Word Problems and Applied Math Alıştırma Soruları — GMAT — Sayfa 3

Soru 41Soru →

A commercial bank's loan portfolio consists of three categories of loans: small business loans, residential mortgages, and commercial real estate loans. Small business loans carry an average annual interest rate of $8.5\%$ and account for $30\%$ of the portfolio's total value. Residential mortgages carry an average annual interest rate of $5.0\%$ , and commercial real estate loans carry an average annual interest rate of $6.5\%$ . If the overall weighted average annual interest rate for the entire portfolio is $6.35\%$ , what percentage of the portfolio's total value is invested in commercial real estate loans?

15\%

20\%

35\%

40\%

50\%

Cevabı ve açıklamayı göster

Cevap: $20\%$

Cevap

The commercial real estate loans account for

20\%

of the portfolio's total value.

The overall portfolio rate of

6.35\%

is a weighted average of small business loans (

8.5\%

rate,

30\%

weight), residential mortgages (

5.0\%

rate,

x\%

weight), and commercial real estate loans (

6.5\%

rate,

(70-x)\%

weight). Solving

0.30(8.5) + 0.05x + 0.065(70 - x) = 6.35

gives

x = 50\%

, which means commercial real estate loans constitute

70\% - 50\% = 20\%

of the total portfolio.

Adım Adım Çözüm

1

Define variables for the unknown portfolio weight components.

Let

w_1 = 0.30

(small business loans),

w_2 = x

(residential mortgages), and

w_3 = 0.70 - x

(commercial real estate loans).

The sum of all weights across the three portfolio categories must equal

100\%

(

1.00

). Since small business loans constitute

30\%

, the remaining two categories must sum to

70\%

(

0.70

).

2

Set up the weighted average formula for the overall portfolio interest rate.

0.30(8.5) + x(5.0) + (0.70 - x)(6.5) = 6.35

The overall weighted average interest rate is the sum of each component's rate multiplied by its respective weight in the portfolio.

3

Expand and simplify the algebraic equation.

2.55 + 5.0x + 4.55 - 6.5x = 6.35 \implies 7.10 - 1.5x = 6.35

Multiplying out terms allows combining like terms to solve for

x

.

4

Solve for

x

to find the weight of residential mortgages, then find the weight of commercial real estate loans.

1.5x = 0.75 \implies x = 0.50

(

50\%

). Thus, commercial real estate weight

= 0.70 - 0.50 = 0.20

(

20\%

).

Subtracting

x = 50\%

from the total remaining

70\%

gives the exact share allocated to commercial real estate loans.

Anahtar Kavram

Weighted Averages in Multi-Component Portfolios

Soru 42Soru →

A private equity firm allocated its total investment capital between two tech startups, Company X and Company Y. Over the first year, the value of the investment in Company X increased by $60\%$ , while the value of the investment in Company Y increased by $20\%$ . Over the second year, the value of Company X decreased by $25\%$ from its Year 1 value, while the value of Company Y increased by $25\%$ from its Year 1 value. If the total combined value of the two investments at the end of the second year was $38\%$ greater than the original total investment capital allocated, what percent of the original total investment capital was allocated to Company X?

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

40%

Evaluating successive percent changes gives growth multipliers of

1.60 \times 0.75 = 1.20

for Company X and

1.20 \times 1.25 = 1.50

for Company Y. Setting the weighted sum of final values equal to

1.38(X + Y)

yields

1.20X + 1.50Y = 1.38X + 1.38Y

, which simplifies to

0.18X = 0.12Y

or

Y = 1.5X

. The proportion allocated to Company X is

\frac{X}{X + 1.5X} = \frac{1}{2.5} = 40\%

.

Adım Adım Çözüm

1

Calculate the cumulative growth multiplier for Company X over the two-year period.

Company X's final value is

1.20X

, representing a net increase of

20\%

.

Successive percent changes are calculated by multiplying the growth factors:

(1 + 0.60) \times (1 - 0.25) = 1.60 \times 0.75 = 1.20

.

2

Calculate the cumulative growth multiplier for Company Y over the two-year period.

Company Y's final value is

1.50Y

, representing a net increase of

50\%

.

Successive percent changes are calculated by multiplying the growth factors:

(1 + 0.20) \times (1 + 0.25) = 1.20 \times 1.25 = 1.50

.

3

Set up an equation for the combined final portfolio value relative to initial capital.

1.20X + 1.50Y = 1.38(X + Y)

.

The problem states that the combined final value is

38\%

greater than the total initial capital

X + Y

.

4

Simplify the equation to express

Y

in terms of

X

.

0.12Y = 0.18X \implies Y = 1.5X

.

Expanding

1.38X + 1.38Y

and collecting like terms yields

1.50Y - 1.38Y = 1.38X - 1.20X

.

5

Determine Company X's proportion of the total initial allocation.

\frac{X}{X + 1.5X} = \frac{1}{2.5} = 40\%

.

The initial percentage allocated to Company X is

\frac{X}{X + Y} \times 100\%

.

Anahtar Kavram

Weighted Successive Percent Change

Soru 43Soru →

A medical distributor purchased a lot of $60$ identical diagnostic kits for a total cost of $\$4,800$ . The distributor marked up the cost price of each kit by $50\%$ to set its regular retail price. After selling $40$ kits at the regular retail price, the distributor sold the remaining $20$ kits at a clearance discount of $30\%$ off the regular retail price. What was the distributor's total net profit, in dollars, from the sale of all $60$ diagnostic kits?

Cevabı ve açıklamayı göster

Cevap: 1680

Cevap

The distributor's total net profit from the sale of all 60 diagnostic kits is $1,680.

To calculate net profit, first determine the unit cost:

\frac{\$4,800}{60} = \$80

. The marked-up regular price is

\$80 \times 1.50 = \$120

. Selling

40

kits at

\$120

yields

\$4,800

. The remaining

20

kits are sold at a

30\%

discount off

\$120

, giving a clearance price of

\$120 \times 0.70 = \$84

per kit and revenue of

20 \times \$84 = \$1,680

. Total revenue is

\$4,800 + \$1,680 = \$6,480

. Subtracting the total cost of

\$4,800

yields a net profit of

\$1,680

.

Adım Adım Çözüm

1

Determine the cost price per unit.

Cost per kit is

\frac{\$4,800}{60} = \$80

.

Dividing total initial cost by total units gives the unit cost price.

2

Calculate the regular retail price per unit.

Regular retail price is

\$80 \times (1 + 0.50) = \$120

.

The

50\%

markup is applied directly to the cost price of

\$80

.

3

Calculate revenue from the full-price sales segment.

Revenue from

40

kits is

40 \times \$120 = \$4,800

.

Multiply quantity sold at full price by regular retail price.

4

Determine clearance price and revenue for remaining units.

Clearance price is

\$120 \times (1 - 0.30) = \$84

. Revenue from

20

kits is

20 \times \$84 = \$1,680

.

The clearance discount of

30\%

is taken off the regular retail price, not cost.

5

Calculate total revenue and net profit.

Total revenue is

\$4,800 + \$1,680 = \$6,480

. Net profit is

\$6,480 - \$4,800 = \$1,680

.

Net profit equals total revenue minus total cost.

Anahtar Kavram

Profit, Loss, and Markup

Soru 44Soru →

A freight boat travels downstream along a river from Dock A to Dock B in $4$ hours. On the return trip upstream from Dock B to Dock A, engine trouble causes the boat's speed in still water to decrease by $25\%$ for the entire return trip. If the speed of the river current is a constant $3$ miles per hour and the upstream return trip takes $8$ hours, what is the distance, in miles, between Dock A and Dock B?

Cevabı ve açıklamayı göster

Cevap: 84

Cevap

The distance between Dock A and Dock B is 84 miles.

The correct distance of 84 miles is determined by accounting for both the constant river current of 3 mph and the 25% reduction in still-water boat speed on the return trip. Setting the downstream distance formula equal to the upstream distance formula yields a still-water initial speed of 18 mph, giving a one-way distance of 84 miles.

Adım Adım Çözüm

1

Define variables for the boat's speed in still water and write expressions for effective speeds.

Let

v

be the initial speed of the boat in still water (in mph). The downstream speed is

(v + 3)

mph. Due to a

25\%

reduction on the return leg, the boat's upstream speed in still water is

0.75v

mph, making the effective upstream speed

(0.75v - 3)

mph.

Effective speed downstream equals still-water speed plus current speed, while effective speed upstream equals still-water speed minus current speed.

2

Set up distance equations for both directions using

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

.

Downstream distance:

D = 4(v + 3)

. Upstream distance:

D = 8(0.75v - 3)

.

The distance between Dock A and Dock B is the same in both directions.

3

Equate the two distance expressions and solve for

v

.

4(v + 3) = 8(0.75v - 3) \implies v + 3 = 1.5v - 6 \implies 0.5v = 9 \implies v = 18

mph.

Equating the two representations of distance allows for solving the single unknown variable

v

.

4

Calculate the total distance

D

.

D = 4(18 + 3) = 4(21) = 84

miles.

Substituting

v = 18

back into the downstream distance formula yields the exact distance.

Anahtar Kavram

Rate, Time, and Distance with Currents and Variable Speeds

Tahmini Süre:2m 30s

Soru 45Soru →

A commercial real estate company manages three office buildings: Building A, Building B, and Building C. Building A contains $20,000$ square feet of rentable space leased at an average rate of $\$45$ per square foot per year. Building B contains $30,000$ square feet of rentable space leased at an average rate of $\$50$ per square foot per year. Building C contains $50,000$ square feet of rentable space. If the overall weighted average rental rate for all three buildings combined is $\$52$ per square foot per year, what is the average annual rental rate per square foot, in dollars, for Building C?

Cevabı ve açıklamayı göster

Cevap: 56

Cevap

The average annual rental rate for Building C is

56

dollars per square foot.

To find the average annual rental rate for Building C, determine the total revenue generated by all three buildings combined. With a total square footage of

20,000 + 30,000 + 50,000 = 100,000

sq ft and an overall weighted average rate of

\$52

/sq ft, the total revenue is

100,000 \times 52 = 5,200,000

dollars. Subtracting the revenue generated by Building A (

20,000 \times 45 = 900,000

dollars) and Building B (

30,000 \times 50 = 1,500,000

dollars) leaves

5,200,000 - 2,400,000 = 2,800,000

dollars that must be contributed by Building C. Dividing this by Building C's

50,000

square feet yields an average rate of

56

dollars per square foot.

Adım Adım Çözüm

1

Determine total combined rentable area and total combined revenue

Total area

= 20,000 + 30,000 + 50,000 = 100,000

sq ft. Total revenue

= 100,000 \times 52 = 5,200,000

dollars.

Weighted average rate multiplied by total square footage gives total annual rental revenue across all properties.

2

Calculate known annual revenues from Building A and Building B

Building A revenue

= 20,000 \times 45 = 900,000

dollars. Building B revenue

= 30,000 \times 50 = 1,500,000

dollars. Combined A and B revenue

= 2,400,000

dollars.

Multiplying individual component weights by their respective rates yields individual group totals.

3

Subtract known revenues from total revenue and divide by Building C's square footage

Building C revenue

= 5,200,000 - 2,400,000 = 2,800,000

dollars. Building C rate

= \frac{2,800,000}{50,000} = 56

dollars.

Isolating the remaining revenue requirement and dividing by Building C's area yields its average rental rate per square foot.

Anahtar Kavram

Weighted Average = Total Weighted Sum / Total Weight Sum

Soru 46Soru →

A commercial machinery supplier purchased a batch of industrial generators. The supplier marked up the wholesale cost of each generator by $40\%$ to set the original retail price. During an end-of-quarter promotion, the supplier offered a $15\%$ discount off the original retail price. If a customer purchased a generator at the promotional price for $\$4,760$ , what was the supplier's profit, in dollars, on that sale?

Cevabı ve açıklamayı göster

Cevap: 760

Cevap

The supplier's profit on the sale was $760.

Let

C

represent the wholesale cost of a generator. A

40\%

markup yields an original retail price of

1.40C

. Applying a

15\%

discount to this retail price produces a promotional selling price of

0.85 \times 1.40C = 1.19C

. Setting

1.19C = 4,760

gives

C = 4,000

. The profit is the promotional selling price minus wholesale cost:

\$4,760 - \$4,000 = \$760

.

Adım Adım Çözüm

1

Define the relationship between wholesale cost and original retail price.

Original Retail Price =

1.40 \times C

A

40\%

markup on wholesale cost

C

increases the price to

140\%

of

C

.

2

Apply the discount to calculate the promotional selling price.

Selling Price =

0.85 \times (1.40 \times C) = 1.19 \times C

A

15\%

discount off the retail price leaves

85\%

of the retail price.

3

Solve for the wholesale cost

C

.

C = \frac{4,760}{1.19} = 4,000

Given that the final selling price is

\$4,760

, set

1.19C = 4,760

and divide.

4

Calculate the profit made on the sale.

Profit =

\$4,760 - \$4,000 = \$760

Profit equals total revenue (selling price) minus total cost.

Anahtar Kavram

Successive markup and discount calculations with distinct base values.

Soru 47Soru →

An investor deposited a principal sum of money into an account that earns simple annual interest at a rate of $r\%$ . At the end of $4$ years, the total accumulated balance in the account (principal plus interest) was $\$7,200$ . If the account had instead earned simple annual interest at a rate of $(r + 2)\%$ over the same $4$ -year period, the total accumulated balance would have been $\$7,680$ . What was the original principal amount deposited, in dollars?

Cevabı ve açıklamayı göster

Cevap: 6000

Cevap

The original principal amount deposited was

6,000

dollars.

The difference in accumulated balances over

4

years is

\$7,680 - \$7,200 = \$480

. Since simple interest is calculated strictly on the original principal

P

, an annual rate increase of

2\%

yields an additional

2\%

of

P

each year. Over

4

years, this total rate increase is

4 \times 2\% = 8\%

. Therefore,

8\%

of

P

equals

\$480

, giving

0.08P = 480

, which yields

P = 6,000

dollars.

Adım Adım Çözüm

1

Calculate the difference between the two final balances.

\$7,680 - \$7,200 = \$480

The additional

\$480

represents the total extra simple interest earned over

4

years due to the higher interest rate.

2

Determine the cumulative percentage rate increase over the 4-year period.

4 \text{ years} \times 2\% \text{ per year} = 8\%

Simple interest is calculated solely on the original principal for each year.

3

Set up and solve the equation for the principal amount

P

.

8\% \text{ of } P = \$480 \implies 0.08P = 480 \implies P = 6,000

Dividing the additional dollar amount by the additional percentage yields the original base principal.

Anahtar Kavram

Simple Interest and Rate Sensitivity

Tahmini Süre:1m 30s

Soru 48Soru →

A boutique investment fund allocates its capital among Technology, Healthcare, and Renewable Energy in the ratio $3 : 4 : 5$ , respectively. After receiving an additional $\$12$ million in new capital, the entire new amount is invested in Healthcare, changing the ratio of capital among Technology, Healthcare, and Renewable Energy to $3 : 6 : 5$ . What was the total initial capital, in millions of dollars, invested in the fund?

24

72

84

36

60

Cevabı ve açıklamayı göster

Cevap: $72$

Cevap

72

million dollars

Let the initial capital allocations be

3x

,

4x

, and

5x

million dollars for Technology, Healthcare, and Renewable Energy, respectively. The initial total capital is

3x + 4x + 5x = 12x

. When

\$12

million is added to Healthcare, its new amount becomes

4x + 12

. Comparing the unchanged Technology amount to the updated Healthcare amount gives the proportion

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

. Cross-multiplying yields

6x = 4x + 12

, so

2x = 12

and

x = 6

. The total initial capital is

12 \times 6 = 72

million dollars.

Adım Adım Çözüm

1

Represent initial allocations using a multiplier

x

Technology

= 3x

, Healthcare

= 4x

, Renewable Energy

= 5x

Ratios represent proportional parts, so multiplying each term by a common constant

x

expresses the actual amounts.

2

Set up the equation following the addition of

\$12

million to Healthcare

New Healthcare allocation

= 4x + 12

. The new ratio is

3x : (4x + 12) : 5x = 3 : 6 : 5

.

Only the Healthcare amount changes, while Technology and Renewable Energy allocations remain

3x

and

5x

respectively.

3

Solve for

x

using the ratio between Technology and Healthcare

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

Equating the algebraic ratio of Technology to Healthcare to the new simplified ratio allows solving for

x

.

4

Calculate the total initial capital

\text{Total initial capital} = 3x + 4x + 5x = 12x = 12(6) = 72

Summing the initial parts (

12x

) and substituting

x = 6

yields the initial total.

Anahtar Kavram

Ratio scaling and constant-part proportion adjustments

Tahmini Süre:1m 30s

Soru 49Soru →

Three industrial machines, $A$ , $B$ , and $C$ , operating independently at their respective constant rates, can complete a production order of $1,200$ units in $12$ hours, $15$ hours, and $20$ hours, respectively. All three machines start working simultaneously on the order at 8:00 AM. At 10:00 AM, Machine $A$ experiences a mechanical failure and stops working permanently. Machine $B$ and Machine $C$ continue working together until 11:00 AM, at which point Machine $B$ 's operating efficiency drops by $50\%$ due to overheating, while Machine $C$ continues at its original rate. At what time will the production order of $1,200$ units be fully completed?

4:48 PM

3:09 PM

5:12 PM

4:18 PM

6:30 PM

Cevabı ve açıklamayı göster

Cevap: 4:48 PM

Cevap

The production order will be completed at 4:48 PM.

The correct answer of 4:48 PM is derived by dividing the job into three distinct stages. From 8:00 AM to 10:00 AM (2 hours), Machines A, B, and C produce 240 units/hour, yielding 480 units. From 10:00 AM to 11:00 AM (1 hour), Machines B and C produce 140 units/hour, yielding 140 units, leaving 580 units remaining. After 11:00 AM, Machine B operates at 40 units/hour and Machine C at 60 units/hour, producing a combined rate of 100 units/hour. Dividing 580 remaining units by 100 units/hour gives 5.8 hours (5 hours 48 minutes), placing completion at exactly 4:48 PM.

Adım Adım Çözüm

1

Determine individual production rates for each machine.

Machine

A

produces

\frac{1,200}{12} = 100

units/hour. Machine

B

produces

\frac{1,200}{15} = 80

units/hour. Machine

C

produces

\frac{1,200}{20} = 60

units/hour.

Establishing hourly rates is necessary to calculate work completed in each time interval.

2

Calculate work completed during Stage 1 (8:00 AM to 10:00 AM).

Combined rate of

A + B + C = 100 + 80 + 60 = 240

units/hour. In

2

hours, work completed

= 240 \times 2 = 480

units. Remaining units

= 1,200 - 480 = 720

units.

All three machines work together for 2 full hours before Machine A breaks down.

3

Calculate work completed during Stage 2 (10:00 AM to 11:00 AM).

Combined rate of

B + C = 80 + 60 = 140

units/hour. In

1

hour, work completed

= 140 \times 1 = 140

units. Remaining units

= 720 - 140 = 580

units.

Machine A is inactive, leaving only Machines B and C operating at full capacity for 1 hour.

4

Calculate the time required for Stage 3 (from 11:00 AM until completion).

Machine

B

's reduced rate

= 80 \times 0.50 = 40

units/hour. New combined rate of

B + C = 40 + 60 = 100

units/hour. Time needed

= \frac{580}{100} = 5.8

hours

= 5

hours and

48

minutes.

Machine B operates at half speed while Machine C continues at full speed to finish the remaining 580 units.

5

Determine the final completion clock time.

11:00 AM

+ 5

hours and

48

minutes

= 4:48

PM.

Adding the duration of the final stage to the starting time of 11:00 AM gives the exact completion time.

Anahtar Kavram

Multi-stage combined work problems require calculating individual rates, tracking partial work completed in each phase, and adjusting combined rates whenever active entities or their individual rates change.

Tahmini Süre:3m 0s

Soru 50Soru →

An investment fund initially divides its total capital among three portfolios—Real Estate, Stocks, and Bonds—in the ratio $3 : 4 : 5$ . During a financial restructuring, a portion of capital is transferred directly from the Bonds portfolio to the Real Estate portfolio while the capital in Stocks remains unchanged, resulting in a new ratio of Real Estate to Stocks of $5 : 4$ . Subsequently, an additional $\$60,000$ in external capital is deposited into the Bonds portfolio, making the ratio of Stocks to Bonds $2 : 3$ . What was the total initial capital in the fund across all three portfolios?

$180,000

$200,000

$240,000

$300,000

$720,000

Cevabı ve açıklamayı göster

Cevap: $240,000

Cevap

The total initial capital in the fund was $240,000.

The correct answer of

240,000 is obtained by tracking each portfolio's balance across the two sequential changes. Initially, Real Estate, Stocks, and Bonds are

3x

,

4x

, and

5x

respectively (total

12x

). When Real Estate increases to

5x

relative to Stocks (

4x

), the required capital comes entirely from Bonds, reducing Bonds from

5x

to

3x

. Adding

\

60,000

to Bonds gives a new balance of

3x + 60,000

. Establishing the new Stocks to Bonds ratio

\frac{4x}{3x + 60,000} = \frac{2}{3}

yields

x = 20,000

. Multiplying by the initial total of 12 parts yields

12 \times 20,000 =

240,000$.

Adım Adım Çözüm

1

Express initial portfolio amounts in terms of a multiplier

x

Real Estate =

3x

, Stocks =

4x

, Bonds =

5x

. Total initial capital =

3x + 4x + 5x = 12x

.

The given initial ratio of Real Estate : Stocks : Bonds is

3 : 4 : 5

.

2

Account for the internal capital transfer between Bonds and Real Estate

Real Estate becomes

5x

, Stocks remains

4x

, and Bonds becomes

3x

.

Stocks remains

4x

. Since the ratio of Real Estate to Stocks becomes

5 : 4

, Real Estate must equal

5x

. Because capital was only moved internally from Bonds to Real Estate, the sum of Real Estate and Bonds remains

3x + 5x = 8x

, leaving Bonds with

8x - 5x = 3x

.

3

Set up an algebraic proportion after adding

\$60,000

to Bonds

\frac{4x}{3x + 60,000} = \frac{2}{3}

The new ratio of Stocks to Bonds is given as

2 : 3

.

4

Solve for the multiplier

x

and determine the total initial capital

3(4x) = 2(3x + 60,000) \Rightarrow 12x = 6x + 120,000 \Rightarrow 6x = 120,000 \Rightarrow x = 20,000

. Total initial capital

= 12x = 12 \times 20,000 =

240,000$.

Cross-multiplying yields the value of

x

, which is then scaled by the total initial ratio sum of 12 parts.

Anahtar Kavram

Multi-Step Ratio Adjustments and Internal vs. External Capital Changes

Tahmini Süre:2m 0s

Soru 51Soru →

Printer Alpha can fabricate a set of architectural prototypes in $4$ hours operating alone at a constant rate. Printer Beta can fabricate the exact same set of prototypes in $12$ hours operating alone at a constant rate. If both printers operate simultaneously at their respective constant rates, how many hours will it take them to fabricate one set of architectural prototypes?

16

hours

8

hours

3

hours

6

hours

4.8

hours

Cevabı ve açıklamayı göster

Cevap: $3$ hours

Cevap

3

hours

The correct answer is

3

hours. The rate of Printer Alpha is

\frac{1}{4}

of the prototype set per hour, and the rate of Printer Beta is

\frac{1}{12}

of the set per hour. Their combined rate when operating simultaneously is

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

set per hour. Therefore, the time required to complete

1

set working together is

\frac{1}{\frac{1}{3}} = 3

hours.

Adım Adım Çözüm

1

Determine the individual work rate for Printer Alpha.

Printer Alpha completes

\frac{1}{4}

of the job per hour.

Work rate is defined as job completed divided by time taken.

2

Determine the individual work rate for Printer Beta.

Printer Beta completes

\frac{1}{12}

of the job per hour.

Work rate is defined as job completed divided by time taken.

3

Add the individual rates to find the combined work rate.

Combined rate =

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

of the job per hour.

When entities work together, their individual rates add up.

4

Calculate the total time required for the combined rate to complete 1 job.

Time =

\frac{1}{\text{Combined Rate}} = \frac{1}{\frac{1}{3}} = 3

hours.

Time is the reciprocal of the combined rate.

Anahtar Kavram

Combined Work Rate

Tahmini Süre:45s

Soru 52Soru →

Data Pipeline A can process a standard batch of records in $12$ hours, while Data Pipeline B can process the same batch of records in $18$ hours. Data Pipeline A begins processing a batch alone and operates for $4$ hours before Data Pipeline B is brought online to assist. Both pipelines then continue processing 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 records from start to finish?

4.8

6.4

8.8

15.0

24.0

Cevabı ve açıklamayı göster

Cevap: $8.8$

Cevap

8.8

hours (or

8\frac{4}{5}

hours)

The rate of Pipeline A is

\frac{1}{12}

per hour, so in

4

hours it completes

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

of the job. The remaining

\frac{2}{3}

of the job is completed by both pipelines working together. Their combined rate is

\frac{1}{12} + \frac{1}{18} = \frac{5}{36}

per hour. The time required for the joint phase is

\frac{2/3}{5/36} = 4.8

hours. Adding the initial

4

hours gives a total time of

8.8

hours.

Adım Adım Çözüm

1

Calculate the work completed by Data Pipeline A alone during the first

4

hours.

Pipeline A's rate is

\frac{1}{12}

batch/hour. Work done in

4

hours =

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

of the batch.

Determining the completed portion establishes how much work remains for the combined phase.

2

Determine the remaining fraction of work to be done.

Remaining work =

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

of the batch.

The combined pipelines only need to finish the remaining portion of the job.

3

Calculate the combined processing rate of both pipelines.

Combined rate =

\frac{1}{12} + \frac{1}{18} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36}

batch/hour.

Rates add when entities work simultaneously on the same task.

4

Calculate the time required for both pipelines to finish the remaining work together.

Joint time =

\frac{\text{Remaining Work}}{\text{Combined Rate}} = \frac{2/3}{5/36} = \frac{2}{3} \times \frac{36}{5} = \frac{24}{5} = 4.8

hours.

Dividing remaining work by the combined rate gives the duration of the second phase.

5

Calculate total time from start to finish.

Total time =

4 \text{ hours (phase 1)} + 4.8 \text{ hours (phase 2)} = 8.8

hours.

Summing both phase durations gives the total time required for the entire process.

Anahtar Kavram

Work Rate and Combined Work

Tahmini Süre:2m 0s

Soru 53Soru →

At a software company, the ratio of senior developers to junior developers was initially $3 : 7$ . During a quarterly hiring push, $12$ senior developers and $18$ junior developers were hired, but $6$ junior developers resigned shortly thereafter. As a result, the ratio of senior developers to junior developers became $4 : 9$ . If a certain number $N$ of senior developers were subsequently promoted out of the developer pool such that the ratio of senior developers to junior developers became $5 : 12$ , what is the value of $N$ ?

8

12

15

18

24

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The value of

N

is

12

.

By defining the initial counts as

3x

and

7x

, accounting for the net additions (

+12

seniors and

+12

juniors), and solving

\frac{3x+12}{7x+12} = \frac{4}{9}

, we find

x = 60

. Thus, there are

192

senior developers and

432

junior developers after hiring. Setting the final ratio

\frac{192-N}{432} = \frac{5}{12}

gives

192 - N = 180

, which yields

N = 12

.

Adım Adım Çözüm

1

Define initial quantities using a common multiplier

x

.

Initial senior developers =

3x

, Initial junior developers =

7x

.

The given initial ratio of senior to junior developers is

3 : 7

.

2

Account for changes in staffing and set up the equation for the new ratio.

Senior developers after hiring =

3x + 12

; Net junior developer change =

+18 - 6 = +12

, so Junior developers after hiring =

7x + 12

. Equation:

\frac{3x + 12}{7x + 12} = \frac{4}{9}

.

12 seniors were added, while 18 juniors were hired and 6 resigned.

3

Solve for the multiplier

x

.

9(3x + 12) = 4(7x + 12) \implies 27x + 108 = 28x + 48 \implies x = 60

.

Cross-multiplication yields the value of the algebraic multiplier

x

.

4

Calculate post-hiring developer counts.

Senior developers post-hiring =

3(60) + 12 = 192

; Junior developers post-hiring =

7(60) + 12 = 432

.

Substitute

x = 60

back into the expressions for post-hiring developer counts.

5

Determine

N

using the final ratio

5 : 12

.

\frac{192 - N}{432} = \frac{5}{12} \implies 192 - N = 5 \times 36 = 180 \implies N = 192 - 180 = 12

.

Junior developer count remains unchanged (

432

) while senior developer count decreases by

N

.

Anahtar Kavram

Multi-step algebraic ratio setup and alteration of part-to-part proportions

Soru 54Soru →

Scanner X can scan a batch of archival documents in $6$ hours operating alone at a constant rate. Scanner Y can scan the exact same batch of documents in $3$ hours operating alone at a constant rate. Working together simultaneously at their respective constant rates, how many hours will it take Scanner X and Scanner Y to scan the entire batch of documents?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

It will take Scanner X and Scanner Y a total of

2

hours to scan the batch of documents together.

To calculate combined completion time, add the rates of each machine rather than their completion times. Scanner X completes

\frac{1}{6}

of the task per hour, and Scanner Y completes

\frac{1}{3}

(or

\frac{2}{6}

) of the task per hour. Together, they complete

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

of the task per hour. The inverse of this rate gives the total time required:

2

hours.

Adım Adım Çözüm

1

Calculate individual work rates per hour.

Scanner X rate =

\frac{1}{6}

batch/hour, Scanner Y rate =

\frac{1}{3}

batch/hour.

Work rate is defined as

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

.

2

Add the individual rates together to find the total combined rate.

Combined rate =

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

batch/hour.

When entities work together simultaneously, their work rates are additive.

3

Compute total time taken for 1 whole batch.

Total Time =

2

hours.

Time is the reciprocal of the combined work rate:

\text{Time} = \frac{1}{\text{Combined Rate}}

.

Anahtar Kavram

Combined Work Rate

Tahmini Süre:45s

Soru 55Soru →

Three automated data-processing pipelines, $A$ , $B$ , and $C$ , process incoming dataset files at constant individual rates. Pipeline $A$ operating alone can complete a standard processing job in $10$ hours, while Pipeline $B$ operating alone can complete the same job in $15$ hours.

The processing begins with Pipelines $A$ and $B$ working together at their standard rates for $2$ hours. At that point, Pipeline $A$ encounters a network slowdown that reduces its processing rate by $50\%$ for the remainder of the job. Pipelines $A$ and $B$ continue working at these modified rates for another $2$ hours, after which Pipeline $C$ is brought online to assist them. If Pipeline $C$ operating alone at its constant rate could finish the entire job in $10$ hours, how many total hours from the start of the processing will it take for the job to be completed?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The total time required to complete the job is 6 hours.

To solve multi-stage work rate problems, determine the rate per unit of time for each participant, multiply by the duration of each operational phase, and track the remaining fraction of the job. In the first 2 hours, Pipelines

A

and

B

complete

\frac{1}{3}

of the job. In the next 2 hours with Pipeline

A

at half speed (rate

\frac{1}{20}

), they complete an additional

\frac{7}{30}

of the job, leaving

\frac{13}{30}

unfinished. When Pipeline

C

(rate

\frac{1}{10}

) joins, the combined rate becomes

\frac{13}{60}

per hour, taking exactly 2 more hours to complete the remaining

\frac{13}{30}

of the job. Adding all three intervals (

2 + 2 + 2

) yields a total time of 6 hours.

Adım Adım Çözüm

1

Determine the individual hourly work rates for each pipeline

Rate of Pipeline

A = \frac{1}{10}

job/hr, Rate of Pipeline

B = \frac{1}{15}

job/hr, Rate of Pipeline

C = \frac{1}{10}

job/hr.

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

2

Calculate work completed during Stage 1 (first 2 hours)

Combined rate =

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

job/hr. Work done =

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

of the job.

Both Pipeline

A

and Pipeline

B

operate together at their standard rates for 2 hours.

3

Calculate work completed during Stage 2 (next 2 hours)

Modified rate of Pipeline

A = 0.50 \times \frac{1}{10} = \frac{1}{20}

job/hr. Combined rate =

\frac{1}{20} + \frac{1}{15} = \frac{7}{60}

job/hr. Work done =

2 \times \frac{7}{60} = \frac{7}{30}

of the job.

Pipeline

A

's efficiency drops by 50%, while Pipeline

B

remains unchanged.

4

Calculate the remaining work after 4 hours

Total work completed =

\frac{1}{3} + \frac{7}{30} = \frac{17}{30}

. Remaining work =

1 - \frac{17}{30} = \frac{13}{30}

.

Subtracting cumulative work done from the total job (1) leaves the remaining portion.

5

Calculate the duration of Stage 3 when Pipeline

C

joins

Combined rate =

\frac{1}{20} + \frac{1}{15} + \frac{1}{10} = \frac{13}{60}

job/hr. Time required =

\frac{13/30}{13/60} = 2

hours.

Dividing remaining work by the combined rate of all three active pipelines gives the remaining time.

6

Sum the durations of all stages to find total elapsed time

Total time =

2\text{ hours} + 2\text{ hours} + 2\text{ hours} = 6\text{ hours}

.

The question asks for total elapsed time from the start of processing.

Anahtar Kavram

Multi-stage combined work rate problems require tracking accumulated work completed across distinct intervals with changing worker rates.

Soru 56Soru →

Pumps X, Y, and Z, operating simultaneously at their respective constant rates, can fill an empty reservoir in $4$ hours. Operating alone at its constant rate, Pump X can fill $\frac{1}{5}$ of the reservoir in $2$ hours. The rate at which Pump Y fills the reservoir is $50\%$ greater than the rate at which Pump Z fills the reservoir. If Pump X and Pump Z work together for $3$ hours, and then Pump Y is turned on so that all three pumps work together until the reservoir is full, how many additional hours will it take to finish filling the remainder of the reservoir?

1 \frac{18}{25}

hours

1 \frac{23}{25}

hours

2 \frac{2}{25}

hours

2 \frac{1}{2}

hours

2 \frac{4}{5}

hours

Cevabı ve açıklamayı göster

Cevap: $2 \frac{2}{25}$ hours

Cevap

2 \frac{2}{25}

hours

The rate of Pump X is

\frac{1}{10}

reservoir/hour, and the combined rate of all three pumps is

\frac{1}{4}

reservoir/hour. Thus, the combined rate of Pumps Y and Z is

\frac{1}{4} - \frac{1}{10} = \frac{3}{20}

reservoir/hour. Given

r_y = 1.5 r_z

, setting

\frac{5}{2} r_z = \frac{3}{20}

yields

r_z = \frac{3}{50}

and

r_y = \frac{9}{100}

. The combined rate of Pumps X and Z is

\frac{1}{10} + \frac{3}{50} = \frac{4}{25}

. In

3

hours, Pumps X and Z complete

3 \times \frac{4}{25} = \frac{12}{25}

of the reservoir, leaving

\frac{13}{25}

un-filled. When all three pumps operate, their combined rate is

\frac{1}{4}

. The additional time required is

\frac{13/25}{1/4} = \frac{52}{25} = 2 \frac{2}{25}

hours.

Adım Adım Çözüm

1

Determine the individual rate of Pump X and the total combined rate of all three pumps.

Pump X's rate

r_x = \frac{1/5}{2} = \frac{1}{10}

of the reservoir per hour. The total combined rate of Pumps X, Y, and Z is

r_x + r_y + r_z = \frac{1}{4}

of the reservoir per hour.

Work rate is equal to work completed divided by time elapsed.

2

Determine the individual work rates of Pump Y and Pump Z.

The combined rate of Y and Z is

r_y + r_z = \frac{1}{4} - \frac{1}{10} = \frac{3}{20}

. Since

r_y = 1.5 r_z = \frac{3}{2} r_z

, we have

\frac{5}{2} r_z = \frac{3}{20}

, which gives

r_z = \frac{3}{50}

and

r_y = \frac{9}{100}

.

Use the given relationship between the rates of Pump Y and Pump Z to solve for their individual values.

3

Calculate the work completed during the first 3 hours by Pumps X and Z working together, and find the remaining work.

The combined rate of X and Z is

r_x + r_z = \frac{1}{10} + \frac{3}{50} = \frac{8}{50} = \frac{4}{25}

. In 3 hours, they complete

3 \times \frac{4}{25} = \frac{12}{25}

of the reservoir. The remaining work is

1 - \frac{12}{25} = \frac{13}{25}

.

Work done equals combined rate multiplied by time worked.

4

Calculate the additional time required for all three pumps working together to finish the remaining work.

Time

t = \frac{\text{Remaining Work}}{\text{Combined Rate of X, Y, Z}} = \frac{13/25}{1/4} = \frac{52}{25} = 2 \frac{2}{25}

hours.

Time needed is remaining work divided by the total combined rate of all active pumps.

Anahtar Kavram

Work Rate and Combined Work

Tahmini Süre:2m 30s

Soru 57Soru →

A beverage recipe requires mixing orange juice and cranberry juice in a ratio of $4 : 1$ . If a caterer uses $12$ liters of orange juice, how many liters of cranberry juice must be added to maintain this ratio?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The caterer must add 3 liters of cranberry juice.

The ratio of orange juice to cranberry juice is

4:1

. Since

12

liters of orange juice represents

4

parts, each part equals

\frac{12}{4} = 3

liters. Therefore,

1

part of cranberry juice equals

3

liters.

Adım Adım Çözüm

1

Set up the ratio equation based on the recipe requirements.

Orange Juice : Cranberry Juice =

4 : 1

The problem states that for every 4 parts of orange juice, 1 part of cranberry juice is required.

2

Calculate the unknown amount of cranberry juice using the given quantity of orange juice.

Cranberry Juice =

\frac{12}{4} = 3

liters

Since 12 liters represents 4 equal parts, dividing 12 by 4 yields the size of 1 part.

Anahtar Kavram

Direct Ratio and Proportion Scaling

Soru 58Soru →

A corporate marketing team initially allocates its quarterly budget among Digital, Print, and Event advertising in the ratio $5 : 3 : 2$ , respectively. Midway through the quarter, $\$12,000$ is reallocated from Digital advertising to Event advertising, while the Print advertising budget remains unchanged. As a result of this transfer, the ratio of the Digital advertising budget to the Event advertising budget becomes $4 : 3$ . What was the total initial quarterly budget, in dollars, allocated across all three advertising categories?

Cevabı ve açıklamayı göster

Cevap: 120000

Cevap

The total initial quarterly budget allocated across all three categories was 120,000 dollars.

By representing the initial amounts as

5x

,

3x

, and

2x

, the total initial budget is

10x

. Accounting for the

\$12,000

shift gives

\frac{5x - 12,000}{2x + 12,000} = \frac{4}{3}

. Solving yields

x = 12,000

, which gives a total initial budget of

10 \times 12,000 = 120,000

dollars.

Adım Adım Çözüm

1

Define initial allocations using a multiplier variable

Digital =

5x

, Print =

3x

, Event =

2x

, Total Initial Budget =

10x

Representing ratio components algebraically allows set up of linear equations.

2

Adjust allocations according to the budget transfer

New Digital =

5x - 12,000

, New Event =

2x + 12,000

Reallocating

\$12,000

reduces Digital by 12,000 and increases Event by 12,000.

3

Set up and solve the ratio equation for

x

\frac{5x - 12,000}{2x + 12,000} = \frac{4}{3} \Rightarrow 15x - 36,000 = 8x + 48,000 \Rightarrow 7x = 84,000 \Rightarrow x = 12,000

Equating the altered ratio of Digital to Event to

4/3

allows solving for multiplier

x

.

4

Compute the total initial budget

Total =

10 \times 12,000 = 120,000

Substituting

x = 12,000

back into the total initial expression

10x

gives the final answer.

Anahtar Kavram

Altering non-isolated ratios through part-to-part modifications

Soru 59Soru →

At a bio-tech research facility, the ratio of male to female scientists is initially $5 : 3$ . After $4$ female scientists leave and $12$ male scientists join the facility, the ratio of male to female scientists becomes $4 : 1$ . If a group of $N$ additional female scientists subsequently joins the facility, changing the ratio of male to female scientists to $8 : 5$ , what is the value of $N$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The value of

N

is 12.

By setting up ratio equations stage-by-stage using multiplier

k

, we determine that

k = 4

, yielding 32 male and 8 female scientists after the first change. Solving

\frac{32}{8+N} = \frac{8}{5}

yields

N = 12

.

Adım Adım Çözüm

1

Define initial variables using a common ratio multiplier

k

.

Male scientists =

5k

, Female scientists =

3k

.

Ratios preserve proportionality through a constant multiplier.

2

Formulate and solve the equation for the first population adjustment.

\frac{5k + 12}{3k - 4} = 4 \implies 5k + 12 = 12k - 16 \implies 7k = 28 \implies k = 4

.

Adding 12 males and removing 4 females sets the new ratio to

4 : 1

.

3

Determine the exact number of male and female scientists at the facility after the first adjustment.

Male scientists =

5(4) + 12 = 32

; Female scientists =

3(4) - 4 = 8

.

Substitute

k = 4

back into the adjusted quantity expressions.

4

Formulate and solve the equation for the second population adjustment involving

N

.

\frac{32}{8 + N} = \frac{8}{5} \implies 8(8 + N) = 160 \implies 64 + 8N = 160 \implies 8N = 96 \implies N = 12

.

The number of male scientists remains 32 while female scientists increase by

N

to achieve an

8 : 5

ratio.

Anahtar Kavram

Multi-step ratio manipulation and algebraic modeling

Tahmini Süre:2m 0s

Soru 60Soru →

A commercial bakery has two industrial ovens, Oven A and Oven B. Working alone at its constant rate, Oven A can bake a full order of $1,800$ pastries in $6$ hours. Working alone at its constant rate, Oven B can bake the same order in $9$ hours. If Oven A begins baking the order alone and is joined by Oven B after $2$ hours, how many total hours will it take from the time Oven A starts until the full order of $1,800$ pastries is baked?

Cevabı ve açıklamayı göster

Cevap: 4.4

Cevap

The total time required to complete the entire order of pastries is 4.4 hours.

The total time of 4.4 hours is derived by determining Oven A's rate (1/6 job/hour) and Oven B's rate (1/9 job/hour). In the first 2 hours, Oven A completes 1/3 of the job, leaving 2/3. Operating together, their combined rate is 5/18 job/hour, which finishes the remaining 2/3 of the job in 2.4 hours. Summing 2 hours and 2.4 hours yields 4.4 total hours.

Adım Adım Çözüm

1

Determine the individual work rates per hour.

Oven A completes

\frac{1}{6}

of the job per hour; Oven B completes

\frac{1}{9}

of the job per hour.

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

2

Calculate the fraction of work completed during the first 2 hours by Oven A.

Oven A completes

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

of the total job.

Work done equals rate multiplied by time spent working.

3

Find the remaining fraction of work to be done.

Remaining work is

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

of the job.

Subtract the completed fraction from 1 (the whole job).

4

Calculate the combined rate of Oven A and Oven B working together.

Combined rate is

\frac{1}{6} + \frac{1}{9} = \frac{3}{18} + \frac{2}{18} = \frac{5}{18}

job per hour.

When entities work together, their individual rates add up.

5

Calculate the time spent by both ovens working together to complete the remaining work.

Time together is

\frac{2/3}{5/18} = \frac{12}{5} = 2.4

hours.

Time equals remaining work divided by the combined rate.

6

Calculate total time from the start.

Total time =

2 + 2.4 = 4.4

hours.

Add the initial single-agent work duration to the combined work duration.

Anahtar Kavram

Work Rate and Combined Work

Word Problems and Applied Math

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