Pre-Algebra Practice Questions — ACT — Page 19

Question 361Question →

An electric vehicle charging station delivers energy at a constant rate of $k$ kilowatt-hours (kWh) per hour. If a vehicle receives a total of $48\text{ kWh}$ of energy after charging for $1.5$ hours, which of the following equations can be solved to find $k$ ?

1.5k = 48

\frac{k}{1.5} = 48

k + 1.5 = 48

k - 1.5 = 48

48k = 1.5

Show answer & explanation

Answer: $1.5k = 48$

Answer

1.5k = 48

Multiplying the charging rate of

k

kWh per hour by the charging duration of

1.5

hours yields the total energy of

48

kWh. Thus,

1.5k = 48

correctly models the situation.

Step-by-Step Solution

1

Identify the relationship between rate, time, and total amount.

Total energy = (charging rate)

\times

(time in hours)

The unit rate in kWh per hour multiplied by hours yields total kWh.

2

Substitute the given values into the relationship.

48 = k \times 1.5

, or

1.5k = 48

The total energy is

48

, the rate is

k

, and time is

1.5

hours.

Key Concept

Formulating one-step multiplication equations from unit rate scenarios

Estimated Time:1m 0s

Question 362Question →

At a community theater performance, the ratio of adult tickets sold to student tickets sold was $5 : 3$ . After $12$ additional student tickets were sold and no additional adult tickets were sold, the ratio of adult tickets to student tickets became $4 : 3$ . How many total tickets (adult and student combined) were sold originally?

48

80

96

128

140

Show answer & explanation

Answer: 128

Answer

128 total tickets were sold originally.

Let the original number of adult tickets be

5x

and student tickets be

3x

. After 12 student tickets are added, the new ratio is

\\frac{5x}{3x + 12} = \\frac{4}{3}

. Cross-multiplying gives

15x = 12x + 48

, which simplifies to

3x = 48

and

x = 16

. The original total number of tickets is

5x + 3x = 8x = 8(16) = 128

.

Step-by-Step Solution

1

Define variables for the original quantities based on the initial ratio.

Let the original number of adult tickets be

5x

and the original number of student tickets be

3x

, where

x

is a common multiplier. The original total is

5x + 3x = 8x

.

Representing a ratio

5 : 3

with a multiplier allows algebraic manipulation when quantities change.

2

Set up an equation representing the new ratio after adding 12 student tickets.

\\frac{5x}{3x + 12} = \\frac{4}{3}

The number of adult tickets remains

5x

, while student tickets increase by 12 to

3x + 12

.

3

Solve for the multiplier

x

using cross-multiplication.

3(5x) = 4(3x + 12) \\implies 15x = 12x + 48 \\implies 3x = 48 \\implies x = 16

.

Equating cross-products isolates

x

to find the value of one ratio unit.

4

Calculate the original total number of tickets.

\\text{Original Total} = 8x = 8(16) = 128

.

Multiplying the common multiplier by 8 yields the combined original total.

Key Concept

Solving multi-step proportion problems involving changing part-to-part ratios.

Estimated Time:1m 15s

Question 363Question →

A nursery technician prepares a liquid fertilizer mixture using nitrogen, phosphorus, and potassium in a volume ratio of $4 : 2 : 1$ , respectively. A storage tank currently holds $35\text{ liters}$ of this mixture. If the technician adds $6\text{ liters}$ of pure nitrogen to the tank, what is the ratio of nitrogen to phosphorus in the resulting mixture?

5 : 1

13 : 5

13 : 10

5 : 4

26 : 41

Show answer & explanation

Answer: $13 : 5$

Answer

The ratio of nitrogen to phosphorus in the resulting mixture is

13 : 5

.

The original

35\text{ liters}

contains

7

equal parts of

5\text{ liters}

each. Nitrogen represents

4

parts (

20\text{ liters}

) and phosphorus represents

2

parts (

10\text{ liters}

). Adding

6\text{ liters}

of nitrogen brings the nitrogen amount to

26\text{ liters}

. Comparing

26\text{ liters}

of nitrogen to

10\text{ liters}

of phosphorus yields the ratio

26 : 10

, which simplifies to

13 : 5

.

Step-by-Step Solution

1

Find the total number of parts in the original ratio.

4 + 2 + 1 = 7\text{ total parts}

.

To determine the volume of each component, calculate the total proportion of the mixture.

2

Calculate the initial volume of nitrogen and phosphorus in

35\text{ liters}

of solution.

\text{Nitrogen} = \frac{4}{7} \times 35 = 20\text{ liters}; \quad \text{Phosphorus} = \frac{2}{7} \times 35 = 10\text{ liters}.

Multiply each component's fraction by the total volume of

35\text{ liters}

.

3

Add

6\text{ liters}

to the initial volume of nitrogen.

\text{New Nitrogen volume} = 20 + 6 = 26\text{ liters}.

Only nitrogen is added to the tank, while phosphorus remains unchanged at

10\text{ liters}

.

4

Write and simplify the new ratio of nitrogen to phosphorus.

\frac{26}{10} = \frac{13}{5} \implies 13 : 5.

Divide both terms by their greatest common factor,

2

.

Key Concept

Partitioning a total quantity into parts using extended ratios and adjusting part quantities.

Question 364Question →

A technology firm allocated its annual equipment budget among three divisions. The engineering division received $\frac{3}{8}$ of the total budget, and the operations division received $30\%$ of the total budget. The remaining $\$52,000$ was allocated to the design division. What was the total equipment budget, in dollars, for the technology firm?

Show answer & explanation

Answer: 160000

Answer

The total equipment budget for the technology firm was $160,000 dollars.

Converting both allocated parts into decimals gives

\frac{3}{8} = 0.375

for engineering and

30\% = 0.30

for operations. Combined, these two divisions account for

0.375 + 0.30 = 0.675

(or

67.5\%

) of the total budget. The remaining fraction for the design division is

1 - 0.675 = 0.325

(or

32.5\%

). Dividing the design budget of

\$52,000

by

0.325

yields the total budget of

\$160,000

.

Step-by-Step Solution

1

Convert the percentage and fraction to equivalent decimal values.

Engineering receives

\frac{3}{8} = 0.375

, and operations receives

30\% = 0.30

.

Expressing all portions in decimal form allows for straightforward addition and subtraction.

2

Sum the portions allocated to the engineering and operations divisions.

0.375 + 0.30 = 0.675

.

This finds the total proportion of the budget that has already been accounted for.

3

Determine the remaining decimal portion allocated to the design division.

1 - 0.675 = 0.325

.

The complete budget represents 1 whole (100%). Subtracting the allocated portions yields the design division's share.

4

Divide the dollar amount of the design division by its decimal portion to calculate the total budget.

\frac{52,000}{0.325} = 160,000

.

Since

\text{Design Portion} \times \text{Total Budget} = \text{Design Amount}

, dividing the amount by the decimal portion gives the total budget.

Key Concept

Solving multi-step applied word problems by converting between fractions, decimals, and percentages to determine an unknown total value.

Question 365Question →

A security system requires a 4-digit passcode created using the digits $1, 2, 3, 4, 5, 6,$ and $7$ . No digit may be repeated in a passcode. If the first digit and the last digit must both be odd numbers, how many different passcodes can be formed?

Show answer & explanation

Answer: 240

Answer

240 passcodes can be formed under the given conditions.

To form a 4-digit passcode with distinct digits where the first and last digits are odd: first choose the first digit from the 4 available odd digits (

1, 3, 5, 7

). Next, choose the fourth digit from the remaining 3 odd digits. Then, choose the second digit from the remaining 5 available digits in the full set, and the third digit from the remaining 4 available digits. Multiplying these choices gives

4 \times 5 \times 4 \times 3 = 240

.

Step-by-Step Solution

1

Categorize available digits into odd and even sets

4 odd digits ({1, 3, 5, 7}) and 3 even digits ({2, 4, 6}) out of 7 total digits

The first and last positions require odd digits, so we need to track odd digits separately.

2

Determine options for the first and last positions

4 options for position 1; 3 options for position 4

Position 1 must be odd (4 choices). Because digits cannot repeat, position 4 has 3 remaining odd choices.

3

Determine options for the second and third positions

5 options for position 2; 4 options for position 3

Two digits have already been assigned. Out of 7 total digits, 5 remain for position 2, leaving 4 for position 3.

4

Multiply choices across all positions

4 * 5 * 4 * 3 = 240

By the Fundamental Counting Principle, the total number of combinations is the product of the number of choices at each step.

Key Concept

Fundamental Counting Principle with Position Restrictions and Non-Replacement

Question 366Question →

An electric commuter train consumes energy at a constant rate of $450\text{ kilowatt-hours (kWh)}$ for every $120\text{ miles}$ it travels. Electricity costs $\$0.14$ per $\text{kWh}$ . What is the total cost of the electricity consumed by the train during a $320\text{ mile}$ trip?

\$23.63

\$63.00

\$168.00

\$448.00

\$1,200.00

Show answer & explanation

Answer: $\$168.00$

Answer

The total cost of electricity consumed during the trip is

\$168.00

.

To solve this problem, first determine the energy consumed per mile by dividing

450\text{ kWh}

by

120\text{ miles}

, which equals

3.75\text{ kWh/mile}

. Next, multiply this unit rate by

320\text{ miles}

to find the total energy required for the trip, which is

1,200\text{ kWh}

. Finally, multiply

1,200\text{ kWh}

by the cost of

\$0.14

per

\text{kWh}

to get the total cost of

\$168.00

.

Step-by-Step Solution

1

Calculate the unit rate of energy consumption per mile

\frac{450\text{ kWh}}{120\text{ miles}} = 3.75\text{ kWh per mile}

Determining the energy consumption rate per mile allows us to calculate energy used for any distance.

2

Find the total energy consumed for

320

miles

320\text{ miles} \times 3.75\text{ kWh/mile} = 1,200\text{ kWh}

Multiply total distance by rate per mile to find total electricity required.

3

Calculate the total electricity cost

1,200\text{ kWh} \times \$0.14\text{/kWh} = \$168.00

Multiply total kilowatt-hours by the cost per kilowatt-hour.

Key Concept

Unit rates and multi-step proportional reasoning

Estimated Time:1m 30s

Question 367Question →

At a university, three sections of an introductory chemistry course took the same midterm examination. Section 1 has $15$ students with a mean score of $80$ points, Section 2 has $25$ students with a mean score of $88$ points, and Section 3 has $10$ students with a mean score of $74$ points. What is the overall mean score for all $50$ students combined?

78.6

80.7

82.8

84.0

1,380.0

Show answer & explanation

Answer: $82.8$

Answer

The overall mean score for all

50

students combined is

82.8

.

To find the combined mean score for all

50

students, calculate the total points earned across all three sections and divide by the total number of students. Section 1 contributes

15 \times 80 = 1,200

points, Section 2 contributes

25 \times 88 = 2,200

points, and Section 3 contributes

10 \times 74 = 740

points. Summing these yields

4,140

total points. Dividing

4,140

by the total student count of

50

gives an overall mean of

82.8

.

Step-by-Step Solution

1

Calculate the total number of points earned by each section.

Section 1:

15 \times 80 = 1,200

points. Section 2:

25 \times 88 = 2,200

points. Section 3:

10 \times 74 = 740

points.

The total points in a group equals the number of items multiplied by the mean of that group.

2

Sum the total points from all three sections to find the combined total points.

1,200 + 2,200 + 740 = 4,140

total points.

To find the overall mean, we need the total score across all students.

3

Divide the total points by the total number of students.

\frac{4,140}{50} = 82.8

.

The weighted mean is the total accumulated score divided by the total sample size (

15 + 25 + 10 = 50

).

Key Concept

Weighted Mean

Estimated Time:1m 15s

Question 368Question →

A botanical garden conservatory uses a specialized organic fertilizer mixture containing $f$ pounds of nutrient concentrate. The concentrate is divided equally into $14$ separate watering reservoirs. If each reservoir receives exactly $2.5$ pounds of concentrate, which of the following equations correctly models this situation?

\frac{f}{14} = 2.5

14f = 2.5

f - 14 = 2.5

\frac{14}{f} = 2.5

f + 2.5 = 14

Show answer & explanation

Answer: $\frac{f}{14} = 2.5$

Answer

The equation

\frac{f}{14} = 2.5

correctly models the situation.

Dividing the total amount of nutrient concentrate (

f

) equally into

14

reservoirs means dividing

f

by

14

. Setting this equal to the amount per reservoir (

2.5

pounds) yields

\frac{f}{14} = 2.5

.

Step-by-Step Solution

1

Identify the total quantity and how it is shared or split.

The total amount of nutrient concentrate is

f

pounds, divided equally into

14

reservoirs.

Equal sharing or division of a total quantity

f

into

14

equal parts is expressed algebraically as

\frac{f}{14}

.

2

Equate the expression for one part to the given value per reservoir.

\frac{f}{14} = 2.5

Each reservoir gets

2.5

pounds, so the portion in one reservoir,

\frac{f}{14}

, must equal

2.5

.

Key Concept

Translating verbal descriptions of equal division into one-step algebraic equations

Question 369Question →

A long-distance runner is training for an upcoming race. On Monday, she completed $\frac{2}{5}$ of her total weekly distance goal. On Tuesday, she completed $0.35$ of her total weekly distance goal. If she has $12$ miles left to complete her weekly goal, what is her total weekly goal distance, in miles?

16

21

30

48

60

Show answer & explanation

Answer: 48

Answer

48 miles

Converting

\frac{2}{5}

to

0.40

gives a total completed portion of

0.40 + 0.35 = 0.75

of the goal. The remaining portion is

1 - 0.75 = 0.25

of the total goal. Setting

0.25 \times \text{total goal} = 12

miles gives a total weekly goal of

\frac{12}{0.25} = 48

miles.

Step-by-Step Solution

1

Convert the fraction completed on Monday to a decimal

\frac{2}{5} = 0.40

Converting all completed portions to decimals allows for direct addition.

2

Calculate the total portion of the weekly goal completed on Monday and Tuesday

0.40 + 0.35 = 0.75

Summing Monday's and Tuesday's completed portions gives the total completed fraction of the goal.

3

Determine the remaining fraction of the weekly goal

1 - 0.75 = 0.25

Subtracting the completed portion from the total whole (1.00) yields the fraction remaining.

4

Set up an equation relating the remaining fraction to the remaining miles to find the total goal distance

x

0.25x = 12 \implies x = \frac{12}{0.25} = 48

Dividing the remaining distance by the remaining fraction yields the total weekly goal distance.

Key Concept

Combining fractions and decimals to determine a total quantity from a remaining part

Estimated Time:1m 15s

Question 370Question →

A craft workshop instructor purchases a large spool of decorative ribbon of length $L$ meters. She cuts the entire spool into $15$ equal-length pieces to make banners for an art project. If each piece measures $2.4$ meters, which of the following equations can be solved to find the total length $L$ , in meters, of the spool of ribbon?

\frac{L}{15} = 2.4

15L = 2.4

L - 15 = 2.4

\frac{15}{L} = 2.4

L + 15 = 2.4

Show answer & explanation

Answer: $\frac{L}{15} = 2.4$

Answer

The equation that can be solved to find the total length

L

is

\frac{L}{15} = 2.4

.

When a total quantity

L

is divided equally into

15

pieces, the length of each individual piece is found by dividing

L

by

15

. Setting this quotient equal to the given single piece length of

2.4

meters yields

\frac{L}{15} = 2.4

.

Step-by-Step Solution

1

Identify the relationship between total length, number of pieces, and length per piece.

Total Length

\div

Number of Pieces

=

Length per Piece

Dividing a total quantity into equal parts corresponds to mathematical division.

2

Substitute the given values and variables into the relationship.

\frac{L}{15} = 2.4

The total length is represented by

L

, the number of pieces is

15

, and each piece is

2.4

meters long.

Key Concept

Translating equal sharing or division word problems into one-step linear equations

Estimated Time:1m 0s

Question 371Question →

An artisan bakery prepared a total batch of dough weighing $60$ pounds. In the morning, the baker used $\frac{2}{5}$ of the total dough to make loaves of bread. In the afternoon, the baker used $35\%$ of the remaining dough to make pastries. How many pounds of dough from the initial batch were left unused at the end of the day?

12.6

15.0

23.4

36.6

47.4

Show answer & explanation

Answer: $23.4$

Answer

The weight of the dough left unused at the end of the day is

23.4

pounds.

First, calculate the dough used in the morning:

\frac{2}{5}

of

60

pounds is

24

pounds. The remaining dough is

60 - 24 = 36

pounds. Next,

35\%

of

36

pounds is

0.35 \times 36 = 12.6

pounds used for pastries. Subtracting this from

36

pounds gives

36 - 12.6 = 23.4

pounds remaining unused.

Step-by-Step Solution

1

Calculate the weight of dough used in the morning.

\frac{2}{5} \times 60 = 24

pounds

The bakery used

\frac{2}{5}

of the total

60

-pound batch.

2

Find the dough remaining after the morning baking.

60 - 24 = 36

pounds

Subtract the amount used in the morning from the initial batch weight.

3

Calculate the percentage of dough remaining after the afternoon baking.

100\% - 35\% = 65\%

of the remaining dough

Since

35\%

of the remaining dough was used for pastries,

65\%

remained unused.

4

Calculate the final unused weight of dough.

0.65 \times 36 = 23.4

pounds

Multiply the remaining weight (

36

pounds) by the percentage left unused (

65\%

).

Key Concept

Multistep calculation involving fractional parts of a whole followed by percentage parts of a remainder.

Estimated Time:1m 30s

Question 372Question →

A local wildlife sanctuary maintains a specialized feeding program for rehabilitated hawks. Each hawk receives $p$ pounds of meat mixture per week. If the sanctuary cares for $18$ hawks and uses a total of $45$ pounds of meat mixture per week, which of the following equations can be used to find $p$ ?

18p = 45

\frac{p}{18} = 45

p + 18 = 45

p - 18 = 45

45p = 18

Show answer & explanation

Answer: $18p = 45$

Answer

18p = 45

The total amount of meat mixture used (

45

pounds) is equal to the number of hawks (

18

) multiplied by the amount of meat mixture each hawk receives (

p

pounds). Setting up this relationship gives the equation

18p = 45

.

Step-by-Step Solution

1

Identify the relationship between total quantity, number of groups, and quantity per group.

Total meat mixture = (Number of hawks)

\times

(Pounds per hawk)

Equal groups being combined call for multiplication.

2

Substitute the given values into the relationship.

45 = 18 \times p

, which simplifies to

18p = 45

.

The total is 45 pounds, the number of hawks is 18, and the rate per hawk is

p

.

Key Concept

Formulating one-step multiplication equations from real-world contexts

Estimated Time:1m 0s

Question 373Question →

A high-speed digital camera records $480$ frames every $3$ seconds. The camera operator increases the recording rate by $25\%$ . At this new recording rate, how many total frames will the camera record in $14$ seconds?

1,680

2,240

2,800

3,920

8,400

Show answer & explanation

Answer: $2,800$

Answer

The camera will record 2,800 frames in 14 seconds at the new recording rate.

To solve this problem, first determine the initial frame rate:

480

frames divided by

3

seconds equals

160

frames per second. Increasing this rate by

25\%

produces

160 \times 1.25 = 200

frames per second. Multiplying the new rate by

14

seconds results in

200 \times 14 = 2,800

frames.

Step-by-Step Solution

1

Calculate the original unit rate in frames per second.

\frac{480 \text{ frames}}{3 \text{ seconds}} = 160 \text{ frames per second}

Dividing the number of frames by the elapsed time yields the frame rate per second.

2

Determine the new unit rate after a

25\%

increase.

160 \times (1 + 0.25) = 160 \times 1.25 = 200 \text{ frames per second}

Increasing a value by

25\%

is equivalent to multiplying the original value by

1.25

.

3

Calculate total frames recorded in

14

seconds at the new rate.

200 \text{ frames per second} \times 14 \text{ seconds} = 2,800 \text{ frames}

Multiplying the new rate by the given duration yields the total frame count.

Key Concept

Unit rates and percentage adjustment in rates

Question 374Question →

A bookstore received a shipment of $50$ new books. Among these books, $24$ are fiction, $20$ are hardcover, and $8$ are hardcover fiction books. If $1$ book is selected at random from the shipment, what is the probability that the selected book is either a fiction book or a hardcover book?

\frac{4}{25}

\frac{14}{25}

\frac{18}{25}

\frac{22}{25}

\frac{9}{11}

Show answer & explanation

Answer: $\frac{18}{25}$

Answer

\frac{18}{25}

To find the probability that a randomly chosen book is either fiction or hardcover, calculate the total number of books satisfying at least one condition. By the Principle of Inclusion-Exclusion, the number of favorable books is

24 + 20 - 8 = 36

. Dividing this by the total of

50

books yields

\frac{36}{50}

, which simplifies to

\frac{18}{25}

.

Step-by-Step Solution

1

Identify the total number of items in the sample space and the counts for each event.

Total books

N = 50

, Fiction

N(F) = 24

, Hardcover

N(H) = 20

, Both

N(F \cap H) = 8

.

The sample space consists of all 50 shipment books.

2

Apply the Inclusion-Exclusion Principle to find the number of favorable outcomes.

N(F \cup H) = N(F) + N(H) - N(F \cap H) = 24 + 20 - 8 = 36

.

Hardcover fiction books are counted in both individual totals, so their intersection must be subtracted once to avoid double-counting.

3

Calculate the probability and simplify the fraction.

P(F \cup H) = \frac{36}{50} = \frac{18}{25}

.

Probability is the ratio of favorable outcomes to total outcomes.

Key Concept

Probability of non-mutually exclusive events using the Addition Rule:

P(A \cup B) = P(A) + P(B) - P(A \cap B)

.

Estimated Time:1m 0s

Question 375Question →

During a laboratory experiment, a liquid cooling solution's temperature dropped by a total of $14.4^\circ\text{C}$ over a duration of $t$ minutes. If the temperature decreased at a constant rate of $1.8^\circ\text{C}$ per minute, which of the following equations can be solved to find $t$ ?

1.8t = 14.4

\frac{t}{1.8} = 14.4

t - 1.8 = 14.4

t + 1.8 = 14.4

14.4t = 1.8

Show answer & explanation

Answer: $1.8t = 14.4$

Answer

The equation

1.8t = 14.4

correctly models the situation.

Because the cooling rate is constant at

1.8^\circ\text{C}

per minute, the total temperature change over

t

minutes is found by multiplying the rate by time (

1.8 \cdot t

). Setting this equal to the total drop of

14.4^\circ\text{C}

gives

1.8t = 14.4

.

Step-by-Step Solution

1

Identify the given quantities and their relationship.

Rate of change =

1.8^\circ\text{C/min}

, Time =

t

minutes, Total change =

14.4^\circ\text{C}

.

The total change in a constant rate scenario is the product of the rate and the time duration.

2

Set up the one-step algebraic equation.

\text{Rate} \times \text{Time} = \text{Total Change} \implies 1.8 \times t = 14.4

Translating the verbal relationship into an algebraic equation yields

1.8t = 14.4

.

Key Concept

Translating constant rate scenarios into one-step linear multiplication equations

Estimated Time:1m 0s

Question 376Question →

A commercial bakery makes a specialty flour blend using whole wheat flour, rye flour, and white flour in a ratio of $3 : 2 : 5$ by weight, respectively. To prepare a batch for a large order, a baker uses $36\text{ pounds}$ of whole wheat flour. If rye flour costs $\$1.50$ per pound and white flour costs $\$0.80$ per pound, what is the total cost of the rye flour and white flour needed for this batch?

\$25.20

\$73.20

\$84.00

\$109.20

\$120.00

Show answer & explanation

Answer: $\$84.00$

Answer

The total cost of the rye flour and white flour needed for this batch is

\$84.00

.

To find the cost, first determine the weight per ratio part. Whole wheat represents

3

parts out of

3 : 2 : 5

, so

3

parts equal

36\text{ pounds}

, making

1\text{ part} = 12\text{ pounds}

. Thus, the batch requires

2 \times 12 = 24\text{ pounds}

of rye flour and

5 \times 12 = 60\text{ pounds}

of white flour. At

\$1.50

per pound, the rye flour costs

24 \times \$1.50 = \$36.00

. At

\$0.80

per pound, the white flour costs

60 \times \$0.80 = \$48.00

. Adding these together gives a total cost of

\$36.00 + \$48.00 = \$84.00

.

Step-by-Step Solution

1

Determine the weight represented by one ratio part

Since whole wheat flour represents

3

parts of the ratio and equals

36\text{ pounds}

,

1\text{ part} = \frac{36}{3} = 12\text{ pounds}

.

Dividing the known quantity by its corresponding ratio weight finds the unit multiplier for the proportional relationship.

2

Calculate the weight of rye flour and white flour needed

Rye flour (

2

parts)

= 2 \times 12 = 24\text{ pounds}

. White flour (

5

parts)

= 5 \times 12 = 60\text{ pounds}

.

Multiply each component's ratio value by the single-part weight value.

3

Calculate the cost of each ingredient and sum them

Cost of rye flour

= 24 \times \$1.50 = \$36.00

. Cost of white flour

= 60 \times \$0.80 = \$48.00

. Total cost

= \$36.00 + \$48.00 = \$84.00

.

Multiply the weight of each ingredient by its unit price and add the totals together.

Key Concept

Solving multi-part ratio problems by finding a common multiplier and using unit rates.

Estimated Time:1m 15s

Question 377Question →

A nutritionist analyzed the proportion of daily recommended fiber provided by four different snack bars. The proportions were recorded as follows: Fruit Bar ( $58\%$ ), Oat Bar ( $\frac{7}{12}$ ), Nut Bar ( $0.585$ ), and Seed Bar ( $\frac{3}{5}$ ). Arrange the four snack bars in order from the LEAST proportion of daily recommended fiber to the GREATEST proportion of daily recommended fiber.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

Fruit Bar (

58\%

), Oat Bar (

\frac{7}{12}

), Nut Bar (

0.585

), Seed Bar (

\frac{3}{5}

)

To order the values from least to greatest, convert each representation into a standard decimal form: Fruit Bar =

58\% = 0.580

, Oat Bar =

\frac{7}{12} \approx 0.5833

, Nut Bar =

0.5850

, and Seed Bar =

\frac{3}{5} = 0.6000

. Arranging these decimals in increasing order yields

0.5800 < 0.5833 < 0.5850 < 0.6000

, which corresponds to Fruit Bar, Oat Bar, Nut Bar, and Seed Bar.

Step-by-Step Solution

1

Convert each numerical value into a common format (decimal representation) to allow direct comparison.

Fruit Bar:

58\% = \frac{58}{100} = 0.580

; Oat Bar:

\frac{7}{12} = 7 \div 12 \approx 0.5833

; Nut Bar:

0.585

; Seed Bar:

\frac{3}{5} = \frac{6}{10} = 0.600

.

Converting all numbers to decimals makes it straightforward to compare place values from left to right.

2

Compare the resulting decimal values up to the thousandths place:

0.5800

,

0.5833

,

0.5850

, and

0.6000

.

In order from smallest to largest decimal value:

0.5800 < 0.5833 < 0.5850 < 0.6000

.

Comparing the hundredths and thousandths digits shows that

0.580

is smallest, followed by

0.5833

, then

0.585

, and finally

0.600

.

3

Map the ordered decimals back to their original snack bar representations.

Fruit Bar (

58\%

) < Oat Bar (

\frac{7}{12}

) < Nut Bar (

0.585

) < Seed Bar (

\frac{3}{5}

).

This establishes the exact required sequence from least to greatest fiber proportion.

Key Concept

Converting and Comparing Mixed Numerical Representations (Fractions, Decimals, and Percentages)

Estimated Time:1m 15s

Question 378Question →

A box contains red, blue, and yellow game tokens. The probability of randomly selecting a red token is $\frac{3}{8}$ , and the probability of randomly selecting a blue token is $\frac{1}{3}$ . If the box contains exactly $14$ yellow tokens, how many total tokens are in the box?

48

24

22

16

72

Show answer & explanation

Answer: $48$

Answer

There are 48 total tokens in the box.

The total probability of all outcomes in the box must equal

1

. Combining the given probabilities for red and blue tokens yields

\frac{3}{8} + \frac{1}{3} = \frac{9}{24} + \frac{8}{24} = \frac{17}{24}

. The remaining probability representing the yellow tokens is

1 - \frac{17}{24} = \frac{7}{24}

. Since the box contains

14

yellow tokens, setting

\frac{7}{24}

of the total number of tokens equal to

14

gives

\frac{7}{24} N = 14

, which simplifies to

N = 48

.

Step-by-Step Solution

1

Find the probability of selecting a red or blue token.

\frac{3}{8} + \frac{1}{3} = \frac{9}{24} + \frac{8}{24} = \frac{17}{24}

Since selecting a red token and a blue token are mutually exclusive events, add their probabilities using a common denominator of

24

.

2

Calculate the probability of selecting a yellow token.

1 - \frac{17}{24} = \frac{7}{24}

The sum of the probabilities of all possible outcomes (red, blue, yellow) must equal

1

.

3

Set up and solve the equation for the total number of tokens

N

.

\frac{7}{24} N = 14 \implies N = 14 \times \frac{24}{7} = 2 \times 24 = 48

Multiply the count of yellow tokens (

14

) by the reciprocal of the yellow token probability (

\frac{24}{7}

).

Key Concept

Determining total sample size using complementary probabilities of mutually exclusive events

Estimated Time:1m 15s

Question 379Question →

A high school planning committee needs to select a subcommittee of $3$ people from a pool of $5$ teachers and $4$ students. How many different subcommittees can be formed consisting of exactly $2$ teachers and $1$ student?

Show answer & explanation

Answer: 40

Answer

The total number of different subcommittees that can be formed is 40.

To find the number of unique subcommittees with 2 teachers and 1 student, calculate the combination for choosing 2 teachers from 5, which equals 10, and 1 student from 4, which equals 4. Multiplying these independent choices together results in 40 possible subcommittees.

Step-by-Step Solution

1

Calculate the number of ways to select 2 teachers from the pool of 5 teachers.

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

ways

Selection order does not matter for committee members, so combinations are used.

2

Calculate the number of ways to select 1 student from the pool of 4 students.

\binom{4}{1} = 4

ways

Choosing 1 item from a set of 4 offers 4 distinct possibilities.

3

Apply the Fundamental Counting Principle to determine total combinations.

10 \times 4 = 40

subcommittees

Multiply the number of choices for teachers by the number of choices for students.

Key Concept

Combinations and Fundamental Counting Principle

Estimated Time:1m 0s

Question 380Question →

An urban hydroponic farm uses a liquid nutrient mixture containing nitrogen, potassium, and calcium in a mass ratio of $3 : 5 : 8$ , respectively. A technician prepares a batch of solution where the amount of calcium exceeds the amount of nitrogen by exactly $15\text{ grams}$ . How many total grams of nutrient solution will be prepared?

24\text{ grams}

30\text{ grams}

45\text{ grams}

48\text{ grams}

80\text{ grams}

Show answer & explanation

Answer: $48\text{ grams}$

Answer

The total mass of the nutrient solution prepared is

48\text{ grams}

.

The given ratio of nitrogen to potassium to calcium is

3 : 5 : 8

. The difference between calcium and nitrogen in the ratio is

8 - 3 = 5

parts. Since calcium exceeds nitrogen by

15\text{ grams}

, each ratio part corresponds to

15 \div 5 = 3\text{ grams}

. The total mixture consists of

3 + 5 + 8 = 16

parts, so the total mass is

16 \times 3\text{ grams} = 48\text{ grams}

.

Step-by-Step Solution

1

Determine the difference in ratio parts between calcium and nitrogen

Calcium has

8

parts and nitrogen has

3

parts, so calcium exceeds nitrogen by

8 - 3 = 5

ratio parts.

The problem states that calcium exceeds nitrogen by

15\text{ grams}

, which corresponds to the difference between their ratio values.

2

Calculate the mass represented by a single ratio part

\frac{15\text{ grams}}{5\text{ parts}} = 3\text{ grams per part}

.

Dividing the total difference in grams by the difference in ratio parts gives the value of one part.

3

Find the total number of ratio parts and compute the total mass

Total ratio parts =

3 + 5 + 8 = 16

parts. Total mass =

16 \text{ parts} \times 3\text{ grams per part} = 48\text{ grams}

.

Multiplying the single part mass by the sum of all components yields the total mass of the solution.

Key Concept

Solving multi-part ratio problems by finding the value of one ratio unit using given part-to-part differences.

Estimated Time:1m 15s

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