Fractions, Decimals, and Percentages Practice Questions — ACT — Page 2

Question 21Question →

In a school auditorium, $\frac{3}{8}$ of the total seats are in the balcony, and the remaining seats are on the main floor. For the spring play, $80\%$ of the balcony seats were occupied, and $60\%$ of the main floor seats were occupied. If there were exactly 117 empty seats in the auditorium, what was the total number of seats in the auditorium?

Show answer & explanation

Answer: 360

Answer

The total number of seats in the auditorium is 360.

The correct answer of 360 is found by expressing the empty seats from each section as a fraction of the total seats. Since the balcony represents

\frac{3}{8}

of the total seats and is

20\%

empty, the empty balcony seats equal

\frac{3}{40}

of the total seats. The main floor represents

\frac{5}{8}

of the total seats and is

40\%

empty, so the empty main floor seats equal

\frac{10}{40}

of the total seats. Summing these yields

\frac{13}{40}

of the total seats, which equals 117. Solving

\frac{13}{40}S = 117

gives a total of 360 seats.

Step-by-Step Solution

1

Define the variable for the total seats and find the fraction of seats in each section.

Let the total number of seats be

S

. The balcony has

\frac{3}{8}S

seats, and the main floor has

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

seats.

Establishing fractional partitions of the total capacity is necessary to set up the equation.

2

Calculate the fraction of total seats that are empty in the balcony.

Empty balcony seats =

20\%

of

\frac{3}{8}S = 0.20 \times \frac{3}{8}S = \frac{1}{5} \times \frac{3}{8}S = \frac{3}{40}S

.

Since 80% of the balcony seats are occupied, the remaining 20% are empty.

3

Calculate the fraction of total seats that are empty on the main floor.

Empty main floor seats =

40\%

of

\frac{5}{8}S = 0.40 \times \frac{5}{8}S = \frac{2}{5} \times \frac{5}{8}S = \frac{10}{40}S

.

Since 60% of the main floor seats are occupied, the remaining 40% are empty.

4

Sum the empty seat fractions and set the sum equal to the total number of empty seats.

Total empty seats =

\frac{3}{40}S + \frac{10}{40}S = \frac{13}{40}S

. The equation is

\frac{13}{40}S = 117

.

Combining the empty seats from both sections represents the total empty seat count.

5

Solve the equation for the total number of seats

S

.

S = 117 \times \frac{40}{13} = 9 \times 40 = 360

.

Multiplying by the reciprocal of the empty seats fraction isolates the variable for the total seats.

Key Concept

Solving multi-step word problems involving fractions, decimal conversions, and percentages.

Question 22Question →

A video streaming service has three subscription tiers: Basic, Standard, and Premium. Originally, $\frac{2}{5}$ of the subscribers are on the Basic tier and $45\%$ are on the Standard tier. During a promotional campaign, $\frac{1}{4}$ of the Basic subscribers upgrade to the Standard tier, and $20\%$ of the original Standard subscribers upgrade to the Premium tier. If no other subscription changes occur, what percent of the total subscribers are now on the Premium tier?

24\%

26\%

35\%

65\%

9\%

Show answer & explanation

Answer: $24\%$

Answer

The correct answer is

24\%

of the total subscribers.

To find the new Premium subscriber percentage, we first convert the basic tier fraction to a percent:

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

. Since the Standard tier is

45\%

, the original Premium tier must contain the remaining subscribers, which is

100\% - 40\% - 45\% = 15\%

of the total. The upgrade represents

20\%

of the original

45\%

Standard subscribers, which is

0.20 \times 45\% = 9\%

of the total subscribers. Adding this

9\%

upgrade to the original

15\%

Premium percentage results in a final percentage of

24\%

.

Step-by-Step Solution

1

Determine the original subscriber percentages for all three tiers.

Basic:

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

. Standard:

45\%

. Premium:

100\% - (40\% + 45\%) = 15\%

.

Establishing the starting percentages allows us to track changes relative to the total subscriber population.

2

Calculate the percentage of total subscribers upgrading from Standard to Premium.

20\%

of the original

45\%

Standard subscribers =

0.20 \times 45\% = 9\%

of the total subscribers.

Since the upgrade rate is relative to the Standard tier, we multiply the rate by the Standard tier's share of the total population.

3

Calculate the final Premium percentage by adding the upgrade percentage to the original Premium percentage.

15\% + 9\% = 24\%

.

The final Premium percentage is the sum of the original Premium percentage and the percentage of subscribers who upgraded from Standard.

Key Concept

Calculating percentages of subpopulations and translating them to percentages of the entire population.

Question 23Question →

Four different brands of a sports drink contain different proportions of electrolytes by weight. The proportions for each brand are listed below:

* Brand P: $\frac{5}{12}$ of its weight is electrolytes
* Brand Q: $41.8\%$ of its weight is electrolytes
* Brand R: $0.416$ of its weight is electrolytes
* Brand S: $\frac{13}{30}$ of its weight is electrolytes

Arrange the brands in order from the least proportion of electrolytes to the greatest proportion of electrolytes.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest proportion of electrolytes is Brand R, Brand P, Brand Q, and Brand S.

To compare the values, converting each to a decimal is the most reliable strategy. Brand R is already a decimal (

0.416

). Converting Brand P yields

\frac{5}{12} \approx 0.4167

. Converting Brand Q yields

41.8\% = 0.418

. Converting Brand S yields

\frac{13}{30} \approx 0.4333

. Comparing these decimals reveals that

0.416 < 0.4167 < 0.418 < 0.4333

, corresponding to the sequence: Brand R, Brand P, Brand Q, Brand S.

Step-by-Step Solution

1

Identify and list all values given for each brand.

Brand P:

\frac{5}{12}

, Brand Q:

41.8\%

, Brand R:

0.416

, Brand S:

\frac{13}{30}

.

This establishes the raw numbers that need comparison.

2

Convert each value into a decimal format to allow direct comparison, rounding to four decimal places where necessary.

Brand R:

0.4160

; Brand P:

5 \div 12 \approx 0.4167

; Brand Q:

41.8\% = 0.4180

; Brand S:

13 \div 30 \approx 0.4333

.

Converting to decimals standardizes the formats and makes close comparisons straightforward.

3

Compare the standardized decimal values from smallest to largest.

0.4160 < 0.4167 < 0.4180 < 0.4333

, which corresponds to the order: Brand R, Brand P, Brand Q, Brand S.

Ordering the values correctly identifies the brands from the least proportion of electrolytes to the greatest.

Key Concept

Comparing and ordering rational numbers by converting fractions, decimals, and percents into a single consistent format.

Alternative Method

Instead of converting to decimals, all values can be converted to percentages: Brand R is

41.6\%

, Brand P is

\frac{5}{12} \times 100\% \approx 41.67\%

, Brand Q is

41.8\%

, and Brand S is

\frac{13}{30} \times 100\% \approx 43.33\%

. Comparing these values gives

41.6\% < 41.67\% < 41.8\% < 43.33\%

.

Estimated Time:1m 30s

Question 24Question →

Four prototype solar cells are tested for their efficiency in converting solar energy into electricity. The efficiency of each solar cell is recorded as follows:

* Cell W: $\frac{4}{25}$ of the incoming energy is converted.
* Cell X: $16.5\%$ of the incoming energy is converted.
* Cell Y: $0.1605$ of the incoming energy is converted.
* Cell Z: $\frac{13}{80}$ of the incoming energy is converted.

Based on this data, how should these solar cells be ordered from least efficient to most efficient?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ordering from least efficient to most efficient is Cell W, Cell Y, Cell Z, and Cell X.

Converting all efficiency values to decimals facilitates comparison. Cell W is

\frac{4}{25} = 0.1600

. Cell Y is

0.1605

. Cell Z is

\frac{13}{80} = 0.1625

. Cell X is

16.5\% = 0.1650

. Comparing these decimals shows that

0.1600 < 0.1605 < 0.1625 < 0.1650

, which matches the order: Cell W, Cell Y, Cell Z, Cell X.

Step-by-Step Solution

1

Convert the efficiency of Cell W to a decimal.

\frac{4}{25} = 0.16 = 0.1600

To easily compare the values, convert the fraction to a decimal by dividing the numerator by the denominator.

2

Convert the efficiency of Cell X to a decimal.

16.5\% = 0.165 = 0.1650

Convert the percentage to a decimal by dividing by 100.

3

Convert the efficiency of Cell Z to a decimal.

\frac{13}{80} = 0.1625

Divide 13 by 80 to obtain its decimal representation.

4

Compare all four decimal values:

0.1600

(Cell W),

0.1650

(Cell X),

0.1605

(Cell Y), and

0.1625

(Cell Z).

0.1600 < 0.1605 < 0.1625 < 0.1650

Align the decimals by place value to determine their correct ascending order.

Key Concept

Comparing and ordering fractions, decimals, and percentages by converting them to a common format (decimals).

Estimated Time:1m 30s

Question 25Question →

A container holds a mixture of sand, gravel, and cement. By weight, the mixture is $30\%$ sand, $\frac{2}{5}$ gravel, and the remaining portion is cement. To adjust the mixture's properties, the amount of sand is doubled, the amount of gravel is increased by $25\%$ , and the amount of cement is decreased by $50\%$ . What percentage of the new mixture, by weight, is cement?

12\%

15\%

20\%

33\frac{1}{3}\%

40\%

Show answer & explanation

Answer: $12\%$

Answer

12%

To find the correct percentage, we establish that out of an initial

100\text{ g}

of the mixture, there are

30\text{ g}

of sand,

40\text{ g}

of gravel, and

30\text{ g}

of cement. After the changes, the new weights are

60\text{ g}

of sand,

50\text{ g}

of gravel, and

15\text{ g}

of cement. The new total weight is

125\text{ g}

. Thus, cement constitutes

\frac{15}{125} = 12\%

of the new mixture.

Step-by-Step Solution

1

Determine the initial proportions and assign a hypothetical total weight of

100\text{ g}

to simplify the calculations.

Sand is

30\text{ g}

. Gravel is

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

. Cement is the remaining weight:

100\text{ g} - (30\text{ g} + 40\text{ g}) = 30\text{ g}

.

Establishing concrete weights makes applying percentage changes straightforward.

2

Apply the specified weight changes to each component to find their new weights.

New sand weight:

30\text{ g} \times 2 = 60\text{ g}

. New gravel weight:

40\text{ g} \times 1.25 = 50\text{ g}

. New cement weight:

30\text{ g} \times 0.50 = 15\text{ g}

.

This calculates the individual component weights after the adjustments.

3

Calculate the new total weight of the mixture and the new percentage of cement.

New total weight:

60\text{ g} + 50\text{ g} + 15\text{ g} = 125\text{ g}

. Cement percentage:

\frac{15\text{ g}}{125\text{ g}} \times 100\% = 12\%

.

The percentage of a component in a mixture is its weight divided by the new total weight of the mixture.

Key Concept

Calculating new percentage concentrations after changes in individual component weights within a mixture.

Estimated Time:2m 0s

Question 26Question →

A smart thermostat is programmed to reduce a home's heating energy usage. In January, the heating energy usage is reduced by $15\%$ compared to the baseline usage. In February, the usage is reduced by an additional $\frac{1}{8}$ of January's usage level. In March, the usage is reduced by $20\%$ of February's usage level. If the baseline heating energy usage was $400$ kilowatt-hours (kWh), what is the total reduction in energy usage from the baseline to the end of March, in kWh?

Show answer & explanation

Answer: 162

Answer

The total reduction in energy usage from the baseline to the end of March is 162 kWh.

The baseline usage of

400

kWh is reduced by

15\%

in January, leaving

340

kWh. In February, a reduction of

\frac{1}{8}

of

340

kWh reduces usage by

42.5

kWh, leaving

297.5

kWh. In March, a reduction of

20\%

of

297.5

kWh reduces usage by

59.5

kWh, leaving a final usage of

238

kWh. The total reduction is the difference between the baseline and final usage levels, which is

162

kWh.

Step-by-Step Solution

1

Calculate January's energy reduction and January's usage level.

January reduction is

60

kWh; January usage level is

340

kWh.

January's reduction is

15\%

of the

400

kWh baseline (

0.15 \times 400 = 60

kWh). Subtracting this reduction from the baseline gives January's usage level (

400 - 60 = 340

kWh).

2

Calculate February's energy reduction and February's usage level.

February reduction is

42.5

kWh; February usage level is

297.5

kWh.

February's reduction is

\frac{1}{8}

of January's usage level (

340

kWh), which is

0.125 \times 340 = 42.5

kWh. Subtracting this from January's level gives February's usage level (

340 - 42.5 = 297.5

kWh).

3

Calculate March's energy reduction and March's usage level.

March reduction is

59.5

kWh; March usage level is

238

kWh.

March's reduction is

20\%

of February's usage level (

297.5

kWh), which is

0.20 \times 297.5 = 59.5

kWh. Subtracting this from February's level gives March's usage level (

297.5 - 59.5 = 238

kWh).

4

Calculate the total energy reduction from the baseline to the end of March.

The total reduction is

162

kWh.

The total reduction can be found by adding the reductions from each of the three months (

60 + 42.5 + 59.5 = 162

kWh) or by subtracting the final usage level from the baseline (

400 - 238 = 162

kWh).

Key Concept

Applying sequential percentage and fraction reductions to changing base values.

Question 27Question →

A chef is preparing a salad dressing. The mixture consists of $\frac{1}{4}$ cup of olive oil, $\frac{1}{5}$ cup of balsamic vinegar, and a certain amount of lemon juice. If the lemon juice accounts for exactly $40\%$ of the total volume of the dressing, what is the total volume, in cups, of the prepared dressing?

\frac{10}{27}

\frac{5}{9}

\frac{3}{4}

\frac{17}{20}

\frac{9}{8}

Show answer & explanation

Answer: $\frac{3}{4}$

Answer

\frac{3}{4}

To find the total volume, first calculate the sum of the volumes of olive oil and balsamic vinegar:

\frac{1}{4} + \frac{1}{5} = \frac{5}{20} + \frac{4}{20} = \frac{9}{20}

cup. Because the lemon juice accounts for

40\%

of the total volume of the dressing, the remaining ingredients (olive oil and balsamic vinegar) must account for the other

60\%

(or

\frac{3}{5}

) of the total volume. Letting

T

represent the total volume, we set up the equation

\frac{3}{5} T = \frac{9}{20}

. Solving for

T

yields

T = \frac{9}{20} \times \frac{5}{3} = \frac{3}{4}

cups. Therefore, the option representing three-fourths of a cup is the correct answer.

Step-by-Step Solution

1

Add the volumes of the two known ingredients, olive oil and balsamic vinegar.

Combined volume is

\frac{1}{4} + \frac{1}{5} = \frac{5}{20} + \frac{4}{20} = \frac{9}{20}

cup (or

0.45

cup).

To find what fraction of the total volume the known ingredients represent, we first need their combined total.

2

Determine the percentage of the total volume that the olive oil and balsamic vinegar represent.

These two ingredients represent

100\% - 40\% = 60\%

(or

\frac{3}{5}

) of the total volume.

Since lemon juice accounts for

40\%

of the total volume, the rest of the ingredients must account for the remaining percentage.

3

Set up a linear equation to solve for the total volume,

T

.

\frac{3}{5} T = \frac{9}{20} \Rightarrow T = \frac{9}{20} \times \frac{5}{3} = \frac{3}{4}

cups (or

0.75

cups).

Solving the equation gives the total volume of the entire dressing.

Key Concept

Solving multi-step word problems involving fractions, decimals, and percentages by relating parts to the whole.

Alternative Method

Convert the fractions to decimals first:

\frac{1}{4} = 0.25

cups of olive oil and

\frac{1}{5} = 0.20

cups of balsamic vinegar. Their combined volume is

0.25 + 0.20 = 0.45

cups. Since lemon juice is

40\%

of the total volume, the other ingredients make up

100\% - 40\% = 60\%

of the total volume. Let the total volume be

T

. We set up the equation

0.60 T = 0.45

, which simplifies to

T = \frac{0.45}{0.60} = 0.75

cups, or

\frac{3}{4}

cups.

Estimated Time:1m 30s

Question 28Question →

A company allocates its annual budget such that $\frac{1}{3}$ is spent on Research and Development, $\frac{1}{4}$ is spent on Marketing, and the remaining portion is spent on Operations. The Operations budget is then split, with $40\%$ of the Operations budget going to employee salaries and the rest to equipment. If the company spends $\$150,000$ on employee salaries in the Operations department, how much money does the company spend on Research and Development?

\$48,000

\$125,000

\$175,000

\$200,000

\$300,000

Show answer & explanation

Answer: $\$300,000$

Answer

\$300,000

The correct answer of

\$300,000

is found by first determining the Operations budget. Since

40\%

of the Operations budget is spent on salaries and this equals

\$150,000

, the Operations budget is

\frac{\$150,000}{0.40} = \$375,000

. The Research and Development (

\frac{1}{3}

) and Marketing (

\frac{1}{4}

) budgets combine for

\frac{7}{12}

of the total budget, leaving

\frac{5}{12}

for Operations. We set

\frac{5}{12}

of the total budget equal to

\$375,000

, which gives a total budget of

\$900,000

. Finally, the Research and Development budget is one-third of the total budget:

\frac{1}{3} \times \$900,000 = \$300,000

.

Step-by-Step Solution

1

Determine the fraction of the total budget allocated to Operations.

Operations receives

\frac{5}{12}

of the total budget.

The sum of the fractions for Research and Development and Marketing is

\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}

. The remaining fraction for Operations is

1 - \frac{7}{12} = \frac{5}{12}

.

2

Calculate the budget allocated to the Operations department.

Operations budget is

\$375,000

.

Employee salaries account for

40\%

(or

0.40

) of the Operations budget and equal

\$150,000

. Dividing

\$150,000

by

0.40

gives the total Operations budget:

\frac{\$150,000}{0.40} = \$375,000

.

3

Calculate the total company budget.

Total budget is

\$900,000

.

Since the Operations budget of

\$375,000

is

\frac{5}{12}

of the total budget

B

, we have

\frac{5}{12} B = \$375,000

. Multiplying both sides by

\frac{12}{5}

yields

B = \$375,000 \times \frac{12}{5} = \$900,000

.

4

Calculate the budget spent on Research and Development.

Research and Development budget is

\$300,000

.

Research and Development is allocated

\frac{1}{3}

of the total budget:

\frac{1}{3} \times \$900,000 = \$300,000

.

Key Concept

Solving multi-step word problems using fractions, decimals, and percentage allocations.

Question 29Question →

A manufacturer of solar panels inspects their production in three stages. In the first stage, $\frac{1}{12}$ of the total panels produced are identified as defective and recycled. In the second stage, $20\%$ of the remaining non-defective panels are found to have cosmetic flaws and are sold at a discount. In the third stage, $\frac{1}{5}$ of the panels that passed the first two stages without any defects or flaws are selected for quality testing. If $1,760$ panels passed the first two stages but were NOT selected for quality testing, what was the total number of panels originally produced?

Show answer & explanation

Answer: 3000

Answer

The total number of panels originally produced was

3,000

.

The correct answer of

3,000

is determined by calculating the cumulative fraction of panels that are not defective, do not have cosmetic flaws, and are not selected for testing. In Stage 1, subtracting

\frac{1}{12}

leaves

\frac{11}{12}

of the original panels. In Stage 2, subtracting

20\%

of those leaves

80\%

(or

\frac{4}{5}

), resulting in

\frac{11}{15}

of the original panels. In Stage 3, subtracting

\frac{1}{5}

of these leaves

\frac{4}{5}

, resulting in

\frac{44}{75}

of the original panels. Setting

\frac{44}{75}

of the original panels equal to

1,760

and solving for the total original panels yields

3,000

.

Step-by-Step Solution

1

Calculate the fraction of panels remaining after the first stage of inspection.

\frac{11}{12}

of the original panels

Since

\frac{1}{12}

of the total panels are defective and recycled, the remaining portion is

1 - \frac{1}{12} = \frac{11}{12}

.

2

Calculate the fraction of the original panels that pass the second stage without defects or flaws.

\frac{11}{15}

of the original panels

Of the remaining

\frac{11}{12}

of the panels,

20\%

have cosmetic flaws, meaning

100\% - 20\% = 80\%

(or

\frac{4}{5}

) do not have flaws. Multiplying these fractions gives

\frac{4}{5} \times \frac{11}{12} = \frac{11}{15}

.

3

Calculate the fraction of original panels that pass the first two stages but are not selected for testing.

\frac{44}{75}

of the original panels

Since

\frac{1}{5}

of the flawless panels are selected for testing, the remaining

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

are not selected. Multiplying this by the fraction from Step 2 gives

\frac{4}{5} \times \frac{11}{15} = \frac{44}{75}

.

4

Solve for the original number of panels,

P

, using the given value of

1,760

.

P = 3,000

panels

Set up the equation

\frac{44}{75} P = 1,760

. Solving for

P

gives

P = 1,760 \times \frac{75}{44} = 40 \times 75 = 3,000

.

Key Concept

Applying successive fractions and percentages to resolve a multi-stage word problem by working backward to the initial value.

Question 30Question →

A public library is organizing its catalog of physical books. Currently, $\frac{3}{8}$ of the books in the collection are categorized as fiction. Of the remaining books, $40\%$ are non-fiction, while the rest are divided between children's literature and reference volumes in a ratio of $3:2$ , respectively. If there are exactly $450$ reference volumes in the library, what is the total number of physical books in the collection?

1,350

1,800

1,875

3,000

5,000

Show answer & explanation

Answer: $3,000$

Answer

The total number of physical books in the library collection is

3,000

.

To find the total number of books, first calculate the remaining fraction after fiction books are removed:

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

. Next, account for non-fiction taking up

40\%

of this remainder, leaving

60\%

(or

\frac{3}{5}

) of

\frac{5}{8}

for children's and reference books, which equals

\frac{3}{8}

of the entire collection. Using the

3:2

ratio, reference books represent

\frac{2}{5}

of this

\frac{3}{8}

portion, which is

\frac{3}{20}

of the total collection. Setting

\frac{3}{20}

of the total equal to

450

gives a total collection of

3,000

books.

Step-by-Step Solution

1

Determine the remaining fraction of books after setting aside fiction books.

Since

\frac{3}{8}

of the books are fiction, the remaining fraction is

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

.

The total collection equals

1

(or

100\%

). Subtracting the fiction fraction leaves the remaining portion.

2

Calculate the fraction of books that are either children's literature or reference volumes.

Since

40\%

of the remaining books are non-fiction, the remaining

60\%

(

0.60

or

\frac{3}{5}

) of the

\frac{5}{8}

portion are children's or reference books:

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

.

Finding

60\%

of the remaining

\frac{5}{8}

identifies the portion shared between children's and reference books.

3

Determine the fraction of total books represented by reference volumes using the given ratio.

With a children's to reference ratio of

3:2

, reference books represent

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

of this subgroup. Thus, reference books account for

\frac{2}{5} \times \frac{3}{8} = \frac{6}{40} = \frac{3}{20}

of the total collection.

A ratio of

3:2

divides the subgroup into

5

equal parts, of which reference books make up

2

parts.

4

Solve for the total collection size using the known number of reference volumes.

Set up the equation

\frac{3}{20} \times T = 450

, where

T

is the total number of books. Solving yields

T = 450 \times \frac{20}{3} = 150 \times 20 = 3,000

.

Dividing the given quantity by its corresponding fraction gives the total population size.

Key Concept

Fractions, Decimals, and Percentages

Estimated Time:2m 0s

Question 31Question →

A retail clothing store applies successive markdowns to a coat during a seasonal clearance sale. In the first week, the original price of the coat is discounted by $30\%$ . In the second week, the price is reduced by an additional $\frac{1}{5}$ of the first-week sale price. Finally, a customer uses a promotional code at checkout to receive an extra $15\%$ off the second-week sale price. If the customer pays a final price of $\$119.00$ before tax, what was the original price of the coat, in dollars?

Show answer & explanation

Answer: 250

Answer

The original price of the coat was

250

dollars.

To find the original price of the coat, express each successive discount as a multiplier representing the portion of the remaining price. The first discount of

30\%

leaves

70\%

(

0.70

). The second discount of

\frac{1}{5}

(

20\%

) leaves

80\%

(

0.80

) of the first sale price. The final discount of

15\%

leaves

85\%

(

0.85

) of the second sale price. Combining these yields an overall price multiplier of

0.70 \times 0.80 \times 0.85 = 0.476

. Dividing the final purchase price of

\$119.00

by

0.476

gives the original price of

\$250.00

.

Step-by-Step Solution

1

Calculate the price multiplier for the first week discount.

Multiplier 1 =

0.70

A

30\%

discount leaves

100\% - 30\% = 70\%

of the original price.

2

Convert the second week fraction discount to a decimal multiplier.

Multiplier 2 =

0.80

A discount of

\frac{1}{5}

is equal to

20\%

, leaving

1 - 0.20 = 0.80

of the week-1 price.

3

Calculate the price multiplier for the promotional code.

Multiplier 3 =

0.85

A

15\%

discount leaves

100\% - 15\% = 85\%

(

0.85

) of the week-2 price.

4

Find the combined price multiplier by taking the product of all three individual multipliers.

Combined Multiplier =

0.476

Successive discounts compound multiplicatively:

0.70 \times 0.80 \times 0.85 = 0.476

.

5

Solve for the original price

P

using the equation

0.476 P = 119

.

P = 250

Dividing the final price

\$119.00

by

0.476

yields

250

.

Key Concept

Compounding successive percentage and fraction discounts and solving for the original value.

Estimated Time:2m 0s

Question 32Question →

A coffee artisan roasts a $120$ -pound batch of green coffee beans. During the roasting process, the beans lose $15\%$ of their total weight due to moisture evaporation. Next, $\frac{1}{8}$ of the roasted weight is removed during quality control sorting. Finally, the remaining high-quality roasted beans are packaged into small bags that each hold $0.75$ pounds of coffee. How many full bags of roasted coffee can be produced from this batch?

116

119

122

136

161

Show answer & explanation

Answer: $119$

Answer

The total number of full bags of roasted coffee that can be produced is

119

.

To find the number of bags, calculate the remaining weight step-by-step. First,

15\%

of

120

pounds is

18

pounds, leaving

102

pounds of roasted coffee. Next,

\frac{1}{8}

of

102

pounds is

12.75

pounds, leaving

89.25

pounds of high-quality coffee. Dividing

89.25

by

0.75

pounds per bag yields exactly

119

full bags.

Step-by-Step Solution

1

Calculate the weight lost during roasting

15\%

of

120\text{ lbs} = 0.15 \times 120 = 18\text{ lbs}

The beans lose

15\%

of their initial weight due to evaporation.

2

Determine the remaining weight of roasted beans

120\text{ lbs} - 18\text{ lbs} = 102\text{ lbs}

Subtract the weight lost during roasting from the starting weight.

3

Calculate the weight removed during quality control

\frac{1}{8} \times 102\text{ lbs} = 12.75\text{ lbs}

Quality control removes

\frac{1}{8}

of the roasted weight.

4

Determine the final usable weight of coffee beans

102\text{ lbs} - 12.75\text{ lbs} = 89.25\text{ lbs}

Subtract the rejected beans from the roasted weight.

5

Calculate the number of

0.75

-pound bags produced

89.25 \div 0.75 = 119\text{ bags}

Divide the total usable weight by the capacity of a single bag.

Key Concept

Sequential application of percentages and fractions to remaining quantities

Estimated Time:2m 0s

Question 33Question →

During a weekly training program, an athlete covers $\frac{2}{5}$ of her total distance by cycling and $35\%$ of her total distance by running. The remaining distance is completed by swimming. If she swims $15$ kilometers that week, what is her total training distance, in kilometers?

20

37.5

45

60

75

Show answer & explanation

Answer: $60$

Answer

The athlete's total training distance for the week is 60 kilometers.

Converting

\frac{2}{5}

to

40\%

shows that cycling (

40\%

) and running (

35\%

) together account for

75\%

of the total training distance. Swimming accounts for the remaining

25\%

(

100\% - 75\%

). Since

25\%

of the total distance is

15

kilometers, dividing

15

by

0.25

yields the total distance of

60

kilometers.

Step-by-Step Solution

1

Convert the cycling fraction to a percentage.

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

Converting all portions to percentages allows for easy summation and subtraction.

2

Calculate the total percentage covered by cycling and running combined.

40\% + 35\% = 75\%

Adding the cycling and running portions yields the covered percentage before swimming.

3

Find the percentage of the total distance that swimming represents.

100\% - 75\% = 25\%

The remaining portion of the whole (

100\%

) must equal the swimming percentage.

4

Set up an equation relating the swimming percentage to the actual distance and solve for total distance

T

.

0.25 \times T = 15 \implies T = \frac{15}{0.25} = 60

Dividing the part by its corresponding percentage rate gives the whole total distance.

Key Concept

Solving multi-step part-to-whole word problems by combining fractions and percentages.

Question 34Question →

Four real estate agents reported the proportion of their home listings that sold above asking price last month as follows: Agent W reported $\frac{31}{50}$ , Agent X reported $\frac{5}{8}$ , Agent Y reported $63\%$ , and Agent Z reported $0.64$ . Place the four agents in order from the smallest proportion of listings sold above asking price to the largest proportion.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from smallest to largest proportion is Agent W (31/50 = 0.62), Agent X (5/8 = 0.625), Agent Y (63% = 0.63), and Agent Z (0.64).

To place the values in ascending order, convert all values to decimals: Agent W = 31/50 = 0.620, Agent X = 5/8 = 0.625, Agent Y = 63% = 0.630, and Agent Z = 0.640. Arranging these from least to greatest gives 0.620 < 0.625 < 0.630 < 0.640, corresponding to the sequence Agent W, Agent X, Agent Y, and Agent Z.

Step-by-Step Solution

1

Convert each fraction, percentage, and decimal to a common decimal representation to facilitate comparison.

All four numbers will be expressed as decimals rounded/extended to three decimal places.

Converting all values to decimals makes comparing their magnitudes straightforward.

2

Convert Agent W's value:

\frac{31}{50}

.

\frac{31}{50} = \frac{62}{100} = 0.620

Multiply numerator and denominator by 2 to convert to hundreds.

3

Convert Agent X's value:

\frac{5}{8}

.

\frac{5}{8} = 5 \div 8 = 0.625

Perform long division of 5 by 8.

4

Convert Agent Y's value:

63\%

.

63\% = \frac{63}{100} = 0.630

Divide percentage value by 100 to obtain decimal format.

5

Note Agent Z's value, which is already a decimal.

0.64 = 0.640

Align decimal places with the other converted values.

6

Order the decimal values from smallest to largest.

0.620 < 0.625 < 0.630 < 0.640

, which corresponds to Agent W, Agent X, Agent Y, and Agent Z.

Comparing digits from left to right establishes the proper sequence.

Key Concept

Comparing and ordering real numbers by converting fractions, decimals, and percentages into a unified numerical format.

Estimated Time:1m 15s

Question 35Question →

A community garden allocates its total area among three crops: vegetables, herbs, and flowers. Exactly $\frac{3}{8}$ of the total area is dedicated to vegetables, and $40\%$ of the remaining area is planted with herbs. If the remaining flower section covers $270$ square feet, what is the total area, in square feet, of the community garden?

432

450

720

1,080

1,200

Show answer & explanation

Answer: $720$

Answer

The total area of the community garden is

720

square feet.

The correct answer is

720

square feet. Vegetables take

\frac{3}{8}

of the total area, leaving

\frac{5}{8}

. Herbs take

40\%

of

\frac{5}{8}

, which equals

\frac{2}{5} \times \frac{5}{8} = \frac{2}{8}

of the total. Subtracting the herb portion from the remaining portion leaves

\frac{5}{8} - \frac{2}{8} = \frac{3}{8}

of the total for flowers. Solving

\frac{3}{8} T = 270

gives

T = 720

square feet.

Step-by-Step Solution

1

Calculate the remaining fractional area after allocating space for vegetables.

Since vegetables occupy

\frac{3}{8}

of the garden,

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

of the total area remains.

The total area represents

1

whole, so subtracting the vegetable fraction gives the remaining fraction.

2

Determine the fractional area allocated to herbs.

Herbs take

40\%

of the remaining

\frac{5}{8}

. Converting

40\%

to a fraction gives

\frac{2}{5}

. Calculating

\frac{2}{5} \times \frac{5}{8} = \frac{2}{8} = \frac{1}{4}

of the total garden area.

Multiplying the percentage by the remaining fraction yields the portion of the whole garden used for herbs.

3

Find the fractional area remaining for flowers.

Subtract the herb fraction from the remaining area after vegetables:

\frac{5}{8} - \frac{2}{8} = \frac{3}{8}

of the total area.

The flower section represents whatever area is left after both vegetables and herbs are accounted for.

4

Set up an equation to solve for the total garden area

T

.

\frac{3}{8} T = 270 \implies T = 270 \times \frac{8}{3} = 720

square feet.

Dividing the actual area of flowers by its fractional share gives the total garden area.

Key Concept

Multi-step word problems involving sequential fractions and percentages of a remaining amount.

Estimated Time:1m 15s

Question 36Question →

An artisanal bakery prepares a specialty flour blend consisting of rye, oat, and wheat flour. By weight, $\frac{3}{10}$ of the blend is rye flour and $0.45$ of the blend is oat flour. The remaining portion of the blend is wheat flour. If a single batch of the blend contains $4.5$ pounds of wheat flour, what is the total weight, in pounds, of the flour blend prepared for the batch?

Show answer & explanation

Answer: 18

Answer

The total weight of the flour blend prepared for the batch is 18 pounds.

To find the total weight of the mixture, first determine the decimal fraction representing wheat flour. Converting

\frac{3}{10}

to

0.30

, the combined proportion of rye and oat flour is

0.30 + 0.45 = 0.75

. Wheat flour makes up the remaining

1 - 0.75 = 0.25

(

25\%

) of the mixture. Setting up the equation

0.25 \times \text{Total Weight} = 4.5

gives

\text{Total Weight} = \frac{4.5}{0.25} = 18

pounds.

Step-by-Step Solution

1

Convert the fraction of rye flour to a decimal and sum it with the oat flour portion.

Rye portion =

\frac{3}{10} = 0.30

; Combined Rye and Oat portion =

0.30 + 0.45 = 0.75

.

Converting all given values to a consistent decimal format allows for straightforward addition.

2

Calculate the remaining portion representing wheat flour.

Wheat portion =

1 - 0.75 = 0.25

.

The sum of all proportional components in a whole mixture equals 1.

3

Divide the weight of the wheat flour by its decimal proportion to solve for the total batch weight.

Total weight =

\frac{4.5}{0.25} = 18

pounds.

Since

25\%

(

0.25

) of the total weight is

4.5

pounds, dividing the part by its decimal rate yields the total whole.

Key Concept

Solving multi-step word problems involving conversions between fractions, decimals, and percentages to find an unknown total.

Question 37Question →

A municipal transit system tracks its monthly revenue from three fare categories: Single-Ride passes, Weekly passes, and Monthly subscriptions. Single-Ride passes account for $\frac{9}{20}$ of the total monthly revenue, and Weekly passes account for $30\%$ of the total monthly revenue. If the remaining revenue from Monthly subscriptions is $\$21,000$ , what was the total monthly revenue, in dollars, for the transit system?

$28,000

$35,000

$63,000

$84,000

$105,000

Show answer & explanation

Answer: $84,000

Answer

The total monthly revenue for the transit system is $84,000.

Single-Ride passes account for

\frac{9}{20} = 45\%

of revenue, and Weekly passes account for

30\%

. Together, they account for

45\% + 30\% = 75\%

of total revenue. The remaining

25\%

(

100\% - 75\%

) corresponds to Monthly subscriptions, which equals

\$21,000

. Dividing

\$21,000

by

0.25

gives the total monthly revenue of

\$84,000

.

Step-by-Step Solution

1

Convert the fraction portion into a percentage.

\frac{9}{20} = \frac{45}{100} = 45\%

Converting all proportions to percentages allows for direct comparison and addition.

2

Calculate the combined percentage accounted for by Single-Ride and Weekly passes.

45\% + 30\% = 75\%

Summing the two given parts determines the total portion of revenue accounted for so far.

3

Find the percentage corresponding to Monthly subscriptions.

100\% - 75\% = 25\%

The three fare categories make up

100\%

of the total monthly revenue.

4

Set up an equation to solve for the total revenue

T

.

0.25 T = 21,000 \implies T = \frac{21,000}{0.25} = 84,000

Dividing the dollar amount of the remaining category by its percentage equivalent yields the total revenue.

Key Concept

Combining fractions and percentages to solve for an unknown total value.

Estimated Time:1m 30s

Question 38Question →

A software development company allocated its annual operating budget across three departments: Engineering, Marketing, and Customer Support. Exactly $\frac{2}{5}$ of the total budget was allocated to Engineering, and $35\%$ of the total budget was allocated to Marketing. If the remaining budget of $\$45,000$ was allocated to Customer Support, what was the company's total annual operating budget?

\$60,000

\$112,500

\$125,000

\$180,000

\$225,000

Show answer & explanation

Answer: $\$180,000$

Answer

The total annual operating budget of the company was

\$180,000

.

To find the total budget, first convert

\frac{2}{5}

to a percentage, which is

40\%

. Adding Engineering's

40\%

and Marketing's

35\%

gives

75\%

of the total budget. This leaves

100\% - 75\% = 25\%

for Customer Support. Since Customer Support received

\$45,000

, setting

0.25 T = 45,000

yields

T = \frac{45,000}{0.25} = \$180,000

.

Step-by-Step Solution

1

Convert the fraction allocated to Engineering into a percentage.

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

Converting all departmental allocations to percentages allows for direct comparison and summation.

2

Calculate the combined percentage allocated to Engineering and Marketing.

40\% + 35\% = 75\%

Adding the two known departmental shares determines the total fraction of the budget accounted for.

3

Find the percentage allocated to Customer Support.

100\% - 75\% = 25\%

The remaining portion of the

100\%

total budget belongs to Customer Support.

4

Set up an equation relating Customer Support's dollar allocation to the total budget

T

and solve for

T

.

0.25 T = 45,000 \implies T = \frac{45,000}{0.25} = 180,000

Dividing the dollar amount by its corresponding decimal percentage yields the original total budget.

Key Concept

Solving word problems involving mixed representations of fractions, decimals, and percentages to find an unknown initial total.

Estimated Time:1m 30s

Question 39Question →

A solar power plant generated a total of $480\text{ megawatt-hours (MWh)}$ of energy in one month. Of this total, $\frac{2}{5}$ was stored in battery banks, $0.35$ was supplied directly to an industrial facility, and the remaining portion was delivered to residential homes. How many megawatt-hours of energy were delivered to residential homes?

168 MWh

192 MWh

120 MWh

216 MWh

360 MWh

Show answer & explanation

Answer: 120 MWh

Answer

120 MWh

Converting

\frac{2}{5}

to

0.40

allows us to sum the battery and industrial portions (

0.40 + 0.35 = 0.75

). The remaining residential portion is

1 - 0.75 = 0.25

, or

25\%

. Taking

25\%

of

480\text{ MWh}

gives

120\text{ MWh}

.

Step-by-Step Solution

1

Convert the fraction to a decimal

\frac{2}{5} = 0.40

Converting all given values to a common format (decimals) simplifies addition.

2

Sum the non-residential portions of energy

0.40 + 0.35 = 0.75

Combining the battery storage fraction and industrial supply fraction determines the total non-residential portion.

3

Find the remaining fraction for residential homes

1.00 - 0.75 = 0.25

Subtracting the combined non-residential portion from the whole (1.00) gives the residential fraction.

4

Calculate the megawatt-hours delivered to residential homes

0.25 \times 480 = 120\text{ MWh}

Multiplying the residential fraction by the total energy generated gives the target quantity.

Key Concept

Combining fractions and decimals to determine a remaining percentage of a whole quantity

Estimated Time:1m 0s

Question 40Question →

A solar power system generated a total of $450$ kilowatt-hours (kWh) of electricity during a 3-day period. On Day 1, the system generated $\frac{2}{9}$ of the total 3-day electricity. On Day 2, it generated $40\%$ of the remaining electricity after Day 1. How many kilowatt-hours of electricity were generated on Day 3?

Show answer & explanation

Answer: 210

Answer

210 kWh

To find the electricity generated on Day 3, first determine the Day 1 portion by multiplying

\frac{2}{9}

by

450

, which gives

100

kWh. The amount remaining for Days 2 and 3 is

450 - 100 = 350

kWh. Day 2 accounts for

40\%

of this remaining amount, which is

0.40 \times 350 = 140

kWh. Finally, subtract Day 2's portion from the remaining amount:

350 - 140 = 210

kWh.

Step-by-Step Solution

1

Calculate the amount of electricity generated on Day 1.

100

kWh

Multiply the fraction

\frac{2}{9}

by the total electricity (

450

kWh).

2

Determine the remaining electricity after Day 1.

350

kWh

Subtract Day 1's generation (

100

kWh) from the total amount (

450

kWh).

3

Calculate the amount of electricity generated on Day 2.

140

kWh

Convert

40\%

to a decimal (

0.40

) and multiply by the remaining

350

kWh.

4

Calculate the amount of electricity generated on Day 3.

210

kWh

Subtract Day 2's generation (

140

kWh) from the

350

kWh remaining after Day 1.

Key Concept

Multi-step word problems involving fractions, percentage calculations, and remaining quantities.

Estimated Time:1m 15s

Fractions, Decimals, and Percentages

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

Alternative Method

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