Ratios, Rates, and Proportions Alıştırma Soruları — ACT

Soru 1Soru →

A recipe for fruit punch calls for orange juice and cranberry juice to be mixed in a ratio of $3:4$ . If a chef uses $12$ cups of orange juice, how many cups of cranberry juice should be used?

4

9

13

16

28

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16 cups

Since the ratio of orange juice to cranberry juice is 3 to 4, we can set up the proportion 3/4 = 12/x, where x is the amount of cranberry juice. Solving for x yields 16 cups.

Adım Adım Çözüm

1

Set up a proportion based on the ratio of orange juice to cranberry juice.

\frac{3}{4} = \frac{12}{x}

where

x

represents the number of cups of cranberry juice.

The ratio of orange juice to cranberry juice is constant at 3 to 4.

2

Solve the proportion for

x

by cross-multiplying.

3x = 12 \times 4

3x = 48

Cross-multiplication is a standard method to solve proportions.

3

Divide both sides by 3 to isolate

x

.

x = 16

Isolating the variable gives the required amount of cranberry juice.

Anahtar Kavram

Solving a part-to-part ratio problem using proportions.

Soru 2Soru →

A basket contains red apples and green apples in a ratio of $2:5$ . If there are 35 apples in the basket, how many green apples are in the basket?

5

10

14

25

7

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

25

To find the number of green apples, first determine the total number of parts in the ratio by adding the two components:

2 + 5 = 7

parts. Next, divide the total number of apples in the basket by the total parts to find the value of each part:

35 \div 7 = 5

apples per part. Finally, multiply the number of parts representing green apples (

5

parts) by the value of each part (

5

):

5 \times 5 = 25

. Therefore, the correct quantity of green apples in the basket is 25.

Adım Adım Çözüm

1

Find the total number of parts in the ratio.

2 + 5 = 7

parts

The ratio of red apples to green apples is

2:5

, so the total quantity is divided into

7

equal parts.

2

Calculate the value of a single part of the ratio.

35 \div 7 = 5

apples per part

Divide the total number of apples (

35

) by the total number of parts (

7

) to find how many apples make up one part.

3

Multiply the value of a single part by the number of parts representing green apples.

5 \text{ parts} \times 5 \text{ apples/part} = 25

green apples

Since green apples represent

5

parts of the ratio, multiply

5

by the value of one part (

5

) to find the total number of green apples.

Anahtar Kavram

Solving word problems involving part-to-part ratios and totals by finding the unit value per ratio part.

Soru 3Soru →

On a map of a city, the scale is $0.5\text{ inches} = 12\text{ miles}$ . If the actual distance between two towns is $96\text{ miles}$ , what is the distance between the two towns on the map, in inches?

2

4

8

16

48

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4 inches

To find the distance on the map, we set up the proportion comparing map inches to actual miles:

\frac{0.5\text{ inches}}{12\text{ miles}} = \frac{x\text{ inches}}{96\text{ miles}}

. Cross-multiplying gives

12x = 48

, which simplifies to

x = 4

. Therefore, the distance on the map is

4\text{ inches}

.

Adım Adım Çözüm

1

Set up a proportion to relate the map distance to the actual distance.

\frac{0.5\text{ inches}}{12\text{ miles}} = \frac{x\text{ inches}}{96\text{ miles}}

where

x

represents the distance between the two towns on the map.

This establishes a direct proportion based on the map's scale.

2

Solve for the unknown variable

x

by cross-multiplying.

12 \cdot x = 0.5 \cdot 96

12x = 48

Cross-multiplication is the standard algebraic method to solve proportions.

3

Divide both sides of the equation by 12 to find

x

.

x = \frac{48}{12} = 4

Isolating the variable gives the final map distance.

Anahtar Kavram

Solving ratio scale problems by setting up and solving proportions.

Soru 4Soru →

A cleaning mixture is prepared by combining bleach and water in a ratio of $1:8$ . If a container holds a total of $36$ cups of this mixture, how many cups of bleach were used?

4.5

8

4

32

28

Cevabı ve açıklamayı göster

Cevap: $4$

Cevap

4 cups

The correct answer is 4. The ratio of bleach to water is

1:8

, which means the entire mixture is divided into

1 + 8 = 9

equal parts. Bleach constitutes

\frac{1}{9}

of the overall mixture. By taking

\frac{1}{9}

of the

36

-cup total volume, we find that the amount of bleach used is

4

cups.

Adım Adım Çözüm

1

Determine the total parts in the mixture's ratio.

1 + 8 = 9

total parts

Since the ratio of bleach to water is

1:8

, the mixture is composed of

1

part bleach for every

8

parts water.

2

Express the bleach portion as a fraction of the total mixture.

\frac{1}{9}

The bleach represents

1

part out of the

9

total parts in the mixture.

3

Find the amount of bleach in the

36

-cup container.

\frac{1}{9} \times 36 = 4

cups

Multiply the fraction of bleach by the total volume of the cleaning mixture.

Anahtar Kavram

Determining a part from a part-to-part ratio and a total quantity

Soru 5Soru →

An artist mixes blue, yellow, and white paint in a ratio of $5:6:3$ , respectively. To create a new shade, she adds $8\text{ ounces}$ of blue paint and $8\text{ ounces}$ of white paint to the mixture, but adds no yellow paint. If the ratio of yellow paint to white paint in the new mixture is $6:5$ , how many ounces of blue paint are in the new mixture?

20

24

28

52

72

Cevabı ve açıklamayı göster

Cevap: 28

Cevap

28

The correct answer is

28

ounces. Let the initial quantities of blue, yellow, and white paint be

5x

,

6x

, and

3x

, respectively. Adding

8

ounces of blue and

8

ounces of white paint gives new amounts of

5x + 8

for blue and

3x + 8

for white. The amount of yellow paint remains

6x

. The new ratio of yellow to white paint is

\frac{6x}{3x + 8} = \frac{6}{5}

. Cross-multiplying yields

30x = 18x + 48

, which simplifies to

12x = 48

, giving

x = 4

. Substituting

x = 4

into the expression for the new amount of blue paint gives

5(4) + 8 = 28

ounces.

Adım Adım Çözüm

1

Define variables for the initial quantities of paint using a common multiplier

x

.

Blue paint =

5x

, yellow paint =

6x

, and white paint =

3x

.

Since the ratio of blue to yellow to white paint is

5:6:3

, the actual amounts are multiples of these ratio parts.

2

Express the new amounts of paint after adding the additional ounces.

New blue paint =

5x + 8

, new yellow paint =

6x

, and new white paint =

3x + 8

.

The artist adds

8\text{ ounces}

of blue and

8\text{ ounces}

of white paint, with no change to the yellow paint.

3

Set up a proportion using the ratio of yellow paint to white paint in the new mixture.

\frac{6x}{3x + 8} = \frac{6}{5}

The problem states that the ratio of yellow paint to white paint in the new mixture is

6:5

.

4

Solve the equation for the multiplier

x

.

30x = 6(3x + 8) \implies 30x = 18x + 48 \implies 12x = 48 \implies x = 4

.

Cross-multiplying allows us to solve the linear equation for the scale factor.

5

Find the final amount of blue paint in the new mixture by substituting

x = 4

.

5(4) + 8 = 28\text{ ounces}

.

The question asks for the ounces of blue paint in the new mixture, which is represented by

5x + 8

.

Anahtar Kavram

Setting up and solving algebraic proportions from multi-part ratios when quantities are modified.

Soru 6Soru →

An animal shelter has a ratio of cats to dogs of $4:7$ . If there are 28 cats currently at the shelter, what is the total number of cats and dogs at the shelter?

16

39

44

49

77

Cevabı ve açıklamayı göster

Cevap: 77

Cevap

The correct answer is 77, representing the total number of cats and dogs at the shelter.

The ratio of cats to dogs is 4:7, which means for every 4 cats there are 7 dogs. Since there are 28 cats, we can find the scaling multiplier by dividing the number of cats by 4: 28 / 4 = 7. Applying this same scaling factor to the dogs gives 7 * 7 = 49 dogs. Adding the cats and dogs together gives a total of 28 + 49 = 77 animals.

Adım Adım Çözüm

1

Determine the scale factor for the cats by dividing the number of cats by their ratio part.

28 \div 4 = 7

To find out how many animals each unit in the ratio represents.

2

Calculate the number of dogs by multiplying the scale factor by the dogs' ratio part.

7 \times 7 = 49

dogs

Since the ratio of cats to dogs is 4:7, there are 7 dogs for every 4 cats.

3

Find the total number of animals by adding the number of cats and dogs.

28 + 49 = 77

The question asks for the combined total of both cats and dogs.

Anahtar Kavram

Solving proportions using scaling factors to find a total quantity from a given ratio.

Alternatif Yöntem

Find the total number of parts in the ratio (4 + 7 = 11 parts). Since 4 parts represent 28 cats, each part represents 28 / 4 = 7 animals. Therefore, the total number of animals is 11 parts * 7 animals/part = 77.

Tahmini Süre:1m 0s

Soru 7Soru →

A machine at a factory seals $15$ boxes every $4$ minutes. At this rate, how many minutes will it take the machine to seal $90$ boxes?

6

10

24

60

360

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

The correct answer is

24

minutes. Since the machine seals

15

boxes every

4

minutes, sealing

90

boxes requires

6

times the number of boxes, which will take

6

times as long:

6 \times 4 = 24

minutes.

The correct answer is

24

. At a constant rate, the ratio of boxes sealed to time remains equivalent. Setting up the proportion

\frac{15}{4} = \frac{90}{x}

and cross-multiplying gives

15x = 360

, which simplifies to

x = 24

. Alternatively, since

90

boxes is

6

times the initial amount of

15

boxes (

90 \div 15 = 6

), the time required must also be

6

times the initial time:

6 \times 4 = 24

minutes.

Adım Adım Çözüm

1

Determine the scaling factor for the number of boxes.

Scaling factor is

6

.

Divide the target number of boxes by the initial rate's number of boxes:

90 \div 15 = 6

.

2

Multiply the time interval by the scaling factor to find the total time required.

24

minutes.

Multiply the

4

minutes per interval by the scaling factor of

6

:

6 \times 4 = 24

.

Anahtar Kavram

Solving direct proportion problems by finding equivalent rates.

Alternatif Yöntem

Find the unit rate first: the machine seals

\frac{15}{4} = 3.75

boxes per minute. To find the time to seal

90

boxes, divide the total boxes by the unit rate:

\frac{90}{3.75} = 24

minutes.

Tahmini Süre:45s

Soru 8Soru →

The rates at which three pipes, $A$ , $B$ , and $C$ , can fill a pool are in the ratio $3:4:6$ , respectively. To fill an empty pool, pipes $A$ and $B$ are turned on. After $2\text{ hours}$ , pipe $A$ is turned off and pipe $C$ is turned on. It takes another $3\text{ hours}$ for pipes $B$ and $C$ to finish filling the pool. What fraction of the pool's total volume was filled by pipe $B$ ?

\frac{4}{13}

\frac{3}{11}

\frac{5}{11}

\frac{8}{17}

\frac{20}{41}

Cevabı ve açıklamayı göster

Cevap: $\frac{5}{11}$

Cevap

The fraction of the pool's total volume filled by Pipe B is

\frac{5}{11}

.

To find the fraction of the pool's volume filled by Pipe B, we determine the total volume of the pool and the volume contributed by Pipe B. If the rates of pipes

A

,

B

, and

C

are

3r

,

4r

, and

6r

per hour, the first

2\text{ hours}

fill

2(3r+4r) = 14r

units, and the next

3\text{ hours}

fill

3(4r+6r) = 30r

units, for a total pool volume of

44r

units. Since Pipe B runs for the entire

5\text{ hours}

, it fills

5 \times 4r = 20r

units. The fraction is

\frac{20r}{44r} = \frac{5}{11}

.

Adım Adım Çözüm

1

Define the rates of the three pipes using a constant multiplier

r

.

Let the filling rates of pipes

A

,

B

, and

C

be

3r

,

4r

, and

6r

units of volume per hour, respectively.

This translates the given ratio

3:4:6

into algebraic expressions for their individual rates.

2

Calculate the volume of the pool filled during the first

2\text{ hours}

when pipes

A

and

B

are active.

Volume 1 =

2\text{ hours} \times (3r + 4r) = 2 \times 7r = 14r

units.

The rate of pipes

A

and

B

combined is the sum of their individual rates, and volume is rate multiplied by time.

3

Calculate the volume of the pool filled during the next

3\text{ hours}

when pipes

B

and

C

are active.

Volume 2 =

3\text{ hours} \times (4r + 6r) = 3 \times 10r = 30r

units.

The rate of pipes

B

and

C

combined is the sum of their individual rates, and volume is rate multiplied by time.

4

Find the total volume of the pool.

Total Volume =

14r + 30r = 44r

units.

The total volume is the sum of the volumes filled in both time intervals.

5

Calculate the total volume filled specifically by Pipe

B

across both periods.

Pipe

B

was active for the entire duration of

2 + 3 = 5\text{ hours}

. Volume filled by Pipe

B

=

5\text{ hours} \times 4r = 20r

units.

To find Pipe B's contribution, multiply its rate by the total time it operated.

6

Divide the volume filled by Pipe

B

by the total volume of the pool to find the fraction.

Fraction =

\frac{20r}{44r} = \frac{20}{44} = \frac{5}{11}

.

The fraction is the ratio of the part filled by Pipe B to the whole volume of the pool.

Anahtar Kavram

Using rates and ratios to find the fraction of work completed by a specific component in a multi-stage work problem.

Alternatif Yöntem

Instead of using a variable

r

, you can assume a concrete rate for the pipes that simplifies the arithmetic. For example, let the rates of pipes

A

,

B

, and

C

be

3

,

4

, and

6

gallons per hour, respectively. In the first

2\text{ hours}

, pipes

A

and

B

fill

2 \times (3 + 4) = 14\text{ gallons}

. In the next

3\text{ hours}

, pipes

B

and

C

fill

3 \times (4 + 6) = 30\text{ gallons}

. The total pool capacity is

14 + 30 = 44\text{ gallons}

. Pipe

B

works for the entire

5\text{ hours}

, filling

5 \times 4 = 20\text{ gallons}

. The fraction of the pool filled by Pipe

B

is

\frac{20}{44} = \frac{5}{11}

.

Tahmini Süre:3m 0s

Soru 9Soru →

A chemist has three different acid solutions: Solution $X$ , Solution $Y$ , and Solution $Z$ . In Solution $X$ , the ratio of acid to water is $1:3$ by volume. In Solution $Y$ , the ratio of acid to water is $3:2$ by volume. In Solution $Z$ , the ratio of acid to water is $5:1$ by volume. If the chemist mixes Solution $X$ , Solution $Y$ , and Solution $Z$ in a volume ratio of $4:5:3$ , respectively, what is the ratio of acid to water in the final mixture?

17:7

34:25

13:11

3:2

101:79

Cevabı ve açıklamayı göster

Cevap: $13:11$

Cevap

The ratio of acid to water in the final mixture is

13:11

.

The correct answer is the ratio

13:11

. To find the ratio of acid to water in the final mixture, first convert the individual ratios to part-to-whole fractions: Solution

X

is

\frac{1}{4}

acid and

\frac{3}{4}

water; Solution

Y

is

\frac{3}{5}

acid and

\frac{2}{5}

water; and Solution

Z

is

\frac{5}{6}

acid and

\frac{1}{6}

water. Next, scale these fractions by the volume mixing ratio of

4:5:3

. Assuming

4\text{ units}

of

X

,

5\text{ units}

of

Y

, and

3\text{ units}

of

Z

, the total acid is

4(\frac{1}{4}) + 5(\frac{3}{5}) + 3(\frac{5}{6}) = 1 + 3 + 2.5 = 6.5\text{ units}

, and the total water is

4(\frac{3}{4}) + 5(\frac{2}{5}) + 3(\frac{1}{6}) = 3 + 2 + 0.5 = 5.5\text{ units}

. The ratio of acid to water is

6.5:5.5

, which simplifies to

13:11

.

Adım Adım Çözüm

1

Determine the part-to-whole fractions of acid and water for each solution.

For Solution

X

, the ratio of acid to water is

1:3

, which means the acid fraction is

\frac{1}{4}

and the water fraction is

\frac{3}{4}

. For Solution

Y

, the ratio is

3:2

, meaning the acid fraction is

\frac{3}{5}

and the water fraction is

\frac{2}{5}

. For Solution

Z

, the ratio is

5:1

, meaning the acid fraction is

\frac{5}{6}

and the water fraction is

\frac{1}{6}

.

Converting part-to-part ratios into fractions of the total volume is necessary to scale each solution correctly.

2

Choose convenient volumes representing the

4:5:3

mixing ratio, and calculate the volume of acid and water contributed by each solution.

Assume we mix

4\text{ liters}

of Solution

X

,

5\text{ liters}

of Solution

Y

, and

3\text{ liters}

of Solution

Z

. Acid from

X

is

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

; water is

3\text{ liters}

. Acid from

Y

is

5 \times \frac{3}{5} = 3\text{ liters}

; water is

2\text{ liters}

. Acid from

Z

is

3 \times \frac{5}{6} = 2.5\text{ liters}

; water is

3 \times \frac{1}{6} = 0.5\text{ liters}

.

Multiplying the part-to-whole fractions by the respective mixing volumes yields the absolute amounts of acid and water contributed.

3

Sum the total volumes of acid and water in the final mixture.

Total acid =

1 + 3 + 2.5 = 6.5\text{ liters}

. Total water =

3 + 2 + 0.5 = 5.5\text{ liters}

.

Finding the total amounts of each component allows us to determine the final composition of the mixture.

4

Find the simplified integer ratio of total acid to total water.

The ratio of acid to water is

6.5 : 5.5

. Multiplying both terms by

2

to convert to integers gives

13:11

.

Ratios are standardly expressed as simplified integers.

Anahtar Kavram

Calculating mixture compositions by converting part-to-part ratios to part-to-whole fractions and applying weighted average proportions.

Alternatif Yöntem

Instead of choosing arbitrary volumes like 4, 5, and 3, you can write the total volume as

V

and use algebraic fractions:

\text{Acid} = 4k(\frac{1}{4}) + 5k(\frac{3}{5}) + 3k(\frac{5}{6}) = 6.5k

, and

\text{Water} = 4k(\frac{3}{4}) + 5k(\frac{2}{5}) + 3k(\frac{1}{6}) = 5.5k

. The ratio remains

6.5k : 5.5k = 13:11

.

Tahmini Süre:3m 0s

Soru 10Soru →

At a marketing firm, the ratio of designers to writers is $3:4$ , and the ratio of writers to managers is $5:2$ . The firm hires $10$ new designers and $4$ new managers, while the number of writers remains the same. If the new ratio of designers to managers is $2:1$ , how many writers are employed at the firm?

8

20

30

40

120

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

There are 40 writers employed at the firm.

To find the number of writers, we combine the ratios of designers to writers (

3:4

) and writers to managers (

5:2

) by finding a common multiple for the number of writers (20). This gives a combined ratio of

15:20:8

for designers, writers, and managers. Representing these quantities as

15x

,

20x

, and

8x

, we can write the equation for the new ratio of designers to managers:

\frac{15x + 10}{8x + 4} = 2

. Solving this equation gives

15x + 10 = 16x + 8

, which simplifies to

x = 2

. The number of writers is

20(2) = 40

.

Adım Adım Çözüm

1

Align the two given ratios to find a common ratio.

The ratio of designers to writers is

3:4 = 15:20

. The ratio of writers to managers is

5:2 = 20:8

. The combined ratio of designers to writers to managers is

15:20:8

.

To analyze the changes across the different categories, we need a single ratio that shares the common term (writers).

2

Express the quantities of employees in terms of a variable

x

.

Let the number of designers be

15x

, writers be

20x

, and managers be

8x

.

This allows us to represent the actual number of employees algebraically.

3

Set up an equation based on the new ratio of designers to managers after the new hires.

\frac{15x + 10}{8x + 4} = \frac{2}{1}

The firm hired 10 designers and 4 managers, and the ratio of these new quantities is

2:1

.

4

Solve the equation for

x

.

15x + 10 = 2(8x + 4) \implies 15x + 10 = 16x + 8 \implies x = 2

.

Solving for

x

gives us the multiplier to find the actual employee counts.

5

Calculate the total number of writers.

W = 20(2) = 40

.

The number of writers is represented by

20x

.

Anahtar Kavram

Combining multiple ratios and setting up algebraic equations from changing rates/proportions.

Alternatif Yöntem

Express the number of designers and managers in terms of the number of writers,

W

. The number of designers is

\frac{3}{4}W

, and the number of managers is

\frac{2}{5}W

. After hiring, the equation is

\frac{\frac{3}{4}W + 10}{\frac{2}{5}W + 4} = 2

. Solving for

W

yields

\frac{3}{4}W + 10 = \frac{4}{5}W + 8 \implies 2 = \frac{1}{20}W \implies W = 40

.

Tahmini Süre:2m 0s

Soru 11Soru →

A school district requires a chaperone-to-student ratio of $3:5$ for all field trips. If there are exactly $15$ chaperones registered for an upcoming field trip, what is the maximum number of students who can attend?

9

17

24

25

40

Cevabı ve açıklamayı göster

Cevap: $25$

Cevap

The maximum number of students who can attend is

25

.

The correct answer is

25

. To maintain the required ratio of

3

chaperones to

5

students, we set up the proportion

\frac{3}{5} = \frac{15}{x}

, where

x

is the number of students. Cross-multiplying gives

3x = 75

, and dividing by

3

gives

x = 25

.

Adım Adım Çözüm

1

Set up the proportion using the given chaperone-to-student ratio.

\frac{\text{Chaperones}}{\text{Students}} = \frac{3}{5} = \frac{15}{x}

, where

x

represents the maximum number of students.

Establishing a direct proportion allows us to find the unknown quantity while maintaining the required ratio.

2

Solve for the unknown variable

x

by cross-multiplying.

3 \times x = 5 \times 15 \implies 3x = 75

Cross-multiplication is a standard algebraic method to solve proportions.

3

Divide both sides of the equation by

3

to isolate

x

.

x = 25

This yields the final value of students who can attend.

Anahtar Kavram

Solving direct proportions from part-to-part ratios

Tahmini Süre:45s

Soru 12Soru →

A metal alloy is composed of copper, zinc, and nickel. In the original alloy, the ratio of the weight of copper to zinc is $3:4$ , and the ratio of the weight of zinc to nickel is $4:5$ . A metallurgist melts this $240$ -gram alloy and adds a mixture of pure copper and pure nickel. The weight of the copper added is twice the weight of the nickel added. In the resulting alloy, the weight of copper is equal to the weight of nickel. What is the ratio of the weight of zinc to the total weight of the new alloy?

\frac{4}{15}

\frac{1}{3}

\frac{2}{9}

\frac{2}{7}

\frac{4}{33}

Cevabı ve açıklamayı göster

Cevap: $\frac{2}{9}$

Cevap

The correct ratio of the weight of zinc to the total weight of the new alloy is

\frac{2}{9}

.

The correct answer is

\frac{2}{9}

. To find this, we first establish the joint ratio of the metals as Copper : Zinc : Nickel =

3 : 4 : 5

. For a

240

-gram alloy, this gives

60

g of copper,

80

g of zinc, and

100

g of nickel. Adding

x

grams of nickel and

2x

grams of copper yields a new copper weight of

60 + 2x

and a new nickel weight of

100 + x

. Since these weights are equal in the new alloy,

60 + 2x = 100 + x \implies x = 40

. The new total weight is

240 + 3(40) = 360

grams. The weight of zinc remains

80

grams, so the ratio of zinc to the total weight is

80/360 = \frac{2}{9}

.

Adım Adım Çözüm

1

Express the ratio of the three metals in the original alloy.

Copper : Zinc : Nickel =

3 : 4 : 5

Since Copper : Zinc =

3:4

and Zinc : Nickel =

4:5

, the zinc terms are already aligned, allowing us to combine them into a single three-part ratio.

2

Calculate the initial weights of copper, zinc, and nickel in the

240

-gram alloy.

Copper =

60

grams, Zinc =

80

grams, Nickel =

100

grams

The sum of the ratio parts is

3 + 4 + 5 = 12

. Each part represents

240 \text{ grams} / 12 = 20

grams. Multiplying each part by 20 gives the initial weights.

3

Define variables for the added metals and set up the equation for the new weights.

New Copper =

60 + 2x

, New Nickel =

100 + x

Let

x

represent the weight of nickel added in grams. Since the weight of the copper added is twice that of the nickel added,

2x

grams of copper are added.

4

Solve for the value of

x

using the condition that the final weights of copper and nickel are equal.

x = 40

Setting the expressions equal gives

60 + 2x = 100 + x

. Subtracting

x

and

60

from both sides yields

x = 40

grams.

5

Determine the new total weight of the alloy and calculate the final ratio of zinc to the total weight.

New Total Weight =

360

grams, Final Ratio =

\frac{2}{9}

The new total weight is the original weight plus the added metals:

240 + 3(40) = 360

grams. Since no zinc was added, the weight of zinc remains

80

grams. The ratio is

80 / 360 = \frac{2}{9}

.

Anahtar Kavram

Combining ratios and setting up algebraic equations to model changing quantities in mixtures.

Soru 13Soru →

A landscaping company uses a grass seed mixture containing perennial ryegrass and Kentucky bluegrass in a ratio of $5:3$ by weight. If a bag of this mixture contains $15$ pounds of Kentucky bluegrass, how many pounds of perennial ryegrass does it contain?

9

24

25

40

75

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

25

The ratio of perennial ryegrass to Kentucky bluegrass is

5:3

, which means for every

5

pounds of ryegrass, there are

3

pounds of bluegrass. Setting up the proportion

\frac{5}{3} = \frac{x}{15}

and solving for

x

gives

3x = 75

, which simplifies to

x = 25

pounds of perennial ryegrass.

Adım Adım Çözüm

1

Set up a proportion using the ratio of perennial ryegrass to Kentucky bluegrass.

\frac{5}{3} = \frac{x}{15}

, where

x

represents the weight of perennial ryegrass.

The ratio of perennial ryegrass to Kentucky bluegrass by weight is

5:3

, and the actual weight of Kentucky bluegrass is

15

pounds.

2

Solve the proportion for

x

by cross-multiplying.

3x = 5 \times 15 \implies 3x = 75

Cross-multiplication allows us to clear the denominators to solve for the unknown value.

3

Divide both sides of the equation by

3

.

x = 25

Isolating

x

gives the weight of perennial ryegrass in pounds.

Anahtar Kavram

Solving proportions based on a given part-to-part ratio

Alternatif Yöntem

Identify the scaling factor: since the ratio of Kentucky bluegrass is

3

parts and the actual weight is

15

pounds, each part of the ratio represents

15 \div 3 = 5

pounds. Multiplying the

5

parts of perennial ryegrass by this factor of

5

gives

5 \times 5 = 25

pounds.

Tahmini Süre:45s

Soru 14Soru →

In a school election for student council president, the ratio of the number of votes received by Candidate A to the number of votes received by Candidate B was $2:3$ . If Candidate B received $120$ votes, how many votes did Candidate A receive?

48

72

80

180

200

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

Candidate A received 80 votes.

The ratio of the number of votes for Candidate A to Candidate B is

2:3

. Since Candidate B received

120

votes, we can set up the proportion

\frac{\text{Votes for A}}{120} = \frac{2}{3}

. Multiplying both sides by

120

gives

\text{Votes for A} = 120 \times \frac{2}{3} = 80

.

Adım Adım Çözüm

1

Identify the given ratio and values.

The ratio of Candidate A's votes to Candidate B's votes is

2:3

, and Candidate B received

120

votes.

This sets up the proportion relationship where the ratio of Candidate A's votes to Candidate B's votes is equal to 2 to 3.

2

Set up a proportion to solve for Candidate A's votes.

\frac{x}{120} = \frac{2}{3}

, where

x

represents the number of votes Candidate A received.

This equates the simplified ratio of votes to the actual number of votes.

3

Solve for

x

by multiplying both sides by

120

.

x = 120 \times \frac{2}{3} = 80

.

This isolates the variable

x

to find the correct number of votes Candidate A received.

Anahtar Kavram

Solving proportions and scaling ratios based on a given part.

Tahmini Süre:45s

Soru 15Soru →

A baker makes a fruit tart using strawberries and blueberries. The ratio of the number of strawberries to the number of blueberries used in the tart is $3:5$ . If the baker uses a total of $40$ berries to make the tart, how many strawberries are used?

8

15

20

24

25

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

15

To find the number of strawberries used, we establish the part-to-whole fraction for strawberries. Given the ratio of strawberries to blueberries is

3:5

, strawberries account for

3

out of every

8

berries (since

3 + 5 = 8

). We then scale this fraction to the total of

40

berries:

\frac{3}{8} \times 40 = 15

. This shows that exactly 15 strawberries were used.

Adım Adım Çözüm

1

Determine the part-to-whole ratio for the strawberries.

The ratio of strawberries to blueberries is

3:5

, which means there are

3

parts strawberries for every

3 + 5 = 8

parts of total berries. The fraction of strawberries is

\frac{3}{8}

.

To find the quantity of a specific part from a total, we must use the part-to-whole ratio.

2

Multiply the part-to-whole fraction by the total number of berries.

\frac{3}{8} \times 40 = 3 \times 5 = 15

.

Multiplying the strawberry fraction by the total quantity yields the absolute number of strawberries used.

Anahtar Kavram

Part-to-whole ratios and scaling proportions

Tahmini Süre:45s

Soru 16Soru →

An agricultural drone is programmed to spray fertilizer on two adjacent fields, Field X and Field Y. The ratio of the area of Field X to the area of Field Y is $3:5$ . The drone sprays Field X at a constant rate of $4$ acres per hour, and it sprays Field Y at a constant rate of $6$ acres per hour. If it takes the drone a total of $76$ hours of continuous operation to spray both fields completely, what is the area, in acres, of Field Y?

20

48

144

240

475

Cevabı ve açıklamayı göster

Cevap: 240

Cevap

240

The area of Field Y is

240

acres. Since the ratio of the area of Field X to Field Y is

3:5

, we can define their areas as

3a

and

5a

. Using the relation

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

, the total time is

\frac{3a}{4} + \frac{5a}{6} = 76

. Finding a common denominator of

12

gives

\frac{9a}{12} + \frac{10a}{12} = 76

, which simplifies to

\frac{19a}{12} = 76

. Solving for

a

yields

a = 48

. The area of Field Y is then

5 \times 48 = 240

acres.

Adım Adım Çözüm

1

Define variables for the areas of the fields based on the given ratio.

Let the area of Field X be

3a

acres and the area of Field Y be

5a

acres, where

a

is a positive constant.

Since the ratio of the area of Field X to Field Y is

3:5

, expressing them in terms of a single variable

a

allows us to set up a solvable equation.

2

Express the time spent spraying each field in terms of the variable

a

.

Time for Field X =

\frac{3a}{4}

hours; Time for Field Y =

\frac{5a}{6}

hours.

Using the relation

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

, we substitute the respective areas and constant rates to get the time expressions.

3

Set up an equation for the total time spent spraying both fields.

\frac{3a}{4} + \frac{5a}{6} = 76

The sum of the time spent on each field must equal the total operational time of

76

hours.

4

Solve the equation for the constant

a

.

Find the common denominator of

12

:

\frac{9a}{12} + \frac{10a}{12} = 76 \implies \frac{19a}{12} = 76 \implies 19a = 912 \implies a = 48

.

Finding the least common multiple of the denominators allows us to clear fractions and isolate the variable

a

.

5

Calculate the area of Field Y.

Area of Field Y =

5 \times 48 = 240

acres.

The area of Field Y was defined as

5a

, so substituting

a = 48

gives the final area.

Anahtar Kavram

Setting up and solving equations involving ratios, rates, and proportions.

Tahmini Süre:2m 0s

Soru 17Soru →

On a certain map, a distance of $3$ inches represents an actual distance of $45$ miles. If the distance between two cities on this map is $8$ inches, what is the actual distance, in miles, between the two cities?

15

50

120

128

360

Cevabı ve açıklamayı göster

Cevap: $120$

Cevap

120

miles

Since

3

inches represents

45

miles, each inch on the map represents

\frac{45}{3} = 15

miles. To find the actual distance represented by

8

inches, multiply the number of inches by this unit rate:

8 \times 15 = 120

miles.

Adım Adım Çözüm

1

Calculate the scale of the map by finding the number of miles represented by

1

inch.

\frac{45}{3} = 15

miles per inch

This establishes the unit rate of miles per inch on the map.

2

Multiply the map distance between the two cities by the unit rate.

8 \times 15 = 120

miles

This scales the unit rate to find the actual distance for

8

inches.

Anahtar Kavram

Direct proportions and unit rates

Tahmini Süre:45s

Soru 18Soru →

Three machines— $X$ , $Y$ , and $Z$ —produce widgets at constant rates. The ratio of the production rate of Machine $X$ to Machine $Y$ to Machine $Z$ is $3:4:6$ , respectively. Working together at their constant rates, the three machines can complete a standard production order in exactly $8$ hours. A new order of the same size is started with all three machines working together. After $3$ hours, Machine $Z$ breaks down and stops working, and the production rate of Machine $Y$ decreases by $25\%$ . If Machine $X$ and the slowed Machine $Y$ continue working to complete the order, what is the total number of hours required to complete this entire second order?

13\frac{5}{6}

10\frac{5}{6}

8

11\frac{1}{3}

20\frac{1}{3}

Cevabı ve açıklamayı göster

Cevap: $13\frac{5}{6}$

Cevap

13\frac{5}{6}

hours

The correct answer is

13\frac{5}{6}

hours. By translating the ratios into algebraic rates, we determine that the total order size is

104r

widgets. In the first

3

hours,

39r

widgets are produced, leaving

65r

widgets. After the breakdown and slowdown, the combined rate of Machines

X

and

Y

is

3r + 3r = 6r

widgets per hour. The remaining work requires

\frac{65r}{6r} = 10\frac{5}{6}

hours, making the total duration

3 + 10\frac{5}{6} = 13\frac{5}{6}

hours.

Adım Adım Çözüm

1

Define the rate of each machine and find the total work required for the order.

Let the rates of Machine

X

,

Y

, and

Z

be

3r

,

4r

, and

6r

widgets per hour, respectively. The combined rate is

3r + 4r + 6r = 13r

widgets per hour. The total work

W

is

13r \times 8 = 104r

widgets.

Establishing rates using the given ratios allows us to compute the size of the job in terms of rate units.

2

Calculate the work completed in the first 3 hours and find the remaining work.

Work completed in the first

3

hours is

13r \times 3 = 39r

widgets. The remaining work is

104r - 39r = 65r

widgets.

We must subtract the work done before the breakdown to find the remaining workload.

3

Determine the new rates after Machine

Z

breaks down and Machine

Y

slows down.

Machine

Z

's rate becomes

0

. Machine

Y

's new rate is

4r \times (1 - 0.25) = 3r

widgets per hour. The new combined rate for Machines

X

and

Y

is

3r + 3r = 6r

widgets per hour.

We apply the rate changes to calculate the rate at which the remaining work will be completed.

4

Calculate the time needed to complete the remaining work and add the initial time.

The remaining work takes

\frac{65r}{6r} = 10\frac{5}{6}

hours. The total time for the entire order is

3 + 10\frac{5}{6} = 13\frac{5}{6}

hours.

Dividing the remaining work by the new combined rate gives the remaining time, which we add to the initial 3 hours to get the total time.

Anahtar Kavram

Rate and ratio application to multi-stage work problems

Soru 19Soru →

A company allocates its annual marketing budget among digital, print, and television advertisements in the ratio $4:3:5$ , respectively. Due to a strategy change, the company increases its digital advertising budget by $25\%$ , decreases its print advertising budget by $40\%$ , and increases its television advertising budget by $15\%$ . If the company's total marketing budget after these changes is $\$376,500$ , what was the company's original print advertising budget?

\$30,000

\$54,000

\$90,000

\$94,125

\$120,000

Cevabı ve açıklamayı göster

Cevap: $\$90,000$

Cevap

The original print advertising budget was

\$90,000

.

The correct answer is

\$90,000

. We define the original digital, print, and television budgets as

4x

,

3x

, and

5x

, respectively. Applying the shifts yields a new digital budget of

4x \times 1.25 = 5x

, a new print budget of

3x \times 0.60 = 1.8x

, and a new television budget of

5x \times 1.15 = 5.75x

. The sum of these new budgets is

5x + 1.8x + 5.75x = 12.55x

. Setting

12.55x = 376,500

gives

x = 30,000

. The original print budget is therefore

3x = 3(30,000) = 90,000

.

Adım Adım Çözüm

1

Represent the original budget allocations using a variable

x

based on the given ratio.

Digital budget =

4x

, Print budget =

3x

, and Television budget =

5x

. The original total budget was

4x + 3x + 5x = 12x

.

Establishing algebraic terms for each part allows us to apply the percentage changes.

2

Calculate the new budget allocations after applying the percentage changes.

New Digital budget =

4x \times 1.25 = 5x

. New Print budget =

3x \times (1 - 0.40) = 1.8x

. New Television budget =

5x \times 1.15 = 5.75x

.

Applying the strategic increases and decreases determines the new individual allocations in terms of

x

.

3

Sum the new allocations and set the equation equal to the new total budget.

New Total =

5x + 1.8x + 5.75x = 12.55x

. Since the new total is

\$376,500

, we write:

12.55x = 376,500

.

This allows us to solve for the multiplier

x

.

4

Solve for

x

and calculate the original print advertising budget.

x = \frac{376,500}{12.55} = 30,000

. The original print advertising budget is

3x = 3 \times 30,000 = 90,000

.

Finding the value of

x

allows us to calculate any of the original or new budget components.

Anahtar Kavram

Solving multi-step word problems involving ratios, percentage changes, and linear equations.

Alternatif Yöntem

Alternatively, you can write the equation directly for the fraction of the budget that print represents: the original print budget is

3x

, and the new total budget is

12.55x

. The ratio of the original print budget to the new total budget is

\frac{3}{12.55}

. Multiplying this ratio by the new total budget yields

\frac{3}{12.55} \times 376,500 = 90,000

.

Tahmini Süre:2m 0s

Soru 20Soru →

A manufacturing plant operates two types of machines, Type A and Type B. The ratio of the number of Type A machines to Type B machines is $3:5$ . Each Type A machine produces widgets at a constant rate that is $60\%$ faster than the constant rate of a Type B machine. When all machines of both types are operating simultaneously, they produce a total of $98$ widgets per hour. If the plant increases the number of Type A machines by $50\%$ and decreases the number of Type B machines by $20\%$ , what is the total number of widgets the new setup will produce in $3$ hours?

112

246

312

336

382

Cevabı ve açıklamayı göster

Cevap: 336

Cevap

336

The correct answer is 336 because setting up the initial production equation based on the ratios gives a compound constant of

xr = 10

. Applying the percentage adjustments yields a new hourly production rate of 112 widgets. Multiplying this hourly rate by the specified 3 hours gives a total of 336 widgets.

Adım Adım Çözüm

1

Represent the machine counts and rates using variables based on the given ratios.

Let the number of Type A machines be

3x

and the number of Type B machines be

5x

. Let the rate of a Type B machine be

r

widgets/hour. The rate of a Type A machine is

1.6r

widgets/hour.

This establishes algebraic expressions for the quantities in terms of common multipliers

x

and

r

.

2

Set up an equation for the initial total hourly production and solve for

xr

.

(3x)(1.6r) + (5x)(r) = 98 \implies 4.8xr + 5xr = 98 \implies 9.8xr = 98 \implies xr = 10

.

This allows us to find the value of the joint constant factor

xr

needed for subsequent calculations.

3

Determine the new machine counts after the percentage changes.

New Type A count:

3x \times 1.50 = 4.5x

. New Type B count:

5x \times 0.80 = 4x

.

This applies the 50% increase to Type A and the 20% decrease to Type B machine counts.

4

Calculate the new total hourly production rate using the value of

xr

.

New hourly rate:

(4.5x)(1.6r) + (4x)(r) = 7.2xr + 4xr = 11.2xr

. Substituting

xr = 10

yields

11.2 \times 10 = 112

widgets/hour.

This finds the rate at which widgets are produced in the new configuration.

5

Multiply the new hourly rate by the specified time of 3 hours.

Total production:

112 \text{ widgets/hour} \times 3 \text{ hours} = 336

widgets.

This yields the final total quantity of widgets produced over the 3-hour duration.

Anahtar Kavram

Solving compound ratio and rate problems with percentage changes

Alternatif Yöntem

Instead of variables, you can plug in convenient numbers that satisfy the ratios. Suppose there are initially

30

machines of Type A and

50

machines of Type B. Let a Type B machine produce at a rate of

1

widget/hour, so a Type A machine produces

1.6

widgets/hour. The total initial hourly rate is

30(1.6) + 50(1) = 48 + 50 = 98

widgets/hour, which matches the prompt. Increasing Type A by 50% gives

45

machines, and decreasing Type B by 20% gives

40

machines. The new hourly rate is

45(1.6) + 40(1) = 72 + 40 = 112

widgets/hour. For 3 hours, this gives

112 \times 3 = 336

widgets.

Tahmini Süre:3m 0s

Ratios, Rates, and Proportions

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