Fractions, Decimals, and Percentages Practice Questions — ACT

Question 1Question →

Four different saline solutions contain varying concentrations of dissolved salt: Solution P contains $\frac{7}{20}$ , Solution Q contains $36.5\%$ , Solution R contains $\frac{3}{8}$ , and Solution S contains $0.38$ salt concentration. Place the four solutions in order from least concentration to greatest concentration.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest concentration is Solution P (7/20), Solution Q (36.5%), Solution R (3/8), and Solution S (0.38).

Converting each quantity to a decimal gives Solution P = 0.350, Solution Q = 0.365, Solution R = 0.375, and Solution S = 0.380. Ordering these decimal values from smallest to largest yields Solution P, Solution Q, Solution R, and Solution S.

Step-by-Step Solution

1

Convert all concentration values into equivalent decimal representations.

Solution P:

\frac{7}{20} = 0.350

; Solution Q:

36.5\% = 0.365

; Solution R:

\frac{3}{8} = 0.375

; Solution S:

0.38 = 0.380

.

Converting all numbers to decimals allows for direct numerical comparison.

2

Compare the decimal values digit by digit from left to right.

0.350 < 0.365 < 0.375 < 0.380

Comparing the hundredths and thousandths places establishes the precise order.

3

Map the ordered decimal values back to their corresponding original solutions.

Solution P (

0.350

), Solution Q (

0.365

), Solution R (

0.375

), Solution S (

0.380

).

To complete the sequence from least to greatest.

Key Concept

Converting fractions, decimals, and percentages to a common numerical form (decimals) to compare and order them.

Question 2Question →

Four students are comparing the portion of their homework assignments they have completed:

* Jamie has completed $35\%$ of his assignment.
* Kayla has completed $\frac{3}{8}$ of her assignment.
* Leo has completed $0.4$ of his assignment.
* Mia has completed $\frac{1}{2}$ of her assignment.

Arrange the students by the portion of their homework completed, from least to greatest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

Jamie (

35\%

), Kayla (

\frac{3}{8}

), Leo (

0.4

), and Mia (

\frac{1}{2}

)

Converting all values to decimals yields

35\% = 0.35

,

\frac{3}{8} = 0.375

,

0.4 = 0.40

, and

\frac{1}{2} = 0.50

. Comparing these decimals gives

0.35 < 0.375 < 0.40 < 0.50

, which corresponds to Jamie, Kayla, Leo, and Mia.

Step-by-Step Solution

1

Convert each portion to a decimal value to make them easy to compare.

Jamie:

35\% = 0.35

Kayla:

\frac{3}{8} = 0.375

Leo:

0.4 = 0.40

Mia:

\frac{1}{2} = 0.50

Converting all values to a common format (decimals) allows for direct numerical comparison.

2

Compare the decimal values from least to greatest.

0.35 < 0.375 < 0.40 < 0.50

This establishes the relative sizes of the portions.

3

Match each decimal back to the student and order them.

Jamie (

35\%

), Kayla (

\frac{3}{8}

), Leo (

0.4

), Mia (

\frac{1}{2}

)

This provides the final ordered sequence.

Key Concept

Converting fractions, decimals, and percents to a single format to compare their values.

Question 3Question →

At Oakridge High School, $\frac{3}{5}$ of the students participate in extracurricular sports. Of the students who participate in extracurricular sports, $25\%$ are members of the track and field team. What percent of the total students at Oakridge High School are on the track and field team?

Show answer & explanation

Answer: 15

Answer

The correct answer is 15.

To find the portion of the total students on the track team, convert the fraction of students in sports to a decimal, which is 0.60. Then, multiply this by the track team percentage expressed as a decimal, which is 0.25. The product is 0.15, which represents 15% of the total student body.

Step-by-Step Solution

1

Convert the fraction of students playing sports to a decimal

0.60

To easily multiply the two portions, convert the fraction to a decimal:

\frac{3}{5} = 0.60

.

2

Multiply the sports-participating portion by the track team percentage

0.15

To find a percentage of a decimal, convert the percentage to a decimal (

25\% = 0.25

) and multiply:

0.60 \times 0.25 = 0.15

.

3

Convert the final decimal back to a percentage

15\%

Multiply the decimal by 100 to express the final portion as a percentage:

0.15 \times 100 = 15\%

.

Key Concept

Fractions, Decimals, and Percentages

Estimated Time:45s

Question 4Question →

A chef uses $\frac{1}{3}$ cup of milk and $\frac{1}{4}$ cup of water for a recipe. What fraction of a cup is the total liquid used in the recipe?

\frac{2}{7}

\frac{1}{12}

\frac{7}{12}

\frac{1}{7}

\frac{1}{6}

Show answer & explanation

Answer: $\frac{7}{12}$

Answer

The correct fraction of a cup of liquid is

\frac{7}{12}

.

The correct answer is found by finding a common denominator for the two fractions. The least common multiple of 3 and 4 is 12. Converting the fractions gives

\frac{4}{12}

and

\frac{3}{12}

. Adding these fractions yields

\frac{7}{12}

.

Step-by-Step Solution

1

Identify the mathematical operation needed to find the total liquid.

The total amount of liquid is the sum of the two fractions:

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

.

To find the combined total of two separate quantities, addition is required.

2

Find a common denominator for the two fractions.

The least common multiple of 3 and 4 is 12. Convert the fractions:

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

and

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

.

Fractions must have the same denominator before they can be added.

3

Add the converted fractions.

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

.

Add the numerators together and keep the common denominator.

Key Concept

Adding fractions with unlike denominators.

Estimated Time:45s

Question 5Question →

A store sells a winter coat that is originally priced at $120. During a weekend clearance sale, the coat's price is reduced by 25%. What is the sale price of the coat, in dollars?

Show answer & explanation

Answer: 90

Answer

The sale price of the coat is $90.

The correct answer is

90. To find the sale price, we compute the discount by taking 25% of

120, which is

30, and then subtract that discount from the original price of

120.

Step-by-Step Solution

1

Find 25% of $120 to determine the discount amount.

$30

A 25% reduction means we calculate

120 multiplied by 0.25, which equals

30.

2

Subtract the discount from the original price.

$90

The sale price is the original price minus the discount:

120 -

30 = $90.

Key Concept

Calculating a percentage markdown to determine a final price

Estimated Time:45s

Question 6Question →

In a local park, $\frac{2}{5}$ of the trees are maple trees and $\frac{1}{3}$ of the trees are oak trees. The remaining trees are pine trees. What fraction of the trees in the park are pine trees?

\frac{3}{8}

\frac{5}{8}

\frac{4}{15}

\frac{11}{15}

\frac{14}{15}

Show answer & explanation

Answer: $\frac{4}{15}$

Answer

\frac{4}{15}

To find the fraction of pine trees, first calculate the total fraction occupied by maple and oak trees. Finding a common denominator of

15

for

\frac{2}{5}

and

\frac{1}{3}

gives

\frac{6}{15} + \frac{5}{15} = \frac{11}{15}

. Subtracting this combined fraction from

1

(represented as

\frac{15}{15}

) yields the remaining fraction of pine trees, which is

\frac{4}{15}

.

Step-by-Step Solution

1

Find the sum of the fractions representing maple and oak trees.

\frac{2}{5} + \frac{1}{3} = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}

Before finding the remaining fraction, we need to know the combined fraction of the park occupied by maple and oak trees.

2

Subtract the combined fraction of maple and oak trees from the whole.

1 - \frac{11}{15} = \frac{15}{15} - \frac{11}{15} = \frac{4}{15}

Since the remaining trees are pine trees, subtracting the combined fraction of other trees from the total of

1

yields the fraction of pine trees.

Key Concept

To find a remaining fractional part of a whole, first sum the known fractional parts using a common denominator, and then subtract this sum from 1.

Question 7Question →

At the start of the fiscal year, a city's municipal budget is divided among three departments: Education, Healthcare, and Infrastructure. Education receives $\frac{3}{8}$ of the total budget, Healthcare receives $40\%$ of the total budget, and Infrastructure receives the remainder. Midyear, the total municipal budget is increased by $25\%$ . The budget allocated to Education is increased by $20\%$ of its original amount, and the budget allocated to Healthcare is increased by $15\%$ of its original amount. What percent of the new total municipal budget is allocated to Infrastructure?

Show answer & explanation

Answer: 27.2

Answer

27.2

The correct answer is

27.2\%

. By representing the initial total budget as

1

, we find the initial Infrastructure allocation is

1 - (0.375 + 0.40) = 0.225

. After the budget increase of

25\%

, the new total budget is

1.25

. The new Education allocation is

0.375 \times 1.20 = 0.45

, and the new Healthcare allocation is

0.40 \times 1.15 = 0.46

. The remaining amount for Infrastructure is

1.25 - 0.45 - 0.46 = 0.34

. Expressing

0.34

as a percentage of the new total budget

1.25

gives

\frac{0.34}{1.25} \times 100\% = 27.2\%

.

Step-by-Step Solution

1

Convert the initial budget allocations to decimal shares of the original total budget.

Education share is

0.375

, Healthcare share is

0.40

, and Infrastructure share is

0.225

.

Expressing all initial shares as decimals relative to a total budget of

1.0

makes successive calculations straightforward.

2

Calculate the updated budget amounts for the entire city, Education, and Healthcare.

The new total budget is

1.25

, the new Education allocation is

0.375 \times 1.20 = 0.45

, and the new Healthcare allocation is

0.40 \times 1.15 = 0.46

.

Adjust each allocation by its respective percentage change to find its share relative to the initial total.

3

Compute the remaining budget share left for Infrastructure.

New Infrastructure share is

1.25 - (0.45 + 0.46) = 0.34

of the initial budget.

The sum of the three department budgets must equal the new total budget of

1.25

.

4

Find the percentage of the new total budget represented by the new Infrastructure allocation.

\frac{0.34}{1.25} \times 100\% = 27.2\%

Divide the new Infrastructure share by the new total budget to find the new ratio, then convert to a percentage.

Key Concept

Converting between fractions, decimals, and percentages, and tracking changes across multiple base values.

Alternative Method

Instead of using

1.0

as the base, you can assume an initial total budget of

800

dollars (since

800

is a multiple of

8

, which simplifies

\frac{3}{8}

). Initial Education =

300

, Healthcare =

320

, Infrastructure =

180

. The new total budget is

800 \times 1.25 = 1000

. New Education =

300 \times 1.20 = 360

. New Healthcare =

320 \times 1.15 = 368

. New Infrastructure =

1000 - (360 + 368) = 272

. Percentage =

\frac{272}{1000} \times 100\% = 27.2\%

.

Estimated Time:3m 0s

Question 8Question →

An investment fund allocates its capital into three portfolios: Real Estate, Technology, and Green Energy. Initially, $\frac{1}{4}$ of the total capital is allocated to Real Estate, and $\frac{2}{5}$ of the total capital is allocated to Technology. The remaining capital is allocated to Green Energy. During the first year, the Real Estate portfolio decreases in value by $16\%$ , the Technology portfolio increases in value by $15\%$ , and the Green Energy portfolio increases in value by $p\%$ . If the total value of the entire investment fund at the end of the first year has increased by $9\%$ of its initial value, what is the value of $p$ ?

10.0

10.5

18.9

20.0

31.4

Show answer & explanation

Answer: $20.0$

Answer

The value of

p

is

20.0

.

The correct answer is

20.0

. By assuming a total capital of

100

units, we find that the Real Estate portfolio starts at

25

units, the Technology portfolio at

40

units, and the Green Energy portfolio at

35

units. The change in Real Estate is

-4

units, and the change in Technology is

+6

units. To reach a total fund increase of

9

units, the change in the Green Energy portfolio must satisfy

-4 + 6 + \text{Change in Green Energy} = 9

, which means the Green Energy portfolio must increase by

7

units. Finding what percent

7

is of

35

yields

\frac{7}{35} = 0.20

, or

20\%

.

Step-by-Step Solution

1

Determine the initial allocation of capital for each portfolio by assuming a total initial capital of

100

units.

Real Estate receives

\frac{1}{4} \times 100 = 25

units. Technology receives

\frac{2}{5} \times 100 = 40

units. The remaining capital allocated to Green Energy is

100 - (25 + 40) = 35

units.

This establishes the weighted share of each portfolio relative to the total capital.

2

Calculate the changes in value for the Real Estate and Technology portfolios based on their given percentage changes.

Real Estate change:

-16\% \text{ of } 25 = -0.16 \times 25 = -4

units. Technology change:

+15\% \text{ of } 40 = 0.15 \times 40 = +6

units.

This determines the absolute change in value contributed by these two portfolios.

3

Calculate the total change in value for the entire fund.

Total fund change:

+9\% \text{ of } 100 = +9

units.

The total change in the fund is the sum of the individual changes in the portfolios.

4

Set up an equation representing the sum of the changes in all three portfolios and solve for

p

.

-4 + 6 + \left(\frac{p}{100} \times 35\right) = 9 \implies 2 + 0.35p = 9 \implies 0.35p = 7 \implies p = 20

.

This solves for the unknown percentage rate of increase required for the Green Energy portfolio.

Key Concept

Weighted average of percentage changes using fractional and decimal representations.

Alternative Method

Instead of choosing an arbitrary starting capital of

100

, you can use decimals directly. Let

C

be the total capital. The weighted changes are

0.25C(-0.16) + 0.40C(0.15) + 0.35C\left(\frac{p}{100}\right) = 0.09C

. Dividing both sides by

C

yields

-0.04 + 0.06 + 0.0035p = 0.09

. Simplifying gives

0.02 + 0.0035p = 0.09 \implies 0.0035p = 0.07 \implies p = 20

.

Estimated Time:3m 0s

Question 9Question →

A laboratory technician measured the concentration of active reagent in four different solutions and recorded the values in various numerical forms: Solution W contains $\frac{7}{16}$ active reagent, Solution X contains $42.5\%$ active reagent, Solution Y contains $0.44$ active reagent, and Solution Z contains $\frac{4}{9}$ active reagent. What is the correct order of the four solutions from the least concentration of active reagent to the greatest concentration?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest concentration is Solution X (

42.5\%

), Solution W (

\frac{7}{16}

), Solution Y (

0.44

), and Solution Z (

\frac{4}{9}

).

To arrange values given in different numerical formats, convert each to decimal form: Solution X equals

0.4250

, Solution W equals

0.4375

, Solution Y equals

0.4400

, and Solution Z equals

0.4444...

. Arranging these decimals from least to greatest yields Solution X, Solution W, Solution Y, then Solution Z.

Step-by-Step Solution

1

Convert all values to decimal format for direct comparison.

Solution W:

\frac{7}{16} = 0.4375

; Solution X:

42.5\% = 0.425

; Solution Y:

0.44 = 0.4400

; Solution Z:

\frac{4}{9} = 0.4444...

Converting fractions, decimals, and percentages to a single decimal format makes order comparisons straightforward.

2

Compare the resulting decimal values place by place.

0.4250 < 0.4375 < 0.4400 < 0.4444...

Comparing digits from left to right establishes the numerical sequence.

3

Match the ordered decimals back to their corresponding solution names.

Solution X (

0.425

) < Solution W (

0.4375

) < Solution Y (

0.44

) < Solution Z (

\frac{4}{9}

)

The question requires ranking the original solution labels.

Key Concept

Converting fractions, decimals, and percentages into a uniform decimal format to compare and order rational numbers

Question 10Question →

A community library has a collection of 250 books. If $44\%$ of these books are classified as fiction, how many fiction books are in the library's collection?

Show answer & explanation

Answer: 110

Answer

There are 110 fiction books in the library's collection.

To find

44\%

of 250 books, first convert

44\%

to a decimal by dividing by 100, which is

0.44

. Then, multiply the decimal by the total number of books:

0.44 \times 250 = 110

. This represents the number of fiction books in the library.

Step-by-Step Solution

1

Convert the percentage of fiction books to a decimal.

0.44

To write a percent as a decimal, divide by 100 or move the decimal point two places to the left.

2

Multiply the decimal by the total number of books to find the number of fiction books.

110

Multiplying the decimal representing the fraction of the total by the total number of items gives the portion representing fiction books.

Key Concept

Calculating a percentage of a total amount

Estimated Time:45s

Question 11Question →

A certain metal alloy is made of copper, zinc, and nickel. By weight, $\frac{1}{4}$ of the alloy is copper, and $30\%$ of the alloy is zinc. What fraction of the alloy is nickel?

\frac{21}{20}

\frac{5}{7}

\frac{9}{20}

\frac{1}{2}

\frac{21}{40}

Show answer & explanation

Answer: $\frac{9}{20}$

Answer

The fraction of the alloy that is nickel is

\frac{9}{20}

.

The correct answer is the fraction representing the remaining part of the alloy after accounting for copper and zinc. First, convert

30\%

to the fraction

\frac{3}{10}

. The combined fraction of copper and zinc is

\frac{1}{4} + \frac{3}{10} = \frac{5}{20} + \frac{6}{20} = \frac{11}{20}

. Subtracting this combined fraction from

1

yields the fraction of nickel:

1 - \frac{11}{20} = \frac{9}{20}

.

Step-by-Step Solution

1

Convert the percentage of zinc to a fraction.

Since

30\% = \frac{30}{100}

, this simplifies to

\frac{3}{10}

.

To add or subtract the proportions of the alloy, they must be expressed in the same form (fractions).

2

Calculate the combined fraction of copper and zinc.

\frac{1}{4} + \frac{3}{10} = \frac{5}{20} + \frac{6}{20} = \frac{11}{20}

.

To find how much of the alloy is made of copper and zinc combined, we add their individual fractions using a common denominator of 20.

3

Subtract the combined fraction from the whole to find the fraction of nickel.

1 - \frac{11}{20} = \frac{20}{20} - \frac{11}{20} = \frac{9}{20}

.

Since the entire alloy represents a whole (

1

), the remaining portion must be nickel.

Key Concept

Expressing percentages as fractions and performing operations with fractions to solve part-to-whole problems.

Question 12Question →

A student took a history exam containing $80$ questions. If the student answered $15\%$ of the questions incorrectly, how many questions did the student answer correctly?

Show answer & explanation

Answer: 68

Answer

The student answered

68

questions correctly.

First, find the percentage of correct answers by subtracting the percentage of incorrect answers (

15\%

) from

100\%

, which gives

85\%

. Convert

85\%

to a decimal (

0.85

) and multiply it by the total number of questions (

80

):

80 \times 0.85 = 68

.

Step-by-Step Solution

1

Calculate the percentage of questions answered correctly.

85\%

Subtract the percentage of incorrect questions (

15\%

) from the total (

100\%

) to find the percentage of correct questions.

2

Convert the percentage of correct questions to a decimal.

0.85

Divide the percentage by

100

to express it as a decimal.

3

Multiply the total questions by the decimal value.

68

Multiplying the total number of questions by the proportion of correct answers gives the total number of correct questions:

80 \times 0.85 = 68

.

Key Concept

Calculating percentages of a whole number

Question 13Question →

During a physical education class, a student ran $\frac{1}{4}$ of a mile and walked $\frac{3}{5}$ of a mile. What total distance, in miles, did the student cover by running and walking?

\frac{4}{9}

\frac{13}{20}

\frac{23}{40}

\frac{17}{20}

\frac{7}{10}

Show answer & explanation

Answer: $\frac{17}{20}$

Answer

The correct answer is

\frac{17}{20}

miles, which represents the sum of the distances covered by running and walking.

The correct answer is found by finding a common denominator for the two fractions representing the distances. The least common multiple of

4

and

5

is

20

. Converting the fractions gives

\frac{5}{20}

and

\frac{12}{20}

. Adding these values yields a total of

\frac{17}{20}

miles.

Step-by-Step Solution

1

Set up the addition expression for the two distances.

\frac{1}{4} + \frac{3}{5}

To find the total distance, the distance run and the distance walked must be added together.

2

Find the least common denominator (LCD) for the fractions.

The LCD for

4

and

5

is

20

.

Fractions can only be added when they share a common denominator.

3

Convert each fraction to an equivalent fraction with the denominator of

20

.

\frac{1 \times 5}{4 \times 5} = \frac{5}{20}

and

\frac{3 \times 4}{5 \times 4} = \frac{12}{20}

This scales the fractions correctly so they can be combined.

4

Add the numerators together while keeping the denominator the same.

\frac{5 + 12}{20} = \frac{17}{20}

Adding the numerators of fractions with like denominators gives the total fractional quantity.

Key Concept

Adding fractions with unlike denominators by finding a common denominator.

Alternative Method

Convert the fractions to decimals before adding.

\frac{1}{4}

is equivalent to

0.25

, and

\frac{3}{5}

is equivalent to

0.60

. Adding these decimals yields

0.25 + 0.60 = 0.85

. Converting the decimal back to a fraction results in

\frac{85}{100} = \frac{17}{20}

.

Estimated Time:45s

Question 14Question →

A laboratory mixture is prepared by combining three liquid solutions: Solution X, Solution Y, and Solution Z. Initially, Solution X makes up $20\%$ of the total volume of the mixture. Solution Y's volume is $\frac{1}{3}$ of the combined volume of Solution X and Solution Z, and Solution Z makes up the remaining portion of the mixture. If $3.0$ liters of Solution Z are added to the mixture, Solution Z then constitutes exactly $58\%$ of the new total volume. What was the original total volume, in liters, of the mixture before Solution Z was added?

15

24

35

42

100

Show answer & explanation

Answer: 42

Answer

The original total volume of the mixture was 42 liters.

The correct option represents the original volume of

42

liters. Setting the original total volume to

V

, we find that Solution X is

0.20V

. Since Solution Y is one-third of the combined volume of X and Z, we write

Y = \frac{1}{3}(V - Y)

, which simplifies to

Y = 0.25V

. The remaining portion, Solution Z, must be

V - 0.20V - 0.25V = 0.55V

. Adding

3.0

liters of Solution Z increases both the volume of Solution Z and the total volume of the mixture, yielding the proportion

(0.55V + 3) / (V + 3) = 0.58

. Solving this linear equation results in

0.03V = 1.26

, which gives

V = 42

.

Step-by-Step Solution

1

Express the initial fraction of Solution X as a decimal.

Solution X makes up

0.20

of the total volume.

Converting

20\%

to a decimal simplifies subsequent algebraic modeling.

2

Set up an equation for the fraction of Solution Y in terms of the total volume

V

.

Solution Y makes up

0.25

of the total volume.

If Solution Y's volume is

\frac{1}{3}

of the combined volume of Solution X and Solution Z, then

Y = \frac{1}{3}(V - Y)

. Multiplying by

3

gives

3Y = V - Y

, which simplifies to

4Y = V

, or

Y = 0.25V

.

3

Determine the initial fraction of Solution Z in the mixture.

Solution Z initially makes up

0.55

of the total volume.

Subtracting the fractions of Solution X and Solution Y from the whole yields

1 - 0.20 - 0.25 = 0.55

.

4

Set up and solve the equation representing the addition of Solution Z.

V = 42

liters

Adding

3.0

liters of Solution Z increases the volume of Solution Z to

0.55V + 3

and the total volume to

V + 3

. The equation is

\frac{0.55V + 3}{V + 3} = 0.58

. Multiplying both sides by

V + 3

yields

0.55V + 3 = 0.58(V + 3) \implies 0.55V + 3 = 0.58V + 1.74

. Rearranging terms gives

3 - 1.74 = 0.58V - 0.55V \implies 1.26 = 0.03V \implies V = 42

.

Key Concept

Solving multi-step mixture word problems using equations involving fractions, decimals, and percentages.

Alternative Method

Instead of variables for the total volume, solve the problem by tracking parts. Initially, the mixture consists of

20\%

Solution X,

25\%

Solution Y, and

55\%

Solution Z. The ratio of the volume of Solution Z to the combined volume of Solution X and Solution Y is

55 : 45

, which simplifies to

11 : 9

. Let the volume of X and Y combined be

9x

liters, and the initial volume of Z be

11x

liters, making the initial total volume

20x

liters. Adding

3.0

liters of Solution Z does not change the combined volume of X and Y (

9x

liters). In the new mixture, Solution Z constitutes

58\%

, meaning the combined volume of X and Y must constitute

100\% - 58\% = 42\%

of the new mixture. Thus, the new total volume is

9x / 0.42 = 150x / 7

liters. The difference between the new and original total volumes is the

3.0

liters added:

(150x / 7) - 20x = 3 \implies 10x / 7 = 3 \implies x = 2.1

. The original total volume was

20x = 20(2.1) = 42

liters.

Estimated Time:3m 0s

Question 15Question →

Arrange the following numbers in order from least to greatest: $\frac{2}{5}$ , $0.38$ , $45\%$ , and $0.42$ .

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order of the values from least to greatest is

0.38

,

\frac{2}{5}

,

0.42

, and

45\%

.

Converting all numbers to decimals allows for direct comparison:

0.38

remains

0.38

; the fraction

\frac{2}{5}

equals

0.40

;

0.42

remains

0.42

; and

45\%

equals

0.45

. Comparing these decimal values, we get

0.38 < 0.40 < 0.42 < 0.45

. Therefore, the correct order from least to greatest is

0.38

,

\frac{2}{5}

,

0.42

, and

45\%

.

Step-by-Step Solution

1

Convert the fraction to a decimal.

\frac{2}{5} = 0.40

Converting all values to decimals makes them easier to compare.

2

Convert the percentage to a decimal.

45\% = 0.45

Percentages represent values out of 100, so

45\% = \frac{45}{100} = 0.45

.

3

Compare the four decimal numbers.

0.38 < 0.40 < 0.42 < 0.45

By comparing the tenths and hundredths place, we find the correct order from smallest to largest.

Key Concept

Comparing and ordering fractions, decimals, and percents by converting them to a common decimal format.

Question 16Question →

A water purification plant processes raw water through three successive filtration stages: Stage A, Stage B, and Stage C. Stage A removes $\frac{3}{8}$ of the impurities present in the raw water. Stage B removes $40\%$ of the remaining impurities. Stage C removes $0.75$ of the impurities that remain after Stage B. If $4.5$ kilograms of impurities are successfully removed in Stage C, how many kilograms of impurities were originally in the raw water?

Show answer & explanation

Answer: 16

Answer

The original amount of impurities in the raw water was

16

kilograms.

The correct answer is

16

kg. To find this, we express the remaining impurities at each stage as a fraction of the initial amount

x

. After Stage A,

\frac{5}{8}x

remains. In Stage B,

40\%

is removed, meaning

60\%

of

\frac{5}{8}x

, or

\frac{3}{8}x

, remains. In Stage C,

0.75

(or

\frac{3}{4}

) of this remainder is removed, which is

\frac{3}{4} \times \frac{3}{8}x = \frac{9}{32}x

. Setting

\frac{9}{32}x = 4.5

gives

x = 16

.

Step-by-Step Solution

1

Represent the initial mass of impurities in the raw water with a variable.

Let

x

be the initial kilograms of impurities.

Establishing a variable allows setting up an equation to track the changes through each stage.

2

Calculate the remaining impurities after Stage A.

Impurities remaining =

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

kg.

Stage A removes

\frac{3}{8}

of the initial impurities, so

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

of the impurities remain.

3

Calculate the impurities removed and remaining after Stage B.

Impurities removed in Stage B =

0.40 \times \frac{5}{8}x = \frac{1}{4}x

kg. Impurities remaining after Stage B = \frac{5}{8}x - \frac{1}{4}x = \frac{3}{8}x$ kg.

Stage B removes

40\%

of the impurities that remained after Stage A. Subtracting the removed portion from the starting amount for this stage yields the remaining fraction.

4

Express the impurities removed in Stage C.

Impurities removed in Stage C =

0.75 \times \frac{3}{8}x = \frac{9}{32}x

kg.

Stage C removes

0.75

(or

\frac{3}{4}

) of the impurities remaining after Stage B.

5

Equate the Stage C expression to the given value and solve for

x

.

\frac{9}{32}x = 4.5 \implies x = 4.5 \times \frac{32}{9} = 16

kg.

The problem states that

4.5

kg of impurities are removed in Stage C, so solving this equation yields the initial value.

Key Concept

Solving multi-step word problems involving successive applications of fractions, decimals, and percentages.

Alternative Method

Working backwards from the final stage can simplify the calculations. Since Stage C removes

0.75

of the impurities remaining after Stage B, and this amount equals

4.5

kg, the amount remaining after Stage B is

\frac{4.5}{0.75} = 6

kg. Since Stage B removes

40\%

of the impurities remaining after Stage A, the

6

kg represents

100\% - 40\% = 60\%

of the impurities remaining after Stage A. Thus, the amount remaining after Stage A is

\frac{6}{0.60} = 10

kg. Finally, since Stage A removes

\frac{3}{8}

of the original impurities, the

10

kg represents

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

of the original impurities. The original amount is therefore

10 \times \frac{8}{5} = 16

kg.

Estimated Time:3m 0s

Question 17Question →

A baker is preparing a recipe that calls for $2$ times the sum of $\frac{1}{4}$ cup of milk and $\frac{2}{5}$ cup of water. What is the total volume, in cups, of these two liquid ingredients combined?

\frac{2}{3}

\frac{9}{10}

\frac{13}{10}

\frac{34}{35}

\frac{13}{20}

Show answer & explanation

Answer: $\frac{13}{10}$

Answer

The correct answer is

\frac{13}{10}

cups.

To find the total volume, first determine the sum of the two fractions by finding a common denominator:

\frac{1}{4} + \frac{2}{5} = \frac{5}{20} + \frac{8}{20} = \frac{13}{20}

cups. Then, multiply this sum by

2

as required by the recipe:

2 \times \frac{13}{20} = \frac{26}{20} = \frac{13}{10}

cups.

Step-by-Step Solution

1

Find a common denominator for the two fractions to be added.

The common denominator for 4 and 5 is 20, converting the fractions to

\frac{5}{20}

and

\frac{8}{20}

.

Fractions must have the same denominator before they can be added.

2

Add the two converted fractions.

\frac{5}{20} + \frac{8}{20} = \frac{13}{20}

.

To find the sum of the ingredients inside the parentheses.

3

Multiply the sum of the fractions by 2.

2 \times \frac{13}{20} = \frac{26}{20} = \frac{13}{10}

.

The recipe calls for 2 times the sum of the two liquid ingredients.

Key Concept

Adding fractions with unlike denominators and performing operations in the correct order.

Question 18Question →

A company allocates its annual budget to three departments: Research, Marketing, and Operations. Initially, the Research department receives $\frac{1}{4}$ of the total budget. Marketing receives $40\%$ of the remaining budget, and Operations receives the rest. Mid-year, the budget is adjusted: Research's budget is decreased by $16\%$ of its initial allocation, and Marketing's budget is increased by $\frac{1}{12}$ of its initial allocation. If the total company budget remains unchanged, by what percentage must the Operations department's budget increase relative to its initial allocation?

1.5\%

3\frac{1}{3}\%

5\frac{1}{3}\%

7\frac{2}{3}\%

14\frac{4}{9}\%

Show answer & explanation

Answer: $3\frac{1}{3}\%$

Answer

The Operations department's budget must increase by

3\frac{1}{3}\%

relative to its initial allocation.

The correct answer is

3\frac{1}{3}\%

. Under a total budget of

1,000

units, Research initially receives

250

units, Marketing receives

300

units (

40\%

of the remaining

750

), and Operations receives the remaining

450

units. The mid-year decrease for Research is

40

units (

16\%

of

250

), and the increase for Marketing is

25

units (

\frac{1}{12}

of

300

). To keep the total budget constant, the Operations budget must increase by

15

units (

40 - 25

). Relative to its initial budget of

450

units, this represents an increase of

\frac{15}{450} \times 100\% = 3\frac{1}{3}\%

.

Step-by-Step Solution

1

Set up a total initial budget and find initial allocations for each department.

Let the total initial budget be

1,000

units. Research receives

\frac{1}{4}

of the budget, which is

250

units. The remaining budget is

1,000 - 250 = 750

units. Marketing receives

40\%

of this remaining budget, which is

0.40 \times 750 = 300

units. Operations receives the rest, which is

1,000 - 250 - 300 = 450

units.

Establishing the initial values of each department's budget provides a base to calculate subsequent changes.

2

Calculate the mid-year budget changes for Research and Marketing.

Research decreases by

16\%

:

-0.16 \times 250 = -40

units. Marketing increases by

\frac{1}{12}

:

+\frac{1}{12} \times 300 = +25

units.

Finding the individual change values shows the net budget surplus or deficit before adjusting Operations.

3

Determine the required budget change for the Operations department.

To keep the total budget unchanged, the net sum of changes must be zero:

-40 + 25 + \Delta O = 0 \Rightarrow \Delta O = +15

units.

The change in the Operations budget must balance out the changes in the other two departments.

4

Calculate the percentage increase for the Operations department relative to its initial budget.

Percentage increase =

\frac{15}{450} \times 100\% = \frac{1}{30} \times 100\% = \frac{10}{3}\% = 3\frac{1}{3}\%

.

Dividing the change in Operations by its initial budget yields the relative percentage increase.

Key Concept

Calculating percentage changes across weighted parts of a whole

Question 19Question →

Four investment options, each with a different starting value (expressed as a fraction, decimal, or percentage of a standard initial budget $B$ ) and a specified first-year growth rate, are described below. Arrange the options in order from the lowest final value to the highest final value after the first year.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

Option Y, Option Z, Option X, Option W

The correct order is Option Y, Option Z, Option X, Option W. Converting each initial value to a decimal fraction of

B

and applying the respective growth multipliers yields: Option Y is

0.90 \times 1.15 = 1.035B

, Option Z is

0.875 \times 1.184 = 1.036B

, Option X is

0.85 \times 1.22 = 1.037B

, and Option W is

0.80 \times 1.30 = 1.040B

. Comparing these decimal values gives

1.035 < 1.036 < 1.037 < 1.040

.

Step-by-Step Solution

1

Calculate the final value of Option W relative to

B

.

0.80 \times 1.30 \times B = 1.040B

Convert the starting fraction

\frac{4}{5}

to the decimal

0.80

. A

30\%

increase corresponds to a multiplier of

1 + 0.30 = 1.30

. Multiply these values together:

0.80 \times 1.30 = 1.040

.

2

Calculate the final value of Option X relative to

B

.

0.85 \times 1.22 \times B = 1.037B

The starting decimal is

0.85

. A

22\%

increase corresponds to a multiplier of

1 + 0.22 = 1.22

. Multiply these values together:

0.85 \times 1.22 = 1.037

.

3

Calculate the final value of Option Y relative to

B

.

0.90 \times 1.15 \times B = 1.035B

Convert the starting percentage

90\%

to the decimal

0.90

. A

15\%

increase corresponds to a multiplier of

1 + 0.15 = 1.15

. Multiply these values together:

0.90 \times 1.15 = 1.035

.

4

Calculate the final value of Option Z relative to

B

.

0.875 \times 1.184 \times B = 1.036B

Convert the starting fraction

\frac{7}{8}

to the decimal

0.875

. An

18.4\%

increase corresponds to a multiplier of

1 + 0.184 = 1.184

. Multiply these values together:

0.875 \times 1.184 = 1.036

.

5

Compare the resulting multipliers to order the final values from lowest to highest.

1.035 < 1.036 < 1.037 < 1.040

, which translates to Option Y < Option Z < Option X < Option W.

By arranging the computed multipliers in ascending order, we find the correct sequence.

Key Concept

Converting fractions, decimals, and percentages to common decimal representations to compute and compare multi-step percentage increases.

Question 20Question →

A school garden has flowers of three colors: red, yellow, and white. Red flowers make up $\frac{2}{5}$ of the garden, yellow flowers make up $\frac{1}{4}$ of the garden, and the remaining flowers are white. What percentage of the flowers in the garden are white?

33%

35%

54%

65%

67%

Show answer & explanation

Answer: 35%

Answer

The percentage of white flowers in the garden is 35%.

To find the percentage of white flowers, we convert the fractional parts of the other flowers to percentages: red flowers make up

\frac{2}{5} = 40\%

of the garden, and yellow flowers make up

\frac{1}{4} = 25\%

of the garden. Adding these together, red and yellow flowers account for

40\% + 25\% = 65\%

of the garden. Subtracting this from the total garden (

100\%

) yields

100\% - 65\% = 35\%

for the white flowers.

Step-by-Step Solution

1

Convert the fraction of red flowers to a percentage.

\frac{2}{5} = 0.40 = 40\%

Converting all components to percentages allows for direct addition and subtraction.

2

Convert the fraction of yellow flowers to a percentage.

\frac{1}{4} = 0.25 = 25\%

This puts the yellow flowers in the same percentage unit as the red flowers.

3

Add the percentages of red and yellow flowers to find their combined share of the garden.

40\% + 25\% = 65\%

This gives the total portion of the garden occupied by non-white flowers.

4

Subtract the combined share from the total garden percentage (100%) to find the percentage of white flowers.

100\% - 65\% = 35\%

Since the entire garden represents 100%, subtracting the non-white portion yields the remaining white portion.

Key Concept

Converting fractions to percentages to solve part-to-whole word problems.

Estimated Time:45s

Fractions, Decimals, and Percentages

Answer

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