Ratios, Rates, and Proportions Alıştırma Soruları — GRE General Test — Sayfa 3

Soru 41Soru →

Pumps $A$ and $B$ work independently at their respective constant rates to empty a water storage reservoir. Working together, Pump $A$ and Pump $B$ can empty the reservoir in $12$ hours. If Pump $A$ 's rate of pumping is $1.5$ times the rate of Pump $B$ , how many hours would it take Pump $B$ working alone at its constant rate to empty the entire reservoir?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

It would take Pump B working alone 30 hours to empty the entire reservoir.

Since Pump A's rate is

1.5

times Pump B's rate, their combined rate is

1.5 r_B + r_B = 2.5 r_B

reservoirs per hour. Working together for

12

hours empties

12 \times 2.5 r_B = 30 r_B

reservoirs. Setting

30 r_B = 1

reservoir yields

r_B = \frac{1}{30}

reservoir per hour. Therefore, Pump B working alone requires

30

hours to empty the reservoir.

Adım Adım Çözüm

1

Express Pump A's rate in terms of Pump B's rate.

Let

r_B

be the rate of Pump B in reservoirs per hour. Then the rate of Pump A is

r_A = 1.5 r_B

.

Expressing both rates in terms of a single variable simplifies setting up the combined rate equation.

2

Calculate the combined pumping rate.

Combined rate =

r_A + r_B = 1.5 r_B + r_B = 2.5 r_B

.

When two machines work simultaneously, their individual rates add together.

3

Set up the work equation using the given combined time of 12 hours.

\text{Total Work} = \text{Combined Rate} \times 12 = 2.5 r_B \times 12 = 30 r_B = 1 \text{ reservoir}

.

The total work performed equals the combined rate multiplied by the total time taken.

4

Solve for the time required for Pump B to complete the job alone.

\text{Time}_B = \frac{1}{r_B} = 30 \text{ hours}

.

The time needed for Pump B alone to empty

1

reservoir at rate

r_B

is the reciprocal of

r_B

.

Anahtar Kavram

Combined Work Rates and Rate Ratios

Alternatif Yöntem

Choose a convenient hypothetical capacity for the reservoir. Let Pump B's rate be

2

units per hour. Then Pump A's rate is

1.5 \times 2 = 3

units per hour. Working together, they empty

2 + 3 = 5

units per hour. In

12

hours, the reservoir holds

5 \times 12 = 60

units. Pump B alone at

2

units per hour takes

60 / 2 = 30

hours.

Tahmini Süre:1m 30s

Soru 42Soru →

A delivery truck completes a trip consisting of two consecutive legs. The ratio of the distance of the first leg to the distance of the second leg is $3 : 1$ . The ratio of the truck's constant speed on the first leg to its constant speed on the second leg is $3 : 2$ . If the average speed of the truck for the entire trip is $40\text{ miles per hour}$ , what was the constant speed of the truck on the second leg, in miles per hour?

16\text{ mph}

25\text{ mph}

30\text{ mph}

32\text{ mph}

45\text{ mph}

Cevabı ve açıklamayı göster

Cevap: $30\text{ mph}$

Cevap

30\text{ mph}

The total distance traveled is

3d + d = 4d

. The time required for the first leg is

\frac{3d}{3v} = \frac{d}{v}

, and the time for the second leg is

\frac{d}{2v}

. Combining these yields a total time of

\frac{3d}{2v}

. Setting total distance over total time equal to the given average speed gives

\frac{4d}{\frac{3d}{2v}} = \frac{8v}{3} = 40\text{ mph}

, which yields

v = 15

. Consequently, the constant speed on the second leg is

2v = 30\text{ mph}

.

Adım Adım Çözüm

1

Represent the distances and speeds of both legs using ratio multipliers

Distance leg 1 =

3d

, distance leg 2 =

d

, total distance =

4d

. Speed leg 1 =

3v

, speed leg 2 =

2v

.

Ratios

3:1

for distance and

3:2

for speed allow expressed quantities in terms of single parameters

d

and

v

.

2

Calculate the travel time for each leg using time = distance / speed

Time for leg 1:

t_1 = \frac{3d}{3v} = \frac{d}{v}

. Time for leg 2:

t_2 = \frac{d}{2v}

.

Average speed depends on total distance divided by total time.

3

Compute total time for the entire trip

T_{\text{total}} = t_1 + t_2 = \frac{d}{v} + \frac{d}{2v} = \frac{3d}{2v}

.

Finding a common denominator of

2v

allows combining the two time fractions.

4

Set up the average speed equation and solve for multiplier v

\text{Average Speed} = \frac{4d}{\frac{3d}{2v}} = \frac{8v}{3} = 40 \implies v = 15.

Dividing total distance by total time gives the average speed expression.

5

Calculate the speed of the second leg

Speed of leg 2 =

2v = 2(15) = 30\text{ mph}

.

The question specifically asks for the constant speed on the second leg.

Anahtar Kavram

Average rate over multi-leg trips is total distance divided by total time, not the simple arithmetic mean of speeds.

Tahmini Süre:1m 30s

Soru 43Soru →

A civil engineer creates a scale drawing of a rectangular park using a scale where $2.5\text{ inches}$ on the drawing represents an actual distance of $12\text{ feet}$ . On this initial blueprint, the park has a length of $15\text{ inches}$ and a width of $10\text{ inches}$ . If the engineer creates a second blueprint of the same park using a scale where $3\text{ inches}$ represents an actual distance of $16\text{ feet}$ , what is the perimeter, in inches, of the park on the second blueprint?

22.5\text{ inches}

37.5\text{ inches}

40.5\text{ inches}

45\text{ inches}

60\text{ inches}

Cevabı ve açıklamayı göster

Cevap: $45\text{ inches}$

Cevap

45 inches

The correct answer is

45\text{ inches}

. The perimeter on the first blueprint is

2 \times (15 + 10) = 50\text{ inches}

. Since

2.5\text{ inches}

represents

12\text{ feet}

, each inch on the first drawing represents

\frac{12}{2.5} = 4.8\text{ feet}

. Thus, the actual perimeter of the park is

50 \times 4.8 = 240\text{ feet}

. On the second blueprint,

16\text{ feet}

is represented by

3\text{ inches}

, meaning

1\text{ foot}

is represented by

\frac{3}{16}\text{ inches}

. Therefore, the perimeter on the second drawing is

240 \times \frac{3}{16} = 45\text{ inches}

.

Adım Adım Çözüm

1

Calculate the perimeter of the park on the first blueprint.

Perimeter on first blueprint =

2 \times (15 + 10) = 50\text{ inches}

.

The perimeter of a rectangle is calculated as

2 \times (\text{length} + \text{width})

.

2

Determine the actual perimeter of the park in feet using the first scale (

2.5\text{ inches} = 12\text{ feet}

).

Actual perimeter =

50\text{ inches} \times \left(\frac{12\text{ feet}}{2.5\text{ inches}}\right) = 240\text{ feet}

.

Multiplying by the ratio of actual feet per blueprint inch gives the true physical perimeter.

3

Convert the actual perimeter to inches on the second blueprint using the second scale (

3\text{ inches} = 16\text{ feet}

).

Perimeter on second blueprint =

240\text{ feet} \times \left(\frac{3\text{ inches}}{16\text{ feet}}\right) = 45\text{ inches}

.

Multiplying the actual distance in feet by the second blueprint scale ratio yields the required blueprint perimeter.

Anahtar Kavram

Scale ratios and linear proportional conversions

Tahmini Süre:1m 30s

Soru 44Soru →

An industrial bakery uses two commercial mixers, Mixer X and Mixer Y, to prepare large batches of dough. Working continuously at its own constant rate, Mixer X can complete 1 full batch of dough in 4 hours. Mixer Y, working continuously at its own constant rate, can complete 1 full batch of dough in 6 hours.

If Mixer X works alone for 1 hour and is then joined by Mixer Y to complete the rest of the batch together, which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

The remaining

\frac{3}{4}

of the batch takes 1 hour and 48 minutes for both mixers working together to complete.

Mixer X completes

70\%

of the total batch of dough.

The ratio of the total amount of dough prepared by Mixer X to that prepared by Mixer Y is

7 : 3

.

Both mixers working together complete the remaining

\frac{3}{4}

of the batch in 3 hours and 45 minutes.

Mixer Y completes

40\%

of the total batch of dough.

Cevabı ve açıklamayı göster

Cevap: The remaining $\frac{3}{4}$ of the batch takes 1 hour and 48 minutes for both mixers working together to complete.; Mixer X completes $70\%$ of the total batch of dough.; The ratio of the total amount of dough prepared by Mixer X to that prepared by Mixer Y is $7 : 3$ .

Cevap

The correct statements are those indicating that the remaining work takes 1 hour and 48 minutes, Mixer X completes 70% of the batch, and the ratio of work done by Mixer X to Mixer Y is 7 : 3.

The rate of Mixer X is 1/4 batch per hour and Mixer Y is 1/6 batch per hour. In the first hour, Mixer X completes 1/4 of the batch, leaving 3/4 of the batch. Working together, their combined rate is 1/4 + 1/6 = 5/12 batch per hour. The time needed for the remaining 3/4 batch is (3/4) / (5/12) = 1.8 hours, which equals 1 hour and 48 minutes. Mixer X works for a total of 2.8 hours, completing 2.8 / 4 = 0.7 (or 70%) of the batch. Mixer Y works for 1.8 hours, completing 1.8 / 6 = 0.3 (or 30%) of the batch. The ratio of dough prepared by Mixer X to Mixer Y is 0.7 : 0.3 = 7 : 3. Therefore, the statements asserting 1 hour and 48 minutes remaining time, 70% completed by Mixer X, and a 7 : 3 work ratio are all correct.

Adım Adım Çözüm

1

Determine individual work rates and initial work done

Mixer X rate =

\frac{1}{4}

batch/hr, Mixer Y rate =

\frac{1}{6}

batch/hr. In 1 hour, Mixer X completes

\frac{1}{4}

of the batch, leaving

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

of the batch remaining.

Establishing the individual rates allows calculation of remaining work after Mixer X operates alone.

2

Calculate combined rate and time to finish remaining work

Combined rate =

\frac{1}{4} + \frac{1}{6} = \frac{3+2}{12} = \frac{5}{12}

batch/hr. Time required =

\frac{3/4}{5/12} = \frac{3}{4} \times \frac{12}{5} = \frac{9}{5} = 1.8

hours = 1 hour 48 minutes.

When both mixers operate together, their rates add.

3

Calculate total work contributions of each mixer

Mixer X total time =

1 + 1.8 = 2.8

hours

\rightarrow

Work by X =

2.8 \times \frac{1}{4} = 0.7

(

70\%

). Mixer Y total time =

1.8

hours

\rightarrow

Work by Y =

1.8 \times \frac{1}{6} = 0.3

(

30\%

). Work ratio X : Y =

0.7 : 0.3 = 7 : 3

.

Multiplying each mixer's active operating time by its rate determines its share of the total batch.

Anahtar Kavram

Combined Work Rates and Ratio of Work Completed

Soru 45Soru →

A laboratory technician prepares a chemical mixture containing water, ethanol, and acid in the volume ratio $5 : 3 : 2$ , respectively. After $12\text{ liters}$ of pure ethanol are added to the mixture, the ratio of the volume of water to the volume of ethanol becomes $1 : 1$ . What was the total volume, in liters, of the chemical mixture before the extra ethanol was added?

60\text{ liters}

48\text{ liters}

72\text{ liters}

18\text{ liters}

30\text{ liters}

Cevabı ve açıklamayı göster

Cevap: $60\text{ liters}$

Cevap

60\text{ liters}

The initial ratio of water, ethanol, and acid is

5 : 3 : 2

, which means the original amounts can be expressed as

5x

,

3x

, and

2x

liters, respectively, giving a total initial volume of

10x

liters. Adding

12\text{ liters}

of ethanol increases the ethanol amount to

3x + 12

, while water remains

5x

. Since the new ratio of water to ethanol is

1 : 1

, we set

5x = 3x + 12

, which solves to

2x = 12

and

x = 6

. The original total volume is therefore

10(6) = 60\text{ liters}

.

Adım Adım Çözüm

1

Define initial volumes using a common ratio multiplier

x

Water volume

= 5x

, Ethanol volume

= 3x

, Acid volume

= 2x

, and Total initial volume

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

.

Ratios represent relative parts of a whole, so multiplying each term by

x

gives the actual quantities.

2

Express the new volumes after adding

12\text{ liters}

of pure ethanol

New Ethanol volume

= 3x + 12

, while Water volume remains

5x

.

Ethanol is increased by

12\text{ liters}

, whereas the amount of water remains unchanged.

3

Set up an equation using the new water-to-ethanol ratio of

1 : 1

5x = 3x + 12 \implies 2x = 12 \implies x = 6

.

A ratio of

1 : 1

means the volumes of water and ethanol are equal.

4

Calculate the original total volume of the mixture

Total initial volume

= 10x = 10(6) = 60\text{ liters}

.

Substituting

x = 6

back into the original total volume expression

10x

yields the required initial volume.

Anahtar Kavram

Multi-component ratio modification and proportional equivalence

Tahmini Süre:1m 30s

Soru 46Soru →

A digital publishing company uses two high-speed printing presses, Press P and Press Q. Press P prints pages at a constant rate that is $40\%$ faster than the rate of Press Q. If Press P and Press Q work simultaneously at their respective constant rates, they can complete a printing job of $18,000$ pages in $5$ hours. How many hours would it take Press Q, working alone at its constant rate, to complete a job of $15,000$ pages?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

Working together, Press P and Press Q complete

18,000

pages in

5

hours, which corresponds to a combined rate of

3,600

pages per hour. Because Press P's rate is

1.4

times Press Q's rate, their combined rate is

2.4

times Press Q's rate. Dividing

3,600

by

2.4

gives Press Q's individual rate of

1,500

pages per hour. Finally, dividing

15,000

pages by

1,500

pages per hour yields

10

hours.

Adım Adım Çözüm

1

Relate the rate of Press P to Press Q

r_P = 1.4 r_Q

Press P is 40% faster than Press Q, so its rate is

1 + 0.40 = 1.4

times the rate of Press Q.

2

Calculate the combined rate expression

r_{\text{combined}} = 2.4 r_Q

When working together, their rates add:

r_P + r_Q = 1.4 r_Q + r_Q = 2.4 r_Q

.

3

Solve for the rate of Press Q (

r_Q

)

r_Q = 1,500

pages per hour

Using

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

, we have

18,000 = (2.4 r_Q) \times 5 = 12 r_Q

. Dividing

18,000

by

12

yields

r_Q = 1,500

.

4

Determine the time required for Press Q to complete

15,000

pages

10

hours

Dividing the target workload by Press Q's rate gives

\frac{15,000\text{ pages}}{1,500\text{ pages/hour}} = 10

hours.

Anahtar Kavram

Combined Work Rates and Direct Proportions

Tahmini Süre:1m 30s

Ratios, Rates, and Proportions

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