : 4 : 8$ , respectively. To meet updated formulation guidelines, a chemist adds $12$ liters of Active Compound Y and $36$ liters of Distilled Water to the solution, leaving the quantity of Active Compound X unchanged. If the new ratio of Active Compound X to Distilled Water in the resulting solution is $1 : 4$ , what was the total volume, in liters, of the solution before any ingredients were added?

Show answer & explanation

Answer: 135

Answer

The total volume of the solution before any ingredients were added was 135 liters.

Representing the initial volumes of Active Compound X, Active Compound Y, and Distilled Water as

3x

,

4x

, and

8x

, the initial total volume is

15x

liters. After adding

36

liters of Distilled Water while keeping Compound X at

3x

liters, the ratio of X to Water becomes

\frac{3x}{8x + 36} = \frac{1}{4}

. Solving

12x = 8x + 36

gives

4x = 36

, so

x = 9

. Multiplying

15

by

9

gives the initial total volume of

135

liters.

Step-by-Step Solution

1

Define the initial component quantities using a common ratio multiplier

The initial volumes of Compound X, Compound Y, and Distilled Water are

3x

,

4x

, and

8x

liters, giving an initial total volume of

15x

liters.

A ratio of

3 : 4 : 8

means the actual quantities are integer multiples of a common constant

x

.

2

Set up the ratio equation reflecting the additions

The equation comparing the unchanged Compound X to the updated volume of Distilled Water is

\frac{3x}{8x + 36} = \frac{1}{4}

.

No Compound X was added, so its volume stays

3x

, whereas

36

liters were added to the initial

8x

liters of Distilled Water.

3

Solve the algebraic equation for the multiplier

x

4(3x) = 1(8x + 36) \implies 12x = 8x + 36 \implies 4x = 36 \implies x = 9

.

Cross-multiplying isolates the terms with

x

and allows solving for the multiplier.

4

Calculate the target initial total volume

15 \times 9 = 135

liters.

Substituting

x = 9

into the initial total volume expression

15x

yields the final answer.

Key Concept

Solving multi-step ratio word problems by setting up unknown multiplier equations based on partial component alterations

Question 86Question →

A container initially holds $100$ liters of an acid solution that is $40\%$ acid by volume. First, $x$ liters of the solution are drained and replaced with an equal volume of pure water. After the mixture is thoroughly stirred, $x$ liters of the new solution are drained and replaced with an equal volume of pure acid. If the final solution is $45.6\%$ acid by volume, what is the value of $x$ ?

Show answer & explanation

Answer: 20

Answer

The value of

x

is

20

liters.

The solution requires tracking the amount of pure acid through two distinct replacement operations. Initially, there are

40

liters of acid in

100

liters of solution. Draining

x

liters removes

0.4x

liters of acid, leaving

(40 - 0.4x)

liters of acid after adding

x

liters of pure water. In the second step, draining

x

liters removes a fraction

\frac{x}{100}

of the remaining acid, leaving

(40 - 0.4x)(1 - \frac{x}{100})

liters of acid. Adding

x

liters of pure acid yields a total acid volume of

(40 - 0.4x)(1 - \frac{x}{100}) + x = 45.6

liters. Expanding and solving the resulting quadratic equation

x^2 + 50x - 1400 = 0

gives

x = 20

(since

x > 0

).

Step-by-Step Solution

1

Determine initial volume of solute (pure acid).

Initial acid volume =

40\% \times 100 = 40

liters.

Establishing the starting amount of pure acid in the

100

-liter container.

2

Model the acid amount after the first replacement (with water).

Acid volume after first replacement =

40\left(1 - \frac{x}{100}\right)

liters.

Draining

x

liters removes

\frac{x}{100}

of the total acid, and adding water adds zero acid.

3

Model the acid amount after the second replacement (with pure acid).

Final acid volume =

40\left(1 - \frac{x}{100}\right)\left(1 - \frac{x}{100}\right) + x

liters.

Draining

x

liters of the new solution removes

\frac{x}{100}

of its acid, and replacing with pure acid adds

x

liters of acid.

4

Set up and solve the quadratic equation given final acid volume of

45.6

liters.

x^2 + 50x - 1400 = 0 \implies (x+70)(x-20) = 0 \implies x = 20

.

Discarding the negative root (

x = -70

) because volume must be positive.

Key Concept

Multi-stage sequential mixture removal and replacement

Question 87Question →

A oceanographic research institute uses three autonomous submersibles��Submersible A, Submersible B, and Submersible C—to perform high-resolution seabed mapping. Working together at their respective constant rates, Submersible A and Submersible B can complete a full mapping mission in $15$ hours. Working together at their respective constant rates, Submersible B and Submersible C can complete the same mapping mission in $24$ hours. During a specialized operation, Submersible A works alone for $6$ hours, after which it is recalled. Submersible B then works alone for $10$ hours, after which Submersible C joins Submersible B, and both work together for an additional $12$ hours to complete the remaining portion of the mission. How many hours would it take Submersible A to complete the entire seabed mapping mission working alone at its constant rate?

18

20

24

30

36

Show answer & explanation

Answer: 24

Answer

24 hours

The option stating 24 is correct because setting up the system of work rates

r_A + r_B = \frac{1}{15}

and

r_B + r_C = \frac{1}{24}

, along with the multi-stage work equation

6r_A + 10r_B + 12(r_B + r_C) = 1

, allows us to eliminate

r_C

and solve for

r_A = \frac{1}{24}

. Taking the reciprocal gives 24 hours.

Step-by-Step Solution

1

Define rate variables and set up combined rate equations.

Let

r_A

,

r_B

, and

r_C

be the hourly rates of Submersibles A, B, and C respectively (in fraction of mission per hour). We are given: (1)

r_A + r_B = \frac{1}{15}

and (2)

r_B + r_C = \frac{1}{24}

.

Work rates are additive reciprocals of completion times for constant-rate work problems.

2

Express total mission work completed in the multi-stage operation.

6r_A + 10r_B + 12(r_B + r_C) = 1

, which simplifies to

6r_A + 22r_B + 12r_C = 1

.

Total work equals the sum of work done across all sequential operational phases.

3

Substitute equation (2) into the multi-stage work equation.

Since

r_C = \frac{1}{24} - r_B

, we get

6r_A + 22r_B + 12\left(\frac{1}{24} - r_B\right) = 1 \implies 6r_A + 10r_B + \frac{1}{2} = 1 \implies 6r_A + 10r_B = \frac{1}{2}

.

Substituting

r_C

eliminates one variable, reducing the system to two equations in

r_A

and

r_B

.

4

Solve the two-variable system for

r_A

.

Multiply equation (1) by

10

to get

10r_A + 10r_B = \frac{10}{15} = \frac{2}{3}

. Subtract

6r_A + 10r_B = \frac{1}{2}

from this to get

4r_A = \frac{2}{3} - \frac{1}{2} = \frac{1}{6} \implies r_A = \frac{1}{24}

.

Eliminating

r_B

isolates Submersible A's rate.

5

Calculate the time taken by Submersible A working alone.

Time

= \frac{1}{r_A} = \frac{1}{1/24} = 24

hours.

Completion time is the reciprocal of the hourly work rate.

Key Concept

System of Work Rate Equations for Multi-Stage Combined Work

Estimated Time:2m 30s

Question 88Question →

A barista has $40$ ounces of a flavored drink mixture that is $30\%$ syrup by volume. How many ounces of pure water must be added to reduce the syrup concentration to $24\%$ by volume?

Show answer & explanation

Answer: 10

Answer

10 ounces of pure water must be added.

Adding 10 ounces of pure water increases the total volume of the mixture from 40 ounces to 50 ounces while keeping the volume of pure syrup constant at 12 ounces. The resulting concentration is 12 / 50 = 0.24, or 24% by volume.

Step-by-Step Solution

1

Calculate the volume of pure syrup present in the initial mixture

12

ounces of syrup

The initial 40-ounce mixture contains 30% syrup by volume (

40 \times 0.30 = 12

).

2

Set up the concentration equation after adding

x

ounces of pure water

\frac{12}{40 + x} = 0.24

Adding pure water increases the total volume to

40 + x

ounces without changing the amount of pure syrup.

3

Solve the linear equation for

x

x = 10

Multiplying both sides by

40 + x

yields

12 = 9.6 + 0.24x

, which simplifies to

2.4 = 0.24x

, giving

x = 10

.

Key Concept

Dilution of a mixture by adding pure solvent

Question 89Question →

In a corporate workshop of $80$ executives, $52$ executives attended the Leadership module and $40$ executives attended the Negotiation module. If $18$ executives attended neither module, how many executives attended both modules?

Show answer & explanation

Answer: 30

Answer

30 executives attended both modules.

Applying the formula

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

yields

80 = 52 + 40 - \text{Both} + 18

. Simplifying gives

80 = 110 - \text{Both}

, so the number of executives who attended both modules is

30

.

Step-by-Step Solution

1

Determine the number of executives who attended at least one of the two modules.

Executives attending at least one module =

80 - 18 = 62

.

Subtracting those who attended neither module from the total gives the union of the two sets.

2

Apply the inclusion-exclusion principle to find the intersection of the two sets.

\text{Both} = \text{Leadership} + \text{Negotiation} - \text{At least one} = 52 + 40 - 62 = 30

.

Summing the participants of each module double-counts those who attended both, so subtracting the number attending at least one module isolates the overlap.

Key Concept

Two-Set Overlapping Sets Formula (Inclusion-Exclusion Principle)

Estimated Time:1m 0s

Question 90Question →

At a software firm, an audit of 200 developers evaluated proficiency in three programming languages: Python, Java, and C++.

- 120 developers are proficient in Python.
- 105 developers are proficient in Java.
- 95 developers are proficient in C++.
- 15 developers are proficient in all three languages.
- 10 developers are proficient in none of the three languages.
- The ratio of the number of developers proficient in BOTH Python and Java ONLY to the number of developers proficient in BOTH Python and C++ ONLY is $3 : 2$ .
- The number of developers proficient in ONLY Java is equal to the number of developers proficient in ONLY C++.

How many developers are proficient in ONLY Python?

Show answer & explanation

Answer: 55

Answer

55 developers are proficient in ONLY Python.

By decomposing the 3-set Venn diagram into 7 mutually exclusive regions, the given ratio

d : e = 3 : 2

gives

d = 3k

and

e = 2k

. Comparing the totals for Java (

105

) and C++ (

95

) under the condition that Java-only equals C++-only shows that

k = 10

. Thus,

30

developers are in Python and Java only,

20

are in Python and C++ only, and

15

are in all three. Subtracting these three regions from the total

120

Python developers yields

120 - 30 - 20 - 15 = 55

proficient in Python only.

Step-by-Step Solution

1

Assign variables to the Venn diagram regions and incorporate given ratios.

Represent

d = |P \cap J \text{ only}| = 3k

and

e = |P \cap C \text{ only}| = 2k

, with

g = 15

and

\text{None} = 10

.

Establishing explicit variables for non-overlapping regions simplifies system solving.

2

Set up set total equations for

|J|

and

|C|

to solve for

k

.

|J| \implies b + 3k + f = 90

and

|C| \implies b + 2k + f = 80

. Subtracting yields

k = 10

.

Since

b = c

(Java only = C++ only), subtracting the two set equations eliminates

b

and

f

, directly giving

k

.

3

Calculate the region values

d

and

e

, then solve for

a = |P \text{ only}|

.

d = 30

,

e = 20

. Then

a = 120 - (30 + 20 + 15) = 55

.

Subtracting all other regions of set

P

from the total proficient in Python isolates those proficient ONLY in Python.

Key Concept

Three-Set Overlapping Sets and Region Decomposition

Question 91Question →

Vessel A contains $40$ kilograms of an organic fertilizer mixture that is $60\%$ Nitrogen by weight. Vessel B contains $60$ kilograms of a fertilizer mixture that is $20\%$ Nitrogen by weight. First, $x$ kilograms of the mixture are removed from Vessel A and transferred into Vessel B, where the contents are thoroughly mixed. Then, $x$ kilograms of the newly formed mixture in Vessel B are transferred back into Vessel A. If the final mixture in Vessel A is $50\%$ Nitrogen by weight, what is the value of $x$ ?

10

12

15

16

20

Show answer & explanation

Answer: $12$

Answer

The value of

x

is

12

.

The correct answer of 12 kg accounts for the dynamic concentration change in Vessel B after the initial transfer. When

x = 12

kg is moved from Vessel A to Vessel B, Vessel B contains 72 kg total mixture with 19.2 kg Nitrogen, giving a concentration of

26.67\%

(or

\frac{4}{15}

). Transferring 12 kg of this mixture back to Vessel A adds

12 \times \frac{4}{15} = 3.2

kg Nitrogen to the 16.8 kg Nitrogen remaining in Vessel A, resulting in exactly 20 kg of Nitrogen in a 40 kg total mixture (

50\%

concentration).

Step-by-Step Solution

1

Calculate the initial mass of Nitrogen in each vessel.

Vessel A initially has

0.60 \times 40 = 24

kg of Nitrogen. Vessel B initially has

0.20 \times 60 = 12

kg of Nitrogen.

Tracking exact solute quantities is required for setting up the mixture conservation equation.

2

Determine the amount of Nitrogen in both vessels after transferring

x

kg from Vessel A to Vessel B.

Vessel A retains

(40 - x)

kg of solution containing

(24 - 0.60x)

kg of Nitrogen. Vessel B now has

(60 + x)

kg of total mixture containing

(12 + 0.60x)

kg of Nitrogen.

The solution removed from Vessel A carries Nitrogen at a concentration of 60%.

3

Express the concentration of Nitrogen in Vessel B prior to the second transfer.

The concentration in Vessel B is

C_B = \frac{12 + 0.60x}{60 + x}

.

Concentration equals total mass of Nitrogen divided by total mass of the mixture in Vessel B.

4

Formulate the equation for the final Nitrogen mass in Vessel A after returning

x

kg from Vessel B.

Final Nitrogen in Vessel A:

(24 - 0.60x) + x \cdot \left(\frac{12 + 0.60x}{60 + x}\right) = 0.50 \times 40 = 20

.

The final volume of Vessel A is restored to 40 kg with a target concentration of 50%.

5

Solve the algebraic equation for

x

.

4 - 0.60x + \frac{12x + 0.60x^2}{60 + x} = 0 \implies (4 - 0.60x)(60 + x) + 12x + 0.60x^2 = 0 \implies 240 + 4x - 36x - 0.60x^2 + 12x + 0.60x^2 = 0 \implies 240 - 20x = 0 \implies x = 12

.

Multiplying through by

(60 + x)

cancels the non-linear

x^2

terms, yielding a linear relation.

Key Concept

Two-Stage Transfer and Dilution

Estimated Time:2m 30s

Question 92Question →

A regional logistics company maintains a fleet consisting of delivery vans, medium trucks, and heavy cargo trucks in the initial ratio of $3 : 4 : 5$ , respectively. After the company retires 4 delivery vans, purchases 12 medium trucks, and purchases 6 heavy cargo trucks, the ratio of delivery vans to heavy cargo trucks becomes $1 : 2$ . What was the total number of vehicles in the fleet initially?

120

144

168

182

196

Show answer & explanation

Answer: 168

Answer

168

The initial ratio of

3 : 4 : 5

implies that the number of vans, medium trucks, and heavy cargo trucks can be represented as

3x

,

4x

, and

5x

for some positive multiplier

x

. The initial total fleet is

3x + 4x + 5x = 12x

. Modifying the counts according to the problem gives

3x - 4

vans and

5x + 6

heavy trucks. Equating their ratio to

1/2

produces

\frac{3x - 4}{5x + 6} = \frac{1}{2}

. Cross-multiplying gives

6x - 8 = 5x + 6

, which simplifies to

x = 14

. Multiplying

14

by the initial sum of ratio units (

12

) yields the correct initial total of

168

.

Step-by-Step Solution

1

Define initial quantities using a common multiplier

x

Delivery vans

= 3x

, Medium trucks

= 4x

, Heavy cargo trucks

= 5x

. Initial total fleet

= 3x + 4x + 5x = 12x

.

Expressing each vehicle count in terms of

x

preserves the given initial ratio of

3 : 4 : 5

.

2

Formulate the equation based on the updated vehicle counts and new ratio

New number of vans

= 3x - 4

. New number of heavy cargo trucks

= 5x + 6

. Equation:

\frac{3x - 4}{5x + 6} = \frac{1}{2}

.

The problem states that after retiring 4 vans and adding 6 heavy cargo trucks, the ratio of vans to heavy cargo trucks becomes

1 : 2

.

3

Solve the algebraic equation for

x

2(3x - 4) = 1(5x + 6) \implies 6x - 8 = 5x + 6 \implies x = 14

.

Cross-multiplying eliminates the fraction and isolates

x

.

4

Calculate the initial total number of vehicles

Initial total

= 12x = 12 \times 14 = 168

.

Substituting

x = 14

into the initial total expression

12x

yields the total fleet size prior to any modifications.

Key Concept

Multi-part ratio scaling and structural algebraic modeling under quantity changes

Question 93Question →

A retailer purchases a jacket for $80. The retailer marks up the purchase price by 25% to establish a list price. During a clearance sale, the jacket is sold at a 10% discount off the list price. What is the retailer's net profit on the sale of the jacket?

$8

$10

$12

$18

$20

Show answer & explanation

Answer: $10

Answer

The retailer's net profit is $10.

To find the net profit, first determine the list price by adding a 25% markup to the cost price of

80, which gives

100. Then apply a 10% discount to the list price of

100, yielding a selling price of

90. Subtracting the initial cost of

80 from

90 gives a net profit of $10.

Step-by-Step Solution

1

Calculate the list price by applying the 25% markup to the cost price.

\text{List Price} = \$80 + (0.25 \times \$80) = \$80 + \$20 = \$100

Markup is calculated as a percentage of the original cost price.

2

Calculate the selling price after applying the 10% discount to the list price.

\text{Selling Price} = \$100 - (0.10 \times \$100) = \$100 - \$10 = \$90

Discounts are calculated based on the list price, not the original cost price.

3

Calculate the net profit by subtracting the cost price from the selling price.

\text{Net Profit} = \text{Selling Price} - \text{Cost Price} = \$90 - \$80 = \$10

Profit is the difference between the final selling price and the initial purchase cost.

Key Concept

Profit, Loss, and Markup with Successive Percentages

Question 94Question →

At a research foundation, annual grant funding was initially allocated among three divisions—Artificial Intelligence, Biotechnology, and Renewable Energy—in the ratio $7 : 4 : 3$ , respectively. At mid-year, the foundation redistributed the funding by transferring $\$10$ million from the Artificial Intelligence division to the Renewable Energy division, while the Biotechnology division's funding remained unchanged. After this transfer, the ratio of Artificial Intelligence funding to Renewable Energy funding became $3 : 2$ . What was the total annual grant funding, in millions of dollars, allocated across all three divisions?

Show answer & explanation

Answer: 140

Answer

The total annual grant funding allocated across all three divisions was

140

million dollars.

By setting the initial funding amounts to

7x

,

4x

, and

3x

million dollars, the total funding is

14x

. The transfer alters Artificial Intelligence funding to

7x - 10

and Renewable Energy funding to

3x + 10

. Setting their ratio to

\frac{3}{2}

yields the equation

2(7x - 10) = 3(3x + 10)

, which simplifies to

5x = 50

, giving

x = 10

. Thus, total funding is

14 \times 10 = 140

million dollars.

Step-by-Step Solution

1

Define variables for the initial allocation based on the ratio

7 : 4 : 3

.

Artificial Intelligence funding

= 7x

, Biotechnology funding

= 4x

, Renewable Energy funding

= 3x

, and total funding

= 7x + 4x + 3x = 14x

.

Ratios represent relative parts of a whole multiplier

x

.

2

Model the redistribution of funds.

New Artificial Intelligence funding

= 7x - 10

; New Renewable Energy funding

= 3x + 10

.

Transferring

\$10

million decreases Artificial Intelligence funding by

10

and increases Renewable Energy funding by

10

.

3

Equate the new ratio to

3 : 2

and solve for

x

.

\frac{7x - 10}{3x + 10} = \frac{3}{2} \implies 14x - 20 = 9x + 30 \implies 5x = 50 \implies x = 10

.

Cross-multiplication converts ratio relationships into a solvable linear equation.

4

Calculate the total annual grant funding.

Total funding

= 14x = 14 \times 10 = 140

million dollars.

Substituting

x = 10

into the total funding expression

14x

gives the final answer.

Key Concept

Algebraic setup of multi-part ratio redistribution problems

Question 95Question →

An agricultural processing plant produces a livestock feed blend consisting of corn, soybeans, and oats in an initial weight ratio of $5 : 3 : 2$ , respectively. To increase the protein content of a $1,200\text{ kg}$ batch of this blend, a technician removes a certain quantity of oats and replaces it with an equal weight of soybeans. If the resulting weight ratio of corn to soybeans to oats becomes $10 : 9 : 1$ , how many kilograms of oats were replaced by soybeans?

120

180

240

540

720

Show answer & explanation

Answer: 180

Answer

180 kg of oats were replaced by soybeans.

The correct answer demonstrates that since corn is unchanged (

600\text{ kg}

), it corresponds to

10

parts in the new

10 : 9 : 1

ratio. This yields

60\text{ kg}

per part in the new ratio. Oats decrease from

240\text{ kg}

(

2

initial parts of

120\text{ kg}

) to

60\text{ kg}

(

1

final part of

60\text{ kg}

), indicating that

180\text{ kg}

of oats were replaced by an equal weight of soybeans.

Step-by-Step Solution

1

Determine the initial weight of each component in the 1,200 kg batch.

The initial ratio is

5 : 3 : 2

, giving a total of

5 + 3 + 2 = 10

parts. Each part corresponds to

\frac{1,200\text{ kg}}{10} = 120\text{ kg}

. Therefore: Corn

= 5 \times 120 = 600\text{ kg}

, Soybeans

= 3 \times 120 = 360\text{ kg}

, Oats

= 2 \times 120 = 240\text{ kg}

.

Finding the absolute initial quantities provides baseline values before the substitution occurs.

2

Analyze the invariant quantity after the substitution.

Since oats are replaced by an equal weight of soybeans, the total weight of the batch remains

1,200\text{ kg}

, and the weight of corn remains unchanged at

600\text{ kg}

.

Identifying unchanged quantities allows direct scaling of the new ratio.

3

Calculate the value of one ratio unit in the new ratio.

The new ratio of corn to soybeans to oats is

10 : 9 : 1

. The

600\text{ kg}

of corn represents

10

parts of the new ratio. Thus,

1\text{ part} = \frac{600\text{ kg}}{10} = 60\text{ kg}

. Alternatively, total parts

= 10 + 9 + 1 = 20

parts, so

1\text{ part} = \frac{1,200\text{ kg}}{20} = 60\text{ kg}

.

Determining the scale of the new ratio allows finding the new weight of each ingredient.

4

Compute the weight of oats replaced.

The new oats weight is

1 \times 60\text{ kg} = 60\text{ kg}

. The replaced quantity is Initial Oats

-

Final Oats

= 240\text{ kg} - 60\text{ kg} = 180\text{ kg}

.

The difference between initial and final oat weights equals the replaced quantity.

Key Concept

Solving multi-part ratio changes using invariant quantities and constant total mass scaling.

Estimated Time:2m 0s

Question 96Question →

A storage tank is completely filled with $150$ liters of a solution containing Compounds X, Y, and Water in a ratio of $2:3:5$ by volume. First, $30$ liters of the mixture are drained and replaced with $30$ liters of a liquid mixture that is $20\%$ Compound Y by volume. Next, $50$ liters of the resulting mixture are drained and replaced with $50$ liters of another liquid mixture that is $70\%$ Compound Y by volume. What is the percentage of Compound Y, by volume, in the final mixture?

Show answer & explanation

Answer: 42

Answer

The final mixture contains 42% of Compound Y by volume.

To find the final concentration, track the exact volume of Compound Y through both stages. Initially, the tank holds

150 \times \frac{3}{10} = 45

liters of Y. In the first replacement, draining

30

liters (

20\%

of the tank) leaves

80\%

of Y, which is

36

liters; adding

30

liters of

20\%

Y mixture adds

6

liters, bringing Y to

42

liters. In the second replacement, draining

50

liters (

\frac{1}{3}

of the tank) leaves

\frac{2}{3}

of Y, which is

28

liters; adding

50

liters of

70\%

Y mixture adds

35

liters, yielding

63

liters of Y. The final concentration is

\frac{63}{150} \times 100\% = 42\%

.

Step-by-Step Solution

1

Calculate initial volume of Compound Y in the tank

Initial Y volume = 45 liters

The ratio X:Y:Water is 2:3:5, giving a total of 2 + 3 + 5 = 10 parts. The fraction of Y is 3/10. For a 150-liter tank, Y = 150 * (3/10) = 45 liters.

2

Calculate Compound Y after the first draining and replacement cycle

Compound Y volume after first replacement = 42 liters

Draining 30 liters removes 30 / 150 = 1/5 of the mixture. Y remaining = 45 * (1 - 1/5) = 36 liters. Replacing with 30 liters of 20% Y adds 30 * 0.20 = 6 liters of Y. Total Y = 36 + 6 = 42 liters.

3

Calculate Compound Y after the second draining and replacement cycle

Compound Y volume after second replacement = 63 liters

Draining 50 liters removes 50 / 150 = 1/3 of the current mixture. Y remaining = 42 * (1 - 1/3) = 28 liters. Replacing with 50 liters of 70% Y adds 50 * 0.70 = 35 liters of Y. Total Y = 28 + 35 = 63 liters.

4

Compute final percentage of Compound Y

42%

The final volume of the tank remains 150 liters. Percentage of Y = (63 / 150) * 100% = 42%.

Key Concept

Multi-Stage Mixture Removal and Replacement

Question 97Question →

A textile manufacturing plant uses two automated weaving looms, Loom $P$ and Loom $Q$ , to produce standardized fabric orders. Working alone at its constant rate, Loom $P$ can complete a full fabric order in $8$ hours, while Loom $Q$ working alone at its constant rate can complete the same order in $12$ hours. Loom $P$ begins working on a full order alone. After $3$ hours, Loom $Q$ joins Loom $P$ , and both looms work together at their respective constant rates until the order is completed. How many total hours does it take, from the moment Loom $P$ begins, to complete the entire fabric order?

3

6

7.5

10

15.5

Show answer & explanation

Answer: $6$

Answer

6

hours

In the first

3

hours, Loom

P

completes

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

of the job, leaving

\frac{5}{8}

of the job remaining. When Loom

Q

joins, their combined rate is

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

of the job per hour. The time needed to complete the remaining

\frac{5}{8}

of the job is

\frac{5/8}{5/24} = 3

hours. Adding the initial

3

hours of solo work gives a total of

6

hours.

Step-by-Step Solution

1

Determine the individual hourly work rates of Loom

P

and Loom

Q

.

Rate of Loom

P = \frac{1}{8}

order/hour; Rate of Loom

Q = \frac{1}{12}

order/hour.

Work rate is the reciprocal of the total time needed to complete one full unit of work.

2

Calculate the fraction of the order completed by Loom

P

during its

3

hours of solo operation.

Work completed by Loom

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

of the order.

Work done equals rate multiplied by time spent working.

3

Find the remaining fraction of the order to be completed.

Remaining work =

1 - \frac{3}{8} = \frac{5}{8}

of the order.

The total job represents

1

whole unit of work.

4

Calculate the combined work rate when both looms operate together.

Combined rate =

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

order/hour.

When entities work together, their individual rates add up.

5

Determine the time required for both looms together to finish the remaining work.

Time together =

\frac{5/8}{5/24} = \frac{5}{8} \times \frac{24}{5} = 3

hours.

Time equals remaining work divided by the combined work rate.

6

Calculate the total time from start to finish.

Total time =

3 \text{ hours (solo)} + 3 \text{ hours (combined)} = 6 \text{ hours}

.

The total elapsed time is the sum of the solo operating time and the combined operating time.

Key Concept

Work Rate and Combined Work

Estimated Time:2m 0s

Question 98Question →

A delivery van travels from Warehouse A to Warehouse B at a constant speed of $40$ miles per hour and returns along the exact same route from Warehouse B to Warehouse A at a constant speed of $60$ miles per hour. If the distance between Warehouse A and Warehouse B is $120$ miles, what is the average speed of the delivery van for the entire round trip, in miles per hour?

Show answer & explanation

Answer: 48

Answer

The average speed for the entire round trip is

48

miles per hour.

To find the average speed for the round trip, divide the total distance (

240

miles) by the total time taken (

5

hours). The time taken for the first leg is

\frac{120}{40} = 3

hours, and for the return leg is

\frac{120}{60} = 2

hours. Thus, the average speed is

\frac{240}{5} = 48

miles per hour.

Step-by-Step Solution

1

Determine the total distance traveled during the round trip.

The distance from Warehouse A to B is

120

miles, making the total round-trip distance

120 + 120 = 240

miles.

Average speed requires the total distance for all legs of the journey.

2

Calculate time taken for each leg and find total time.

Time taken at

40

mph is

\frac{120}{40} = 3

hours. Time taken at

60

mph is

\frac{120}{60} = 2

hours. Total time =

3 + 2 = 5

hours.

Time equals distance divided by rate (

t = \frac{d}{r}

).

3

Divide total distance by total time to obtain average speed.

Average speed =

\frac{240 \text{ miles}}{5 \text{ hours}} = 48

miles per hour.

The defining formula for average speed is

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

.

Key Concept

Average speed for a multi-leg journey is always total distance divided by total time, not the arithmetic mean of the speeds.

Question 99Question →

At an intellectual property law firm, a senior partner reviewed $180$ patent applications across three technical domains: Artificial Intelligence, Biotechnology, and Clean Energy. The audit revealed that $25$ applications belonged to none of these three domains. Furthermore, $90$ applications involved Artificial Intelligence, $85$ involved Biotechnology, and $80$ involved Clean Energy. If the number of applications that belonged to exactly two of these domains was three times the number of applications that belonged to all three domains, how many patent applications belonged to exactly one domain?

Show answer & explanation

Answer: 75

Answer

The total number of patent applications that belonged to exactly one domain is

75

.

The total number of applications in at least one domain is

180 - 25 = 155

. Let

E_1

be the number of applications in exactly one domain,

E_2

in exactly two domains, and

x

in all three domains. The total union is

E_1 + E_2 + x = 155

, while the sum of individual set sizes is

|A| + |B| + |C| = E_1 + 2E_2 + 3x = 90 + 85 + 80 = 255

. Subtracting the union equation from the sum equation gives

E_2 + 2x = 100

. Given

E_2 = 3x

, substituting gives

5x = 100 \implies x = 20

, which implies

E_2 = 60

. Finally, subtracting

E_2

and

x

from the total union yields

E_1 = 155 - 60 - 20 = 75

.

Step-by-Step Solution

1

Calculate the total number of applications in at least one domain

|A \cup B \cup C| = 180 - 25 = 155

Subtracting the applications belonging to none of the domains from the overall total yields the total number of unique applications covered by the three domains combined.

2

Formulate regional Venn diagram equations

E_1 + E_2 + x = 155

and

E_1 + 2E_2 + 3x = 255

Summing individual set counts counts elements in exactly two domains twice and elements in all three domains three times.

3

Deduce the relationship between

E_2

and

x

E_2 + 2x = 100

Subtracting the equation for total union from the sum of individual sets isolates the overcounted regions.

4

Use the given proportion

E_2 = 3x

to solve for

x

and

E_2

x = 20

and

E_2 = 60

Substituting

E_2 = 3x

into

E_2 + 2x = 100

gives

5x = 100

, so

x = 20

and

E_2 = 60

.

5

Find the number of applications belonging to exactly one domain (

E_1

)

E_1 = 75

Subtracting the count of applications in exactly two domains (

60

) and all three domains (

20

) from the total union (

155

) gives

155 - 60 - 20 = 75

.

Key Concept

Three-Set Venn Diagram Region Partitioning

Question 100Question →

In a software development department, the numbers of Frontend developers, Backend developers, and DevOps engineers were initially in the ratio of $4 : 5 : 3$ , respectively. After the department hired $6$ additional DevOps engineers, with no changes to the number of Frontend or Backend developers, the ratio of Backend developers to DevOps engineers became $1 : 1$ . What is the total number of developers and engineers in the department after these new hires?

30

36

42

48

78

Show answer & explanation

Answer: 42

Answer

The total number of developers and engineers in the department after the new hires is 42.

By setting the initial counts of Frontend developers, Backend developers, and DevOps engineers as

4x

,

5x

, and

3x

, adding 6 to DevOps yields

3x + 6

. Equating Backend and DevOps counts gives

5x = 3x + 6

, so

x = 3

. Substituting

x = 3

into the post-hire counts gives 12 Frontend, 15 Backend, and 15 DevOps, summing to a total of 42 employees.

Step-by-Step Solution

1

Define initial quantities using a common ratio multiplier.

Let the initial numbers of Frontend developers, Backend developers, and DevOps engineers be

4x

,

5x

, and

3x

, respectively.

Ratios allow representing unknown totals in terms of a single multiplier

x

.

2

Express the updated counts after hiring new DevOps engineers.

Frontend =

4x

, Backend =

5x

, and DevOps =

3x + 6

.

Only the DevOps group increases by 6 while other group sizes remain unchanged.

3

Set up an equation based on the new Backend-to-DevOps ratio of

1 : 1

.

5x = 3x + 6 \implies 2x = 6 \implies x = 3

.

A

1 : 1

ratio means the number of Backend developers equals the new number of DevOps engineers.

4

Calculate the final total number of employees.

Frontend =

4(3) = 12

, Backend =

5(3) = 15

, DevOps =

3(3) + 6 = 15

. Total =

12 + 15 + 15 = 42

.

Summing all three roles after adding the 6 new hires gives the final count.

Key Concept

Ratio scaling and setting up algebraic equations from modified multi-part ratios.

Estimated Time:1m 30s

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