Ratios, Rates, and Proportions Alıştırma Soruları — GRE General Test — Sayfa 2

Soru 21Soru →

A textile mill uses two weaving looms, Loom $P$ and Loom $Q$ , to produce a fabric blend composed of silk, wool, and cotton.

- Loom $P$ produces silk, wool, and cotton in the ratio $2 : 3 : 5$ by weight, operating at a constant output rate of $120\text{ kg per hour}$ .
- Loom $Q$ produces silk, wool, and cotton in the ratio $1 : 4 : 3$ by weight, operating at a constant output rate of $160\text{ kg per hour}$ .

If both looms operate simultaneously for 5 hours, what is the ratio of the total weight of wool produced to the total weight of cotton produced in the combined output?

29 : 30

7 : 8

32 : 35

29 : 70

30 : 29

Cevabı ve açıklamayı göster

Cevap: $29 : 30$

Cevap

The ratio of the total weight of wool produced to the total weight of cotton produced is

29 : 30

.

To find the overall ratio of wool to cotton, determine the actual mass of each material produced per hour by each machine. Loom

P

outputs

\frac{3}{10} \times 120 = 36\text{ kg}

of wool and

\frac{5}{10} \times 120 = 60\text{ kg}

of cotton per hour. Loom

Q

outputs

\frac{4}{8} \times 160 = 80\text{ kg}

of wool and

\frac{3}{8} \times 160 = 60\text{ kg}

of cotton per hour. Combining both looms yields

116\text{ kg}

of wool and

120\text{ kg}

of cotton per hour. Over 5 hours, the combined output is

580\text{ kg}

of wool and

600\text{ kg}

of cotton. The ratio of total wool to total cotton is

580 : 600

, which simplifies to

29 : 30

.

Adım Adım Çözüm

1

Calculate the hourly production rates of wool and cotton for Loom

P

.

Loom

P

produces fabric at

120\text{ kg/hr}

with ratio parts

2 + 3 + 5 = 10

.
- Wool rate from

P = \frac{3}{10} \times 120 = 36\text{ kg/hr}

- Cotton rate from

P = \frac{5}{10} \times 120 = 60\text{ kg/hr}

Converting the ratio into component rates by multiplying each component's fraction by Loom

P

's total hourly output.

2

Calculate the hourly production rates of wool and cotton for Loom

Q

.

Loom

Q

produces fabric at

160\text{ kg/hr}

with ratio parts

1 + 4 + 3 = 8

.
- Wool rate from

Q = \frac{4}{8} \times 160 = 80\text{ kg/hr}

- Cotton rate from

Q = \frac{3}{8} \times 160 = 60\text{ kg/hr}

Converting the ratio into component rates by multiplying each component's fraction by Loom

Q

's total hourly output.

3

Determine total quantities produced over the 5-hour period.

Total wool

= (36 + 80) \times 5 = 116 \times 5 = 580\text{ kg}

.
Total cotton

= (60 + 60) \times 5 = 120 \times 5 = 600\text{ kg}

.

Combining the hourly outputs from both machines and multiplying by the total duration of 5 hours.

4

Compute and simplify the ratio of total wool to total cotton.

\text{Ratio} = \frac{580}{600} = \frac{29}{30}

, which is expressed as

29 : 30$.

Dividing both terms of the ratio by their greatest common divisor, 20.

Anahtar Kavram

Combining weighted rates across multiple ratio-based sub-components

Tahmini Süre:2m 0s

Soru 22Soru →

A chemical processing tank receives two liquid solutions, Solution $X$ and Solution $Y$ , from separate inlet pipes.

- Solution $X$ contains chemical $A$ and water in a volume ratio of $3:2$ and enters the tank at a constant rate of $150$ liters per hour.
- Solution $Y$ contains chemical $A$ and water in a volume ratio of $1:4$ and enters the tank at a constant rate of $250$ liters per hour.

Both inlet pipes run simultaneously into an initially empty tank for $4$ hours. After $4$ hours, the inlet pipes are shut off. To adjust the mixture, pure chemical $A$ is added to the tank at a constant rate of $50$ liters per hour, while water is continuously drained from the tank at a constant rate of $30$ liters per hour.

How many hours must this adjustment process run until the volume of chemical $A$ in the tank is equal to the volume of water in the tank?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

To find the time

t

when the volumes of chemical A and water in the tank are equal, first determine the initial quantities contributed by both solutions during the 4-hour filling period. Solution X provides

150 \times 4 = 600

liters total, containing

\frac{3}{5} \times 600 = 360

liters of chemical A and

\frac{2}{5} \times 600 = 240

liters of water. Solution Y provides

250 \times 4 = 1000

liters total, containing

\frac{1}{5} \times 1000 = 200

liters of chemical A and

\frac{4}{5} \times 1000 = 800

liters of water. Adding these amounts yields

360 + 200 = 560

liters of chemical A and

240 + 800 = 1040

liters of water. In the adjustment phase of

t

hours, chemical A increases at

50

L/hr to

560 + 50t

, while water decreases at

30

L/hr to

1040 - 30t

. Equating the two expressions gives

560 + 50t = 1040 - 30t

, which simplifies to

80t = 480

, resulting in

t = 6

hours.

Adım Adım Çözüm

1

Determine the volumes of chemical A and water supplied by Solution X during the first 4 hours.

Solution X delivers 600 liters in total, consisting of 360 liters of chemical A and 240 liters of water.

Solution X flows at 150 L/hr for 4 hours (150 * 4 = 600 L) with a 3:2 chemical A to water ratio, meaning chemical A represents 3/5 of the total volume and water represents 2/5.

2

Determine the volumes of chemical A and water supplied by Solution Y during the first 4 hours.

Solution Y delivers 1000 liters in total, consisting of 200 liters of chemical A and 800 liters of water.

Solution Y flows at 250 L/hr for 4 hours (250 * 4 = 1000 L) with a 1:4 chemical A to water ratio, meaning chemical A represents 1/5 of the total volume and water represents 4/5.

3

Calculate the total initial quantities of chemical A and water present in the tank prior to the adjustment phase.

Total chemical A = 560 liters; Total water = 1040 liters.

Sum the quantities from both solutions: Chemical A = 360 + 200 = 560 L; Water = 240 + 800 = 1040 L.

4

Formulate linear expressions representing the total volume of chemical A and water after t hours of adjustment.

Chemical A volume = 560 + 50t; Water volume = 1040 - 30t.

Pure chemical A is added at 50 L/hr, increasing its total volume, while water is drained at 30 L/hr, reducing its total volume.

5

Set the two component volume expressions equal to each other and solve for t.

t = 6 hours.

Solving 560 + 50t = 1040 - 30t leads to 80t = 480, which yields t = 6.

Anahtar Kavram

Multi-stream mixture rate integration and ratio equality modeling

Soru 23Soru →

A municipal authority allocates water from a central reservoir to three sectors: Agriculture, Industry, and Residential. Initially, the ratio of the volume of water allocated to Agriculture to that of Industry is $5 : 3$ , and the ratio of the volume allocated to Industry to that of Residential is $4 : 5$ . During a drought, the total supply is reallocated such that Agriculture's allocation is decreased by $20\%$ , Industry's allocation is decreased by $10\%$ , and Residential's allocation is increased by $12\%$ . After these adjustments, what is the ratio of Agriculture's new allocation to Residential's new allocation?

20 : 21

5 : 7

4 : 3

21 : 20

15 : 14

Cevabı ve açıklamayı göster

Cevap: $20 : 21$

Cevap

The ratio of Agriculture's new allocation to Residential's new allocation is

20 : 21

.

To find the new ratio, first unify the given ratios Agriculture : Industry (

5 : 3

) and Industry : Residential (

4 : 5

) by finding a common multiplier for Industry (12). This yields an overall initial ratio of

20 : 12 : 15

. Applying a

20\%

decrease to Agriculture gives

20 \times 0.80 = 16

, and applying a

12\%

increase to Residential gives

15 \times 1.12 = 16.8

. The ratio of the new allocations is

16 : 16.8

, which simplifies to

160 : 168 = 20 : 21

.

Adım Adım Çözüm

1

Express the three initial allocations in a unified three-part ratio.

Agriculture : Industry =

5 : 3 = 20 : 12

, Industry : Residential =

4 : 5 = 12 : 15

. Therefore, Agriculture : Industry : Residential =

20 : 12 : 15

.

Industry is the common element linking both ratios, so its ratio component must be equalized (LCM of 3 and 4 is 12).

2

Assign algebraic representations to the initial allocations based on the unified ratio.

Let Agriculture's initial allocation be

20x

, Industry's initial allocation be

12x

, and Residential's initial allocation be

15x

.

This allows for exact percentage calculations on consistent base values.

3

Calculate the updated allocations after applying the specified percentage adjustments.

Agriculture's new allocation =

20x \times (1 - 0.20) = 16x

. Residential's new allocation =

15x \times (1 + 0.12) = 16.8x

.

A

20\%

decrease reduces a quantity to

80\%

of its original value, and a

12\%

increase expands it to

112\%

of its original value.

4

Compute and simplify the ratio of Agriculture's new allocation to Residential's new allocation.

\frac{\text{Agriculture}_{\text{new}}}{\text{Residential}_{\text{new}}} = \frac{16x}{16.8x} = \frac{160}{168} = \frac{20}{21}.

Multiplying both terms by 10 eliminates decimals, and dividing both by their greatest common divisor (8) yields the simplified integer ratio

20 : 21

.

Anahtar Kavram

Combining relative ratios through a common quantity and calculating proportional adjustments.

Alternatif Yöntem

Instead of setting a variable

x

, assume a concrete initial volume for Industry equal to

12

units. Consequently, Agriculture is

20

units and Residential is

15

units. Agriculture's new volume is

20 - 4 = 16

units, and Residential's new volume is

15 + 1.8 = 16.8

units. The ratio

\frac{16}{16.8} = \frac{20}{21}

is obtained immediately.

Tahmini Süre:2m 0s

Soru 24Soru →

A biomanufacturing facility operates two harvesting lines, Line A and Line B, to collect a refined protein compound suspended in a liquid growth medium into a single storage tank.

- Line A processes liquid at a constant rate of $120\text{ liters per hour}$ , and its output contains protein and medium in a volume ratio of $1 : 4$ .
- Line B processes liquid at a constant rate of $180\text{ liters per hour}$ , and its output contains protein and medium in a volume ratio of $1 : 9$ .

Both lines run simultaneously for exactly $5\text{ hours}$ into the empty storage tank. Which of the following statements about the resulting liquid mixture in the storage tank must be true? Indicate all such statements.

Geçerli olan tümünü seçin

The total volume of protein collected in the storage tank is

210\text{ liters}

.

Protein accounts for exactly

14\%

of the total liquid volume in the storage tank.

The ratio of total protein to total medium in the storage tank is

7 : 43

.

The total volume of protein collected in the storage tank is

250\text{ liters}

.

Protein accounts for exactly

15\%

of the total liquid volume in the storage tank.

Cevabı ve açıklamayı göster

Cevap: The total volume of protein collected in the storage tank is $210\text{ liters}$ .; Protein accounts for exactly $14\%$ of the total liquid volume in the storage tank.; The ratio of total protein to total medium in the storage tank is $7 : 43$ .

Cevap

The correct statements are those asserting that the total volume of protein collected is 210 liters, that protein accounts for exactly 14% of the total liquid volume, and that the ratio of total protein to total medium is 7 to 43.

The total protein collected is

210\text{ liters}

(

120\text{ L}

from Line A and

90\text{ L}

from Line B). Dividing this by the overall volume of

1500\text{ liters}

gives

14\%

. Subtracting protein from total volume gives

1290\text{ liters}

of medium, yielding a protein-to-medium ratio of

210 : 1290 = 7 : 43

. Therefore, the three statements asserting

210\text{ liters}

of protein, a

14\%

concentration, and a

7:43

ratio are all correct.

Adım Adım Çözüm

1

Calculate the total liquid volume produced by each line in 5 hours.

Line A volume =

120\text{ L/hr} \times 5\text{ hr} = 600\text{ liters}

. Line B volume =

180\text{ L/hr} \times 5\text{ hr} = 900\text{ liters}

. Total combined volume =

600 + 900 = 1500\text{ liters}

.

Total volume per line is the product of its constant flow rate and duration.

2

Convert the part-to-part ratios to part-to-whole fractions to find the protein volume from each line.

Line A ratio

1:4

means protein is

\frac{1}{1+4} = \frac{1}{5}

of the volume. Protein A =

\frac{1}{5} \times 600 = 120\text{ L}

. Line B ratio

1:9

means protein is

\frac{1}{1+9} = \frac{1}{10}

of the volume. Protein B =

\frac{1}{10} \times 900 = 90\text{ L}

. Total protein =

120 + 90 = 210\text{ liters}

.

Ratios of

a:b

correspond to a component fraction of

\frac{a}{a+b}

of the total mixture.

3

Determine the percentage concentration of protein in the final mixture.

\text{Percentage} = \frac{210\text{ L}}{1500\text{ L}} \times 100\% = 14\%

.

The overall concentration is total protein volume divided by overall mixture volume.

4

Determine the simplified ratio of total protein to total medium.

Total medium volume =

1500 - 210 = 1290\text{ L}

. Ratio of protein to medium =

210 : 1290 = 7 : 43

.

Dividing both parts of

210 : 1290

by their greatest common divisor (

30

) yields

7 : 43

.

Anahtar Kavram

Combining rates and converting part-to-part ratios to part-to-whole fractions

Soru 25Soru →

Three water pumps, $P$ , $Q$ , and $R$ , operate at constant individual rates. The ratio of the rate of pump $P$ to the rate of pump $Q$ is $2 : 3$ . When all three pumps operate simultaneously, their combined rate is $3$ times the rate of pump $P$ alone. If pump $R$ working alone can drain a full reservoir in $24$ hours, how many hours would pump $Q$ working alone take to drain the same full reservoir?

8

hours

12

hours

16

hours

32

hours

56

hours

Cevabı ve açıklamayı göster

Cevap: $8$ hours

Cevap

8 hours

The correct answer is 8 hours. By setting the rate of pump Q as

\frac{3}{2}

times the rate of pump P, the combined rate of all three pumps is

r_P + \frac{3}{2} r_P + r_R = \frac{5}{2} r_P + r_R

. Setting this equal to

3 r_P

shows that pump R's rate is

\frac{1}{2} r_P

. Since pump R takes 24 hours (

r_R = \frac{1}{24}

), pump P's rate is

\frac{1}{12}

(taking 12 hours), and pump Q's rate is

\frac{3}{2} \times \frac{1}{12} = \frac{1}{8}

(taking 8 hours).

Adım Adım Çözüm

1

Express the rates of pumps P and Q in terms of a common variable.

Let

r_P

,

r_Q

, and

r_R

be the rates of pumps

P

,

Q

, and

R

in reservoirs per hour. Given

r_P : r_Q = 2 : 3

, we have

r_Q = \frac{3}{2} r_P

.

Relating pump rates using the given ratio simplifies the system of equations to one variable.

2

Set up the combined rate equation and solve for

r_R

in terms of

r_P

.

r_P + r_Q + r_R = 3 r_P \implies r_P + \frac{3}{2} r_P + r_R = 3 r_P \implies \frac{5}{2} r_P + r_R = 3 r_P \implies r_R = \frac{1}{2} r_P

.

The total rate is the sum of individual rates, allowing us to express pump R's rate in terms of pump P's rate.

3

Calculate

r_P

and

r_Q

using the given rate for pump R.

Since pump

R

takes

24

hours alone,

r_R = \frac{1}{24}

. Therefore,

\frac{1}{2} r_P = \frac{1}{24} \implies r_P = \frac{1}{12}

. Then

r_Q = \frac{3}{2} \times \frac{1}{12} = \frac{1}{8}

.

Knowing pump R's explicit numerical rate allows finding the numerical rates for pumps P and Q.

4

Determine the time required for pump Q alone to drain the reservoir.

\text{Time for } Q = \frac{1}{r_Q} = \frac{1}{1/8} = 8 \text{ hours}.

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

Anahtar Kavram

Combined Work Rates and Ratio Relationships

Tahmini Süre:2m 0s

Soru 26Soru →

A data processing center uses two server clusters, Cluster $X$ and Cluster $Y$ , operating at constant individual processing rates. The ratio of the rate of Cluster $X$ to the rate of Cluster $Y$ is $3 : 5$ . Cluster $X$ alone can process a standard dataset of size $D$ gigabytes in 20 hours.

If Cluster $X$ and Cluster $Y$ work together for 4 hours at their initial rates, and then Cluster $X$ 's rate is increased by $33\frac{1}{3}\%$ while Cluster $Y$ 's rate is increased by $20\%$ , how many additional hours will it take for the two clusters working together at their new rates to complete the remaining portion of dataset $D$ ?

Cevabı ve açıklamayı göster

Cevap: 2.8

Cevap

It will take 2.8 additional hours for the two clusters working together at their new rates to complete the remaining portion of dataset

D

.

Representing Cluster

X

's rate as

3k

and Cluster

Y

's rate as

5k

establishes the dataset size

D = 20 \times 3k = 60k

. During the first 4 hours, both clusters process

4 \times (3k + 5k) = 32k

GB, leaving

28k

GB remaining. After rate increases, Cluster

X

's rate becomes

4k

and Cluster

Y

's rate becomes

6k

, resulting in a new combined rate of

10k

. Dividing the remaining

28k

GB by

10k

GB/hr yields

2.8

hours.

Adım Adım Çözüm

1

Define variables for the initial rates and dataset size based on the given ratio.

Let the processing rate of Cluster

X

be

r_X = 3k

GB/hr and Cluster

Y

be

r_Y = 5k

GB/hr for some constant

k > 0

. Since Cluster

X

alone completes dataset

D

in 20 hours,

D = 3k \times 20 = 60k

GB.

Relating the ratio of individual rates to total work defines all quantities in terms of a single parameter

k

.

2

Calculate the amount of work finished during the initial joint operation.

The combined initial rate is

r_X + r_Y = 3k + 5k = 8k

GB/hr. Working together for 4 hours completes

8k \times 4 = 32k

GB.

Working simultaneously means their processing rates add together.

3

Find the remaining work and the updated processing rates after adjustments.

Remaining dataset volume =

60k - 32k = 28k

GB. Cluster

X

's new rate =

3k \times \left(1 + \frac{1}{3}\right) = 4k

GB/hr. Cluster

Y

's new rate =

5k \times 1.20 = 6k

GB/hr. New combined rate =

4k + 6k = 10k

GB/hr.

Modifying the individual rates changes the combined throughput for the remaining task.

4

Compute the additional time required to process the remaining dataset.

Additional time =

\frac{28k \text{ GB}}{10k \text{ GB/hr}} = 2.8

hours.

Dividing the remaining work volume by the new combined rate gives the exact required time.

Anahtar Kavram

Combined work rates, ratio proportionality, and percentage rate adjustments.

Soru 27Soru →

A specialty paint manufacturing facility creates a custom dye by mixing three liquid concentrates: Red, Yellow, and Blue. In the formulation, the ratio of the volume of Red concentrate to Yellow concentrate is $2 : 3$ , and the ratio of the volume of Yellow concentrate to Blue concentrate is $4 : 5$ . If a single storage vat contains $140\text{ liters}$ of this fully mixed custom dye, how many liters of Yellow concentrate are in the vat?

32\text{ liters}

48\text{ liters}

60\text{ liters}

70\text{ liters}

84\text{ liters}

Cevabı ve açıklamayı göster

Cevap: $48\text{ liters}$

Cevap

48\text{ liters}

of Yellow concentrate

To find the volume of Yellow concentrate, first combine the given ratios (Red : Yellow = 2 : 3 and Yellow : Blue = 4 : 5) into a unified ratio by expressing Yellow with a common term. Multiplying the first ratio by 4 gives Red : Yellow = 8 : 12, and multiplying the second ratio by 3 gives Yellow : Blue = 12 : 15. The combined ratio Red : Yellow : Blue is 8 : 12 : 15, yielding a total of 8 + 12 + 15 = 35 parts. Yellow accounts for 12 out of 35 parts. Multiplying 12/35 by the total volume of 140 liters yields (12/35) * 140 = 48 liters.

Adım Adım Çözüm

1

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

Red : Yellow =

2 : 3 = 8 : 12

and Yellow : Blue =

4 : 5 = 12 : 15

. Thus, Red : Yellow : Blue =

8 : 12 : 15

.

Yellow is the common component in both ratios. Finding a common multiple for Yellow's ratio term (LCM of 3 and 4 is 12) allows us to express all three quantities in a unified ratio scale.

2

Calculate the total number of ratio parts and determine Yellow's fraction of the total mixture.

Total parts =

8 + 12 + 15 = 35

. Yellow's fraction of the total volume =

\frac{12}{35}

.

To convert a part-to-part ratio into a part-to-whole ratio, divide the target component's ratio parts by the sum of all ratio parts.

3

Multiply Yellow's fraction by the total volume of the mixture.

Yellow volume =

\frac{12}{35} \times 140 = 12 \times 4 = 48\text{ liters}

.

Multiplying the part-to-whole fraction by the total volume yields the exact volume of Yellow concentrate present.

Anahtar Kavram

Combining pairwise ratios into a unified three-part ratio to determine part-to-whole proportions.

Tahmini Süre:1m 30s

Soru 28Soru →

A commercial bakery prepares a specialty grain blend using oats, wheat, and rye. Initially, the ratio of oats to wheat to rye by weight in the blend is $4 : 3 : 2$ . After $15\text{ kg}$ of oats and $15\text{ kg}$ of rye are added to the mixture while the amount of wheat remains unchanged, the ratio of oats to rye in the new blend becomes $3 : 2$ . Which of the following statements about the final grain blend must be true? Select all such statements.

Geçerli olan tümünü seçin

The weight of wheat in the final blend is

22.5\text{ kg}

.

The total weight of the final grain blend is

97.5\text{ kg}

.

The ratio of wheat to rye in the final grain blend is

3 : 4

.

Oats account for exactly

50\%

of the total weight of the final grain blend.

The total weight of the blend increased by

50\%

from its initial weight.

Cevabı ve açıklamayı göster

Cevap: The weight of wheat in the final blend is $22.5\text{ kg}$ .; The total weight of the final grain blend is $97.5\text{ kg}$ .; The ratio of wheat to rye in the final grain blend is $3 : 4$ .

Cevap

The weight of wheat in the final blend is

22.5\text{ kg}

, the total weight of the final grain blend is

97.5\text{ kg}

, and the ratio of wheat to rye in the final grain blend is

3 : 4

.

Solving the ratio proportion equation yields a multiplier constant of

x = 7.5

. Substituting

x = 7.5

yields final weights of

45\text{ kg}

for oats,

22.5\text{ kg}

for wheat, and

30\text{ kg}

for rye. The weight of wheat is indeed

22.5\text{ kg}

, the total final weight is

45 + 22.5 + 30 = 97.5\text{ kg}

, and the ratio of wheat to rye is

22.5 : 30 = 3 : 4

. Thus, these three statements are correct.

Adım Adım Çözüm

1

Define variables using the initial ratio.

Let the initial weight of oats be

4x

, wheat be

3x

, and rye be

2x

.

Expressing quantities in terms of a common ratio multiplier

x

ensures proportional relationships are preserved.

2

Set up an equation using the updated quantities and given new ratio.

\frac{4x + 15}{2x + 15} = \frac{3}{2}

15 kg was added to both oats and rye, establishing a new ratio of 3 to 2 between oats and rye.

3

Solve for the multiplier

x

.

2(4x + 15) = 3(2x + 15) \implies 8x + 30 = 6x + 45 \implies 2x = 15 \implies x = 7.5

.

Cross-multiplying yields the unique value for the ratio constant

x

.

4

Calculate the final weight of each component and the total weight.

Initial oats =

30\text{ kg}

, final oats =

45\text{ kg}

. Wheat (unchanged) =

22.5\text{ kg}

. Initial rye =

15\text{ kg}

, final rye =

30\text{ kg}

. Final total weight =

45 + 22.5 + 30 = 97.5\text{ kg}

.

Evaluating each component confirms all specific properties of the final mixture.

5

Verify each offered statement against calculated values.

Wheat weight is

22.5\text{ kg}

(True). Total weight is

97.5\text{ kg}

(True). Ratio of Wheat to Rye is

22.5 : 30 = 3 : 4

(True). Oats percentage is

\frac{45}{97.5} \approx 46.15\%

(False). Weight percent increase is

\frac{30}{67.5} \approx 44.44\%

(False).

Comparing calculated facts directly determines which statements must be true.

Anahtar Kavram

Ratio adjustment and algebraic formulation of multi-part mixture problems

Soru 29Soru →

Working alone at its constant rate, Printer $X$ can complete a printing job in $4\text{ hours}$ . Working alone at its constant rate, Printer $Y$ can complete the same printing job in $6\text{ hours}$ . Printer $X$ begins working on the job alone and works for $1\text{ hour}$ . At that point, Printer $Y$ joins Printer $X$ , and both printers work together at their respective constant rates until the job is completed. What is the total time, in hours, required to complete the entire job from start to finish?

1.8\text{ hours}

2.4\text{ hours}

2.8\text{ hours}

3.2\text{ hours}

3.5\text{ hours}

Cevabı ve açıklamayı göster

Cevap: $2.8\text{ hours}$

Cevap

2.8\text{ hours}

The correct answer is

2.8\text{ hours}

. Printer

X

works alone for

1\text{ hour}

at a rate of

\frac{1}{4}

job per hour, completing

\frac{1}{4}

of the total job. This leaves

\frac{3}{4}

of the job unfinished. When Printer

Y

joins, their combined rate is

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

job per hour. Dividing the remaining

\frac{3}{4}

of the job by

\frac{5}{12}

yields

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

for the joint work phase. Adding the initial

1\text{ hour}

of solo work yields a total of

1 + 1.8 = 2.8\text{ hours}

.

Adım Adım Çözüm

1

Calculate individual work rates and the portion of the job completed in the first hour

Printer

X

's rate is

\frac{1}{4}

job/hour and Printer

Y

's rate is

\frac{1}{6}

job/hour. In the first hour, Printer

X

completes

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

of the job.

Printer

X

works alone for the first hour before Printer

Y

joins.

2

Determine the remaining fraction of the job

Remaining job =

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

.

The entire job is represented by

1

, so subtracting the completed portion yields the remaining portion.

3

Calculate the combined rate of both printers working together

Combined rate =

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

job/hour.

Rates add when workers or machines work simultaneously.

4

Calculate the time required for both printers to finish the remaining job and find total time

Time together =

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

. Total time =

1 + 1.8 = 2.8\text{ hours}

.

Time equals remaining work divided by combined rate, plus the

1\text{ hour}

already elapsed.

Anahtar Kavram

Combined Work Rates and Multi-Stage Work Problems

Alternatif Yöntem

Convert the job into arbitrary work units. Let the job equal

12\text{ units}

(the LCM of

4

and

6

). Printer

X

produces

12 / 4 = 3\text{ units/hour}

and Printer

Y

produces

12 / 6 = 2\text{ units/hour}

. In the first hour, Printer

X

produces

3\text{ units}

, leaving

12 - 3 = 9\text{ units}

. Working together, their combined rate is

3 + 2 = 5\text{ units/hour}

. The remaining

9\text{ units}

take

9 / 5 = 1.8\text{ hours}

. Total time is

1 + 1.8 = 2.8\text{ hours}

.

Tahmini Süre:1m 30s

Soru 30Soru →

A logistics facility utilizes two automated sorting lines, Line A and Line B, to process incoming shipments. Line A processes shipments at a constant rate that is 20 percent greater than the constant rate of Line B. When both lines operate simultaneously, they process a combined total of 3,300 shipments in 3 hours. Working alone at its constant rate, how many hours will it take Line B to process 2,250 shipments?

Cevabı ve açıklamayı göster

Cevap: 4.5

Cevap

4.5 hours

Let

r

represent the rate of Line B in shipments per hour. Because Line A processes 20% faster than Line B, Line A's rate is

1.20r

. Together, their combined processing rate is

r + 1.20r = 2.20r

shipments per hour. Operating for 3 hours, the total shipments processed is

3 \times 2.20r = 6.60r = 3,300

. Solving for

r

gives

r = 500

shipments per hour. To process 2,250 shipments alone, Line B requires

\frac{2,250}{500} = 4.5

hours.

Adım Adım Çözüm

1

Define the relationship between the individual processing rates.

If Line B processes at rate

r

shipments per hour, Line A processes at

1.20r

shipments per hour.

Line A's rate is 20 percent greater than Line B's rate.

2

Find the combined processing rate.

Combined rate =

r + 1.20r = 2.20r

shipments per hour.

When working together, individual rates add up.

3

Determine Line B's rate (

r

).

3 \times 2.20r = 3,300 \implies 6.60r = 3,300 \implies r = 500

shipments per hour.

Total work equals combined rate multiplied by time.

4

Calculate the time required for Line B to complete 2,250 shipments.

\text{Time} = \frac{2,250}{500} = 4.5

hours.

Time taken equals total work divided by the individual rate.

Anahtar Kavram

Combined Rates and Ratio Relationships

Soru 31Soru →

An agricultural facility uses three conveyor belts— $X$ , $Y$ , and $Z$ —to transfer harvested grain into a storage silo. Working together at their respective constant rates, Belt $X$ and Belt $Y$ can fill the empty silo in $6\text{ hours}$ . Working together at their respective constant rates, Belt $Y$ and Belt $Z$ can fill the empty silo in $8\text{ hours}$ . If Belt $X$ operates at twice the rate of Belt $Z$ , how many hours would it take Belt $Y$ working alone at its constant rate to fill the empty silo?

10\text{ hours}

12\text{ hours}

16\text{ hours}

18\text{ hours}

24\text{ hours}

Cevabı ve açıklamayı göster

Cevap: $12\text{ hours}$

Cevap

12\text{ hours}

The correct answer is

12\text{ hours}

. Subtracting the combined rate equation for Belts

Y

and

Z

(

r_Y + r_Z = 1/8

) from the equation for Belts

X

and

Y

(

r_X + r_Y = 1/6

) yields

r_X - r_Z = 1/24

. Since Belt

X

works at twice the rate of Belt

Z

(

r_X = 2r_Z

), substituting gives

r_Z = 1/24

. Substituting

r_Z

back into

r_Y + r_Z = 1/8

gives

r_Y = 1/8 - 1/24 = 1/12

. Therefore, Belt

Y

operating alone takes

12\text{ hours}

to fill the silo.

Adım Adım Çözüm

1

Define the work rates of each conveyor belt.

Let

r_X

,

r_Y

, and

r_Z

be the fraction of the silo filled per hour by Belts

X

,

Y

, and

Z

, respectively.

Establishing rates per unit of time allows linear combination of work performed.

2

Set up equations based on the given combined times and rate relationships.

r_X + r_Y = \frac{1}{6}

,

r_Y + r_Z = \frac{1}{8}

, and

r_X = 2r_Z

.

Combined rates equal the reciprocal of the total time required for combined work.

3

Subtract the second equation from the first to isolate

r_X - r_Z

.

(r_X + r_Y) - (r_Y + r_Z) = \frac{1}{6} - \frac{1}{8} \implies r_X - r_Z = \frac{4 - 3}{24} = \frac{1}{24}

.

Eliminating

r_Y

gives a direct linear relationship between

r_X

and

r_Z

.

4

Substitute

r_X = 2r_Z

into

r_X - r_Z = \frac{1}{24}

to solve for

r_Z

.

2r_Z - r_Z = \frac{1}{24} \implies r_Z = \frac{1}{24}

.

Determines the individual rate of Belt

Z

.

5

Substitute

r_Z = \frac{1}{24}

back into the equation

r_Y + r_Z = \frac{1}{8}

to find

r_Y

.

r_Y = \frac{1}{8} - \frac{1}{24} = \frac{3}{24} - \frac{1}{24} = \frac{2}{24} = \frac{1}{12}

.

Finds the individual rate of Belt

Y

.

6

Calculate the time required for Belt

Y

alone.

\text{Time} = \frac{1}{r_Y} = 12\text{ hours}.

The time to complete one full job is the reciprocal of the individual rate.

Anahtar Kavram

Work Rates and Combined Rate Equations

Soru 32Soru →

A civil engineering laboratory prepares a composite soil mixture using three material grades: Grade $X$ , Grade $Y$ , and Grade $Z$ . Initially, the ratio of Grade $X$ to Grade $Y$ by weight is $1 : 2$ , and the ratio of Grade $Y$ to Grade $Z$ by weight is $3 : 4$ . A technician adds $30\text{ kg}$ of Grade $X$ to the batch, causing Grade $X$ to constitute exactly $\frac{1}{4}$ of the total weight of the new mixture. Which of the following statements about the batch must be true? Select all such statements.

Geçerli olan tümünü seçin

The initial total weight of the mixture before Grade

X

was added was

306\text{ kg}

.

The weight of Grade

Z

in the mixture is

144\text{ kg}

.

In the final mixture, the ratio of Grade

Y

to Grade

Z

by weight is

3 : 4

.

The initial weight of Grade

X

in the mixture was

72\text{ kg}

.

Grade

Y

accounts for

35\%

of the total weight of the final mixture.

Cevabı ve açıklamayı göster

Cevap: The initial total weight of the mixture before Grade $X$ was added was $306\text{ kg}$ .; The weight of Grade $Z$ in the mixture is $144\text{ kg}$ .; In the final mixture, the ratio of Grade $Y$ to Grade $Z$ by weight is $3 : 4$ .

Cevap

The correct statements are those indicating that the initial total weight of the mixture was 306 kg, the weight of Grade Z in the mixture is 144 kg, and the ratio of Grade Y to Grade Z in the final mixture is 3 : 4.

The unified ratio of X : Y : Z is 3 : 6 : 8. Setting up the fraction of Grade X after adding 30 kg gives (3k + 30) / (17k + 30) = 1/4, which yields k = 18. This gives an initial total weight of 17 * 18 = 306 kg, a Grade Z weight of 8 * 18 = 144 kg, and an unchanged Grade Y to Grade Z ratio of 108 : 144 = 3 : 4. Therefore, the statements asserting the initial total weight as 306 kg, the Grade Z weight as 144 kg, and the final Y to Z ratio as 3 : 4 are all correct.

Adım Adım Çözüm

1

Unify the two given ratio relationships into a single 3-part ratio.

Grade X : Grade Y = 1 : 2 = 3 : 6, and Grade Y : Grade Z = 3 : 4 = 6 : 8. Unified ratio Grade X : Grade Y : Grade Z = 3 : 6 : 8.

Matching the term for Grade Y across both ratios enables setting up unified algebraic variable parts.

2

Define initial quantities using a common multiplier k.

Initial weight of X = 3k, Y = 6k, Z = 8k. Initial total weight = 3k + 6k + 8k = 17k.

Expressing quantities in terms of k allows setting up an equation after adding material.

3

Formulate and solve the equation based on the addition of Grade X.

(3k + 30) / (17k + 30) = 1/4 => 4(3k + 30) = 17k + 30 => 12k + 120 = 17k + 30 => 5k = 90 => k = 18.

Setting the new weight of X over the new total weight equal to 1/4 determines the exact value of k.

4

Evaluate the individual component weights and statements.

Initial total weight = 17(18) = 306 kg. Initial X = 3(18) = 54 kg. Weight of Y = 6(18) = 108 kg. Weight of Z = 8(18) = 144 kg. Final total weight = 306 + 30 = 336 kg. Final Y : Z ratio = 108 : 144 = 3 : 4.

Determining all numerical values allows verifying which statements are true.

Anahtar Kavram

Combining three-variable ratio streams into a unified ratio and solving linear rate/proportion equations upon single-component addition.

Soru 33Soru →

At a manufacturing facility, the ratio of the daily output of Machine $X$ to Machine $Y$ is $4 : 5$ , and the ratio of the daily output of Machine $Y$ to Machine $Z$ is $3 : 2$ . If the daily output of Machine $X$ is decreased by $10\%$ and the daily output of Machine $Z$ is increased by $25\%$ , what is the new ratio of the daily output of Machine $X$ to the daily output of Machine $Z$ ?

6 : 5

27 : 25

108 : 125

9 : 10

48 : 125

Cevabı ve açıklamayı göster

Cevap: $108 : 125$

Cevap

The new ratio of the daily output of Machine

X

to Machine

Z

is

108 : 125

.

The unified ratio of the outputs of the three machines is

X : Y : Z = 12 : 15 : 10

. Decreasing Machine

X

's output by

10\%

changes its relative units to

12 \times 0.90 = 10.8

. Increasing Machine

Z

's output by

25\%

changes its relative units to

10 \times 1.25 = 12.5

. Comparing the updated values yields

10.8 : 12.5

, which simplifies to

108 : 125

.

Adım Adım Çözüm

1

Unify the two separate ratios into a single three-part ratio

X : Y : Z

.

Since

X : Y = 4 : 5

and

Y : Z = 3 : 2

, multiply

X : Y

by

3

to get

12 : 15

and multiply

Y : Z

by

5

to get

15 : 10

. Thus,

X : Y : Z = 12 : 15 : 10

.

Machine

Y

is the common element linking both ratios, so its ratio component must be equalized.

2

Calculate the updated values for

X

and

Z

after applying their respective percentage changes.

Machine

X_{new} = 12 \times (1 - 0.10) = 12 \times 0.90 = 10.8

. Machine

Z_{new} = 10 \times (1 + 0.25) = 10 \times 1.25 = 12.5

.

Applying a

10\%

decrease scales a quantity by

0.90

, while a

25\%

increase scales it by

1.25

.

3

Form the ratio

X_{new} : Z_{new}

and convert to lowest integer terms.

10.8 : 12.5 = \frac{10.8}{12.5} = \frac{108}{125}

, which gives the ratio

108 : 125

.

Multiplying both terms by

10

eliminates decimals, yielding the coprime integer ratio

108 : 125

.

Anahtar Kavram

Three-Part Ratio Unification and Relative Percentage Modification

Tahmini Süre:1m 30s

Soru 34Soru →

A commercial facility prepares a fruit blend by mixing fruit concentrate with water in a ratio of $3:7$ by volume. After $15\text{ liters}$ of water evaporate from the mixture during processing, the ratio of fruit concentrate to water in the remaining mixture becomes $1:2$ . What was the total volume, in liters, of the original mixture before evaporation?

Cevabı ve açıklamayı göster

Cevap: 150

Cevap

The total volume of the original mixture before evaporation was 150 liters.

Represent the initial concentrate volume as

3x

liters and the initial water volume as

7x

liters, making the initial total volume

10x

liters. Evaporating

15

liters of water leaves

7x - 15

liters of water while the concentrate remains

3x

liters. Setting the ratio

\frac{3x}{7x - 15}

equal to

\frac{1}{2}

yields

6x = 7x - 15

, so

x = 15

. Substituting

x = 15

into the total volume expression

10x

gives

10(15) = 150

liters.

Adım Adım Çözüm

1

Define initial component volumes using ratio multiplier x

Concentrate volume =

3x

, Water volume =

7x

, Total volume =

10x

The given initial ratio of concentrate to water is

3:7

.

2

Formulate equation based on water evaporation and the new ratio

\frac{3x}{7x - 15} = \frac{1}{2}

Evaporation reduces only the water volume by 15 liters, establishing a new ratio of

1:2

.

3

Solve the algebraic equation for x

6x = 7x - 15 \implies x = 15

Cross-multiplication simplifies the proportional relationship into a linear equation.

4

Calculate original total volume

10 \times 15 = 150

liters

The original total volume is represented by

10x

.

Anahtar Kavram

Solving component adjustment problems using ratio multipliers

Soru 35Soru →

A chemical processing plant uses two storage tanks, Tank $X$ and Tank $Y$ . Initially, the ratio of the volume of liquid in Tank $X$ to Tank $Y$ is $3 : 7$ . After $16\text{ liters}$ of liquid are transferred from Tank $Y$ to Tank $X$ , and an additional $8\text{ liters}$ of liquid are added to Tank $X$ from an external supply, the ratio of the volume of liquid in Tank $X$ to Tank $Y$ becomes $6 : 5$ . What was the original volume, in liters, of liquid in Tank $Y$ ?

24

49

56

80

96

Cevabı ve açıklamayı göster

Cevap: 56

Cevap

56 liters

By setting the initial volumes of Tank X and Tank Y to 3k and 7k respectively, the modified volumes after transfer and addition are (3k + 24) and (7k - 16). Setting their ratio equal to 6/5 yields 5(3k + 24) = 6(7k - 16), which simplifies to 27k = 216, or k = 8. Multiplying 8 by 7 gives the original volume of Tank Y as 56 liters.

Adım Adım Çözüm

1

Define initial quantities using a common ratio multiplier

k

.

Let the initial volume of Tank

X

be

3k

liters and the initial volume of Tank

Y

be

7k

liters.

The given initial ratio of

X : Y

is

3 : 7

.

2

Express the new volumes after the liquid transfers.

Tank

X

volume becomes

3k + 16 + 8 = 3k + 24

liters. Tank

Y

volume becomes

7k - 16

liters.

Transferring

16\text{ liters}

from Tank

Y

to Tank

X

decreases Tank

Y

by

16

and increases Tank

X

by

16

. Adding an extra

8\text{ liters}

to Tank

X

brings its total increase to

24\text{ liters}

.

3

Set up the proportion with the new ratio and solve for

k

.

\begin{aligned} \frac{3k + 24}{7k - 16} &= \frac{6}{5} \\[6pt] 5(3k + 24) &= 6(7k - 16) \\[6pt] 15k + 120 &= 42k - 96 \\[6pt] 216 &= 27k \\[6pt] k &= 8 \end{aligned}

The new ratio of Tank

X

to Tank

Y

is given as

6 : 5

.

4

Calculate the original volume of Tank

Y

.

Original volume of Tank

Y = 7k = 7 \times 8 = 56\text{ liters}

.

Tank

Y

initially contained

7k

liters.

Anahtar Kavram

Algebraic setup of part-to-part ratios undergoing quantitative adjustments

Tahmini Süre:1m 30s

Soru 36Soru →

A courier travels along a straight route from Office P to Office Q at a constant speed $r_1$ , and immediately returns along the exact same route from Office Q to Office P at a constant speed $r_2$ . The ratio of the outbound speed to the return speed is $r_1 : r_2 = 3 : 2$ . Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

The time spent on the return trip is

50\%

greater than the time spent on the outbound trip.

The ratio of the time spent on the outbound trip to the time spent on the return trip is

3 : 2

.

The average speed for the entire round trip is equal to

80\%

of the outbound speed.

The average speed for the entire round trip is equal to the arithmetic average of the outbound speed and the return speed.

The outbound trip accounts for

60\%

of the total travel time for the round trip.

Cevabı ve açıklamayı göster

Cevap: The time spent on the return trip is $50\%$ greater than the time spent on the outbound trip.; The average speed for the entire round trip is equal to $80\%$ of the outbound speed.

Cevap

The statement that the return trip time is 50 percent greater than the outbound trip time, and the statement that the average speed for the entire round trip is equal to 80 percent of the outbound speed.

The return trip time is

50\%

greater than the outbound trip time because speed and time are inversely proportional over equal distances (

t_1 : t_2 = 2 : 3

). Additionally, the overall average speed is

\frac{12k}{5} = 2.4k

, which is precisely

80\%

of the outbound speed (

3k

).

Adım Adım Çözüm

1

Express travel times in terms of distance

D

and rate multiplier

k

.

Let outbound speed

r_1 = 3k

and return speed

r_2 = 2k

. Outbound time

t_1 = \frac{D}{3k}

and return time

t_2 = \frac{D}{2k}

. The ratio of times

t_1 : t_2 = \frac{D/3k}{D/2k} = 2 : 3

.

Time equals distance divided by rate, establishing an inverse relationship between speed and time for constant distance.

2

Evaluate the relative difference between return time and outbound time.

Return time

t_2 = 1.5 t_1

, which means

t_2

is

50\%

greater than

t_1

.

A factor of

1.5

represents a

50\%

increase over the base value

t_1

.

3

Calculate the average speed

R_{\text{avg}}

for the round trip.

Total distance is

2D

, and total time is

t_1 + t_2 = \frac{D}{3k} + \frac{D}{2k} = \frac{5D}{6k}

. Thus,

R_{\text{avg}} = \frac{2D}{\frac{5D}{6k}} = \frac{12k}{5} = 2.4k

.

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

4

Compare

R_{\text{avg}}

to the outbound speed

r_1

and the arithmetic mean of rates.

\frac{R_{\text{avg}}}{r_1} = \frac{2.4k}{3k} = 0.80 = 80\%

. The arithmetic mean is

\frac{3k + 2k}{2} = 2.5k \neq 2.4k

. Outbound time fraction is

\frac{t_1}{t_1 + t_2} = \frac{2}{5} = 40\%

.

Unweighted arithmetic averaging fails for rates over equal distances because more time is spent traveling at the lower speed.

Anahtar Kavram

Inverse proportion between speed and time, harmonic mean for round-trip average speed, and part-to-whole time ratios.

Tahmini Süre:2m 0s

Soru 37Soru →

Working alone at their respective constant rates, Machine $A$ can complete a production order in $6\text{ hours}$ , Machine $B$ in $8\text{ hours}$ , and Machine $C$ in $12\text{ hours}$ . All three machines start working together on the order at 9:00 AM. At 10:00 AM, Machine $A$ breaks down and stops working, while Machines $B$ and $C$ continue working together at their constant rates until the order is completed. At what time will the production order be completed?

11:40 AM

12:00 PM

1:00 PM

1:48 PM

2:12 PM

Cevabı ve açıklamayı göster

Cevap: 1:00 PM

Cevap

The production order will be completed at 1:00 PM.

The option specifying 1:00 PM is correct because during the first hour (9:00 AM to 10:00 AM), all three machines work together and complete

\frac{1}{6} + \frac{1}{8} + \frac{1}{12} = \frac{3}{8}

of the job. This leaves

\frac{5}{8}

of the job remaining at 10:00 AM. Machines B and C work together at a rate of

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

of the job per hour. Dividing the remaining work (

\frac{5}{8}

) by this combined rate (

\frac{5}{24}

) gives exactly 3 hours. Adding 3 hours to 10:00 AM yields 1:00 PM.

Adım Adım Çözüm

1

Calculate individual work rates for each machine.

Machine A rate =

\frac{1}{6}

order/hr, Machine B rate =

\frac{1}{8}

order/hr, Machine C rate =

\frac{1}{12}

order/hr.

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

2

Calculate the work completed by all three machines from 9:00 AM to 10:00 AM (1 hour).

Combined rate =

\frac{1}{6} + \frac{1}{8} + \frac{1}{12} = \frac{4 + 3 + 2}{24} = \frac{9}{24} = \frac{3}{8}

of the order.

All three machines work together for exactly 1 hour.

3

Determine the remaining fraction of the order after 10:00 AM.

Remaining work =

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

of the order.

Subtract completed work from the total work (1 whole order).

4

Calculate the combined rate of Machines B and C.

Combined rate of B and C =

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

order/hr.

Machine A stops, leaving only B and C working.

5

Calculate the additional time needed to complete the remaining work.

Time =

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

.

Divide remaining work by the combined rate of the remaining active machines.

6

Add the additional time to the breakdown time (10:00 AM).

Completion time = 10:00 AM + 3 hours = 1:00 PM.

The remaining work begins at 10:00 AM when Machine A breaks down.

Anahtar Kavram

Combined Work Rates and Staggered Work Times

Soru 38Soru →

A liquid chemical solution is composed of Chemical A, Chemical B, and Water in the volume ratio $2 : 3 : 5$ , respectively. If $40\text{ liters}$ of Chemical A and $20\text{ liters}$ of Water are added to the mixture, the ratio of Chemical A to Water becomes $1 : 2$ . What was the total volume, in liters, of the original solution?

120

200

300

420

600

Cevabı ve açıklamayı göster

Cevap: $600$

Cevap

600\text{ liters}

Let the original volumes of Chemical A, Chemical B, and Water be

2x

,

3x

, and

5x

liters, respectively. The total volume of the original solution is

2x + 3x + 5x = 10x

liters. After adding

40\text{ liters}

of Chemical A and

20\text{ liters}

of Water, the ratio of Chemical A to Water is given as

1 : 2

. Setting up the proportion

\frac{2x + 40}{5x + 20} = \frac{1}{2}

and cross-multiplying yields

2(2x + 40) = 5x + 20

, which simplifies to

4x + 80 = 5x + 20

. Solving for

x

gives

x = 60

. Substituting

x = 60

into the total volume formula gives

10(60) = 600\text{ liters}

. Thus,

600

is the correct answer.

Adım Adım Çözüm

1

Represent the initial volumes of each component in terms of a common multiplier

x

.

Chemical A

= 2x

, Chemical B

= 3x

, Water

= 5x

. Total volume

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

.

Ratios

2 : 3 : 5

mean the actual quantities are proportional to these ratio parts.

2

Set up an equation based on the new volumes of Chemical A and Water.

\frac{2x + 40}{5x + 20} = \frac{1}{2}

Adding

40\text{ L}

of Chemical A and

20\text{ L}

of Water changes their ratio to

1 : 2

.

3

Cross-multiply and solve for

x

.

2(2x + 40) = 1(5x + 20) \implies 4x + 80 = 5x + 20 \implies x = 60.

Solving the linear equation yields the common ratio unit value.

4

Calculate the total original volume using

x = 60

.

\text{Total Volume} = 10x = 10(60) = 600\text{ liters}.

The total solution consists of

10

parts in total.

Anahtar Kavram

Three-part ratio algebra and proportion adjustments

Tahmini Süre:1m 30s

Soru 39Soru →

An event production company initially allocates its total equipment budget among Stage Lights, Ambient LED Strips, and Spotlight Towers in the ratio $4 : 5 : 3$ , respectively. Due to venue price adjustments, the budget allocated to Stage Lights is increased by $25\%$ , while the budget allocated to Spotlight Towers is reduced by $20\%$ . If the total overall budget remains unchanged, what is the new ratio of the budget allocated to Stage Lights to Ambient LED Strips to Spotlight Towers?

25 : 23 : 12

25 : 25 : 12

25 : 17 : 12

35 : 22 : 3

5 : 4 : 3

Cevabı ve açıklamayı göster

Cevap: $25 : 23 : 12$

Cevap

The new ratio of the budget allocated to Stage Lights to Ambient LED Strips to Spotlight Towers is

25 : 23 : 12

.

By setting the initial allocations to

4x

,

5x

, and

3x

, the total budget is

12x

. The new allocation for Stage Lights becomes

4x \times 1.25 = 5x

, and for Spotlight Towers it becomes

3x \times 0.80 = 2.4x

. To keep the total budget at

12x

, the Ambient LED Strips allocation must be

12x - 5x - 2.4x = 4.6x

. The ratio

5 : 4.6 : 2.4

scales up to

50 : 46 : 24

, which simplifies to

25 : 23 : 12

.

Adım Adım Çözüm

1

Represent the initial component allocations and the total budget in terms of a common variable.

Let the initial allocations be Stage Lights

= 4x

, Ambient LED Strips

= 5x

, and Spotlight Towers

= 3x

. Total budget

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

.

Establishing algebraic quantities allows precise tracking of individual changes relative to the constant total.

2

Calculate the updated allocations for Stage Lights and Spotlight Towers after their respective percentage changes.

New Stage Lights

= 4x \times (1 + 0.25) = 5x

.
New Spotlight Towers

= 3x \times (1 - 0.20) = 2.4x

.

Applying the specified percentage shifts directly to their corresponding original component values.

3

Determine the new allocation for Ambient LED Strips using the fixed total budget condition.

New Ambient LED Strips

= 12x - (5x + 2.4x) = 12x - 7.4x = 4.6x

.

Because the total overall budget remains unchanged at

12x

, the sum of all three new allocations must equal

12x

.

4

Write the new ratio and simplify to integer terms.

Ratio

= 5x : 4.6x : 2.4x = 5 : 4.6 : 2.4 = 50 : 46 : 24 = 25 : 23 : 12

.

Multiplying by

10

clears decimals, and dividing by

2

reduces the ratio to its simplest integer form.

Anahtar Kavram

Multi-part ratio adjustment with percentage changes under a constant total constraint

Tahmini Süre:1m 30s

Soru 40Soru →

A coffee roaster blends Arabica, Robusta, and Liberica beans in the ratio $4 : 3 : 2$ by weight to prepare a master batch. To adjust the flavor profile, the roaster adds $15\text{ kg}$ of Robusta beans and $15\text{ kg}$ of Liberica beans to the batch without changing the amount of Arabica beans. In the modified batch, Arabica beans constitute exactly $\frac{1}{3}$ of the total weight. Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

The initial weight of Arabica beans in the master batch was

40\text{ kg}

.

In the modified batch, the ratio of Robusta beans to Liberica beans is

9 : 7

.

In the initial master batch, Robusta beans constituted

\frac{1}{3}

of the total weight.

The total weight of the batch increased by

25\%

after the additions.

In the modified batch, Liberica beans account for

35\%

of the total weight.

Cevabı ve açıklamayı göster

Cevap: The initial weight of Arabica beans in the master batch was $40\text{ kg}$ .; In the modified batch, the ratio of Robusta beans to Liberica beans is $9 : 7$ .; In the initial master batch, Robusta beans constituted $\frac{1}{3}$ of the total weight.

Cevap

The correct statements are those asserting that the initial weight of Arabica beans was 40 kg, that the modified ratio of Robusta to Liberica beans is 9 : 7, and that Robusta beans constituted 1/3 of the total weight in the initial batch.

Solving the proportion equation reveals that the scaling factor for the initial ratio is 10, giving an initial batch weight of 90 kg with 40 kg Arabica, 30 kg Robusta, and 20 kg Liberica. Consequently, Arabica originally weighed 40 kg, Robusta originally made up 30/90 = 1/3 of the batch, and the new Robusta-to-Liberica ratio is 45 : 35 = 9 : 7. All three of these statements are mathematically true.

Adım Adım Çözüm

1

Represent initial component weights and total weight using a multiplier variable.

For ratio

4 : 3 : 2

, Arabica =

4x

, Robusta =

3x

, Liberica =

2x

, and initial total weight

W = 9x

.

Ratios define proportional components in terms of a common scalar variable.

2

Set up the equation for the modified batch based on the new total and Arabica ratio.

Added weight =

15 + 15 = 30\text{ kg}

. New total weight =

9x + 30

. Arabica weight remains

4x

. Since Arabica is

\frac{1}{3}

of the new total:

\frac{4x}{9x + 30} = \frac{1}{3} \implies 12x = 9x + 30 \implies 3x = 30 \implies x = 10

.

Equating the unchanged part to the given fraction of the new total allows solving for the scale factor.

3

Calculate all initial and modified quantities.

Initial total

W = 90\text{ kg}

. Initial Arabica =

40\text{ kg}

, Robusta =

30\text{ kg}

, Liberica =

20\text{ kg}

. Modified total =

120\text{ kg}

. Modified Robusta =

45\text{ kg}

, Modified Liberica =

35\text{ kg}

.

Determining exact values enables verification of each statement.

4

Evaluate each given statement.

Initial Arabica =

40\text{ kg}

(True). Modified Robusta : Liberica =

45 : 35 = 9 : 7

(True). Initial Robusta fraction =

\frac{30}{90} = \frac{1}{3}

(True). Percent weight increase =

\frac{30}{90} = 33.33\%

(False). Modified Liberica percentage =

\frac{35}{120} = 29.17\%

(False).

Direct comparison of calculated quantities with statement claims identifies all correct choices.

Anahtar Kavram

Ratio scale factors and part-to-whole proportions under quantity modifications

Tahmini Süre:2m 0s

Ratios, Rates, and Proportions

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