Pre-Algebra Practice Questions — ACT — Page 12

Question 221Question →

A manufacturing assembly line produces a batch of $B$ electronic components in $h$ hours. To increase efficiency, a new assembly line is installed that can produce the same batch of components in $\frac{1}{3}$ of the time. Which of the following expressions represents the rate, in components per hour, at which the new assembly line produces these components?

\frac{3B}{h}

\frac{B}{3h}

\frac{3h}{B}

3Bh

\frac{h}{3B}

Show answer & explanation

Answer: $\frac{3B}{h}$

Answer

The expression

\frac{3B}{h}

represents the rate of production of the new assembly line.

The correct answer shows that the rate of production is the number of components divided by the new time, which is one-third of the original time. Since the original time is

h

hours, the new time is

\frac{h}{3}

hours. Dividing the number of components

B

by

\frac{h}{3}

yields

B \div \frac{h}{3} = B \times \frac{3}{h} = \frac{3B}{h}

components per hour.

Step-by-Step Solution

1

Determine the time it takes for the new assembly line to produce the batch of components.

The new assembly line takes

\frac{h}{3}

hours.

The problem states the new line produces the batch in

\frac{1}{3}

of the original time,

h

.

2

Set up the algebraic expression for the rate of production in components per hour.

\text{Rate} = \frac{\text{Total Components}}{\text{Total Time}} = \frac{B}{\frac{h}{3}}

Rate is defined as quantity produced divided by time elapsed.

3

Simplify the fraction to find the final expression.

\text{Rate} = B \times \frac{3}{h} = \frac{3B}{h}

Dividing by a fraction is equivalent to multiplying by its reciprocal.

Key Concept

Formulating algebraic rate expressions from word problems

Question 222Question →

The table below shows the distribution of the number of books read by a group of students during a summer reading program.

Number of Books Read	Number of Students
1	1
2	4
3	8
4	$x$
5	4

If the median number of books read by the students is 3, what is the maximum possible value for the mean number of books read by these students?

3.0

3.2

3.4

3.6

3.8

Show answer & explanation

Answer: 3.4

Answer

The maximum possible value for the mean number of books read is 3.4.

To maximize the mean, we want to maximize the number of students who read 4 books, which is

x

. However,

x

is constrained by the requirement that the median must remain 3. The total number of students is

17 + x

. The 3s occupy sorted positions 6 through 13. For the median to be 3, the middle position(s) of the sorted list must not exceed position 13. When

N = 17+x

is odd (so

x

is even), the median position is

\frac{18+x}{2}

. Setting

\frac{18+x}{2} \le 13

yields

x \le 8

. For

x = 8

, the total number of students is 25, and the 13th student is the last one who read 3 books, making the median 3. If

x = 9

, the total is 26, and the median is the average of the 13th (3) and 14th (4) values, which is 3.5. Therefore, the maximum integer value of

x

is 8. The mean with

x=8

is the sum of all books read divided by the total number of students:

(1 \times 1 + 2 \times 4 + 3 \times 8 + 4 \times 8 + 5 \times 4) / 25 = 85 / 25 = 3.4

.

Step-by-Step Solution

1

Express the total number of students,

N

, in terms of

x

.

N = 1 + 4 + 8 + x + 4 = 17 + x

To calculate the mean and locate the median, we need the total count of data points.

2

Determine the range of positions that the value 3 occupies in the sorted dataset.

Positions 6 through 13.

Since the ratings are sorted, the single 1 is at position 1, the four 2s are at positions 2–5, and the eight 3s are at positions 6–13.

3

Find the maximum integer value of

x

that keeps the median at 3.

x = 8

If

x = 8

, the total number of students is

25

. The median is the 13th value, which is 3. If

x = 9

, the total is

26

, and the median is the average of the 13th (3) and 14th (4) values, which is 3.5.

4

Calculate the mean of the dataset when

x = 8

.

Mean =

85 / 25 = 3.4

The sum of the books read is

1(1) + 2(4) + 3(8) + 4(8) + 5(4) = 85

. Dividing this by the total of 25 students gives the maximum mean.

Key Concept

Finding the maximum mean of a frequency distribution given a median constraint by setting up inequalities for the median position.

Question 223Question →

A scientist is preparing a mixture by combining liquid A, liquid B, and liquid C. Initially, the ratio of the volume of liquid A to the volume of liquid B is $2:3$ , and the ratio of the volume of liquid B to the volume of liquid C is $4:5$ . The scientist then adds $120$ milliliters of liquid A to the mixture, while the volumes of liquid B and liquid C remain unchanged. This addition changes the ratio of the volume of liquid A to the volume of liquid B to $5:6$ . What was the original total volume of the mixture, in milliliters?

900

990

1,200

2,100

2,220

Show answer & explanation

Answer: 2,100

Answer

The correct total original volume of the mixture is 2,100 milliliters.

The correct answer shows the sum of the initial volumes of all three liquids. By aligning the two ratios, the original ratio is established as

A:B:C = 8:12:15

, representing a total of

35

parts. Setting up the equation for adding

120

milliliters of liquid A relative to liquid B reveals that each part corresponds to

60

milliliters. Multiplying the total of

35

parts by

60

milliliters yields

2,100

milliliters.

Step-by-Step Solution

1

Align the two given ratios by finding a common multiple for the shared component, liquid B.

The ratio of liquid A to liquid B is

2:3 = 8:12

, and the ratio of liquid B to liquid C is

4:5 = 12:15

. This yields the combined ratio

A:B:C = 8:12:15

.

To analyze changes in mixtures with multiple components, all ratio terms must be defined relative to a common unit value.

2

Define the initial volumes using a variable

x

and set up a proportion representing the addition of liquid A.

Let the initial volumes be

A = 8x

,

B = 12x

, and

C = 15x

. Adding

120

milliliters of liquid A gives the new volume

8x + 120

. The new ratio equation is

\frac{8x + 120}{12x} = \frac{5}{6}

.

The new ratio of liquid A to liquid B is given as

5:6

while the volume of liquid B remains unchanged.

3

Solve the proportion for

x

by cross-multiplying.

6(8x + 120) = 5(12x) \implies 48x + 720 = 60x \implies 12x = 720 \implies x = 60

.

Finding the value of

x

allows us to calculate the actual volumes of all three components.

4

Calculate the original total volume of the mixture.

Original total volume

= 8x + 12x + 15x = 35x = 35(60) = 2,100

milliliters.

The question asks for the sum of the initial volumes of all three liquids.

Key Concept

Combining multiple ratios to solve multi-step mixture and rate problems.

Alternative Method

Observe that the volume of liquid B does not change. Initially, the ratio of B to A is

3:2

, which can be scaled to

12:8

. After adding liquid A, the ratio of B to A becomes

6:5

, which is equivalent to

12:10

. The change in liquid A's ratio units is

10 - 8 = 2

units. Since the actual change is

120

milliliters, each unit represents

120 \div 2 = 60

milliliters. The original mixture has

8 + 12 + 15 = 35

units of volume, giving a total of

35 \times 60 = 2,100

milliliters.

Estimated Time:1m 30s

Question 224Question →

The frequency table below shows the number of siblings reported by a group of $10$ students in a class.

Number of Siblings	Number of Students
$0$	$2$
$1$	$4$
$2$	$3$
$3$	$1$

What is the mean number of siblings per student for this group?

1.0

1.5

1.3

2.5

3.25

Show answer & explanation

Answer: $1.3$

Answer

The mean number of siblings per student is

1.3

.

To find the mean number of siblings per student, first determine the total number of siblings by multiplying each sibling category by its frequency:

(0 \times 2) + (1 \times 4) + (2 \times 3) + (3 \times 1) = 13

. Next, determine the total number of students by summing the frequencies:

2 + 4 + 3 + 1 = 10

. Finally, divide the total siblings by the total students:

\frac{13}{10} = 1.3

.

Step-by-Step Solution

1

Find the total number of students by adding the frequencies in the second column.

2 + 4 + 3 + 1 = 10

students

This sum represents the total number of observations in the dataset.

2

Calculate the total number of siblings by multiplying each sibling count by the number of students reporting that count, then summing the products.

(0 \times 2) + (1 \times 4) + (2 \times 3) + (3 \times 1) = 0 + 4 + 6 + 3 = 13

siblings

This gives the sum of all data values across the entire group.

3

Divide the total number of siblings by the total number of students to find the mean.

\frac{13}{10} = 1.3

The mean is calculated as the sum of all values divided by the number of observations.

Key Concept

Calculating the mean from a frequency table.

Estimated Time:1m 0s

Question 225Question →

A school club is raising money by selling rolls of wrapping paper. The club keeps $40\%$ of the total sales revenue as profit. If the club made a profit of $\$220$ from selling $r$ rolls of wrapping paper that cost $\$10$ each, which of the following equations can be used to find $r$ ?

0.40r = 220

10r = 220

4r = 220

10r + 0.40 = 220

25r = 220

Show answer & explanation

Answer: $4r = 220$

Answer

The equation expressing the profit is the profit per roll multiplied by the number of rolls, which yields

4r = 220

.

To find the correct equation, we calculate the profit per roll of wrapping paper. Since each roll is sold for

\$10

and the club keeps

40\%

of the revenue as profit, the profit per roll is

0.40 \times 10 = 4

dollars. For

r

rolls sold, the total profit is

4r

dollars. Setting this equal to the given profit of

\$220

yields the equation

4r = 220

.

Step-by-Step Solution

1

Determine the expression for the total sales revenue.

The total revenue is

10r

dollars.

Each of the

r

rolls of wrapping paper is sold for

\$10

.

2

Express the profit as a fraction of the total sales revenue.

The profit is

0.40 \times 10r = 4r

dollars.

The club keeps

40\%

(

0.40

) of the total revenue as profit.

3

Set the profit expression equal to the actual profit made to form the equation.

4r = 220

The total profit made by the club is given as

\$220

.

Key Concept

Translating a multi-step real-world scenario into a one-step linear equation

Alternative Method

Find the profit per roll first:

40\%

of

\$10

is

\$4

. Thus, the profit for

r

rolls is

4r

. Since the total profit is

\$220

, the equation is

4r = 220

.

Estimated Time:1m 30s

Question 226Question →

What is the value of the expression $\frac{\sqrt{3^8 \times 2^6}}{6^2}$ ?

11

18

36

108

144

Show answer & explanation

Answer: 18

Answer

18

To evaluate the expression, first simplify the square root in the numerator. Since the square root of a product is the product of the square roots of its factors, we have

\sqrt{3^8 \times 2^6} = 3^{8/2} \times 2^{6/2} = 3^4 \times 2^3

. Next, rewrite the base in the denominator in terms of its prime factors:

6^2 = (3 \times 2)^2 = 3^2 \times 2^2

. Now, divide the simplified terms using the quotient rule for exponents:

\frac{3^4 \times 2^3}{3^2 \times 2^2} = 3^{4-2} \times 2^{3-2} = 3^2 \times 2^1

. Evaluating this gives

9 \times 2 = 18

.

Step-by-Step Solution

1

Simplify the square root in the numerator.

\sqrt{3^8 \times 2^6} = \sqrt{3^8} \times \sqrt{2^6} = 3^{8/2} \times 2^{6/2} = 3^4 \times 2^3

.

Applying the square root to factors with even exponents reduces their exponents by half.

2

Rewrite the denominator

6^2

in terms of prime bases

2

and

3

.

6^2 = (2 \times 3)^2 = 2^2 \times 3^2

.

Expressing the bases in prime factors allows the use of exponent rules to simplify the fraction.

3

Divide the simplified numerator by the simplified denominator using the quotient rule.

\frac{3^4 \times 2^3}{3^2 \times 2^2} = 3^{4-2} \times 2^{3-2} = 3^2 \times 2^1 = 9 \times 2 = 18

.

Subtracting exponents of like bases simplifies the rational expression to a final numerical value.

Key Concept

Simplifying radical expressions and applying laws of exponents for multiplication and division of powers.

Estimated Time:1m 0s

Question 227Question →

A tablet battery charges at a constant rate of $r$ percentage points per hour. If the battery charge increases by $48$ percentage points in $3$ hours, which of the following equations can be used to find $r$ ?

r + 3 = 48

3r = 48

\frac{r}{3} = 48

r - 3 = 48

\frac{3}{r} = 48

Show answer & explanation

Answer: $3r = 48$

Answer

The equation

3r = 48

can be used to find the charging rate.

The battery charges at a constant rate of

r

percentage points per hour. Over

3

hours, the total increase is the rate multiplied by the time, which is

3r

. Since the total increase is given as

48

percentage points, the correct relationship is expressed by the equation

3r = 48

.

Step-by-Step Solution

1

Identify the relationship between the charging rate, time, and total charge increase.

The total charge increase is equal to the charging rate multiplied by the number of hours.

Since the battery charges at a constant rate of

r

percentage points per hour, in

3

hours it will charge a total of

3 \times r

or

3r

percentage points.

2

Set the expression for the total charge increase equal to the given total increase value.

3r = 48

The total charge increase is given as

48

percentage points, so we set the expression

3r

equal to

48

.

Key Concept

Translating a real-world scenario involving a constant rate into a one-step linear equation.

Alternative Method

You can also solve for the rate first by dividing the total increase of

48

by

3

hours to get

r = 16

percentage points per hour. Substituting

r = 16

into the equations shows that only

3(16) = 48

is a true statement.

Estimated Time:45s

Question 228Question →

If $a$ and $b$ are integers such that $-5 < a < -1$ and $2 < b < 6$ , what is the difference between the maximum possible value of $|a - b|$ and the minimum possible value of $|a + b|$ ?

2

4

8

9

11

Show answer & explanation

Answer: 9

Answer

The difference between the maximum possible value of

|a - b|

and the minimum possible value of

|a + b|

is 9.

The maximum possible value of

|a - b|

is 9, which occurs when

a = -4

and

b = 5

, giving

|-4 - 5| = 9

. The minimum possible value of

|a + b|

is 0, which occurs when

a = -3

and

b = 3

(or

a = -4

and

b = 4

), giving

|-3 + 3| = 0

. The difference between these two values is

9 - 0 = 9

.

Step-by-Step Solution

1

Identify the possible integer values for

a

and

b

from the strict inequalities.

The integers satisfying

-5 < a < -1

are

a \in \{-4, -3, -2\}

. The integers satisfying

2 < b < 6

are

b \in \{3, 4, 5\}

.

Since the inequalities are strict (

<

), the endpoints

-5, -1, 2,

and

6

are excluded.

2

Determine the maximum possible value of

|a - b|

by selecting the values of

a

and

b

that maximize their distance.

Using

a = -4

and

b = 5

yields

|-4 - 5| = |-9| = 9

.

The absolute value of the difference is maximized when

a

is as small (most negative) as possible and

b

is as large (most positive) as possible.

3

Determine the minimum possible value of

|a + b|

by finding values of

a

and

b

that are closest to being additive opposites.

Using

a = -3

and

b = 3

(or

a = -4

and

b = 4

) yields

|-3 + 3| = |0| = 0

.

The absolute value of any real number is at least 0. Since we can choose integers that sum to exactly 0, the minimum possible value is 0.

4

Calculate the difference between the two extreme values found in the previous steps.

The difference is

9 - 0 = 9

.

We subtract the minimum value of the second expression from the maximum value of the first expression.

Key Concept

Absolute value represents distance from zero, and finding extreme values of absolute value expressions involving restricted integer sets requires testing boundary combinations and understanding additive inverses.

Alternative Method

Instead of checking every pair, we can analyze the extreme values of the intervals. Since

a \in [-4, -2]

and

b \in [3, 5]

, the difference

a - b

ranges from

-4 - 5 = -9

to

-2 - 3 = -5

. The absolute value

|a - b|

therefore ranges from 5 to 9, making the maximum value 9. For the sum, since the interval of

-a

is

[2, 4]

and overlaps with the interval of

b

which is

[3, 5]

, they can be equal (specifically at 3 and 4). When

-a = b

, we have

a + b = 0

, so the minimum value of

|a + b|

must be 0. Subtracting the two values gives

9 - 0 = 9

.

Estimated Time:1m 30s

Question 229Question →

During a science experiment, a student measured the temperature changes of a liquid in degrees Celsius at seven different intervals. The recorded changes were $15^\circ\text{C}$ , $6^\circ\text{C}$ , $20^\circ\text{C}$ , $24^\circ\text{C}$ , $7^\circ\text{C}$ , $4^\circ\text{C}$ , and $8^\circ\text{C}$ . What is the positive difference between the median and the mean of these seven recorded temperatures?

4

6

8

12

16

Show answer & explanation

Answer: 4

Answer

4

To find the correct positive difference, we first find the mean by dividing the sum of the temperatures by 7:

(15 + 6 + 20 + 24 + 7 + 4 + 8) / 7 = 84 / 7 = 12

. We then sort the temperatures in ascending order:

4, 6, 7, 8, 15, 20, 24

. The median is the middle value of this sorted list, which is 8. The positive difference between the median and the mean is

|8 - 12| = 4

.

Step-by-Step Solution

1

Calculate the mean of the temperatures.

Mean = 12

Sum the seven values (

15 + 6 + 20 + 24 + 7 + 4 + 8 = 84

) and divide by the total number of data points (7) to get

84 / 7 = 12

.

2

Find the median of the temperatures.

Median = 8

Sort the values in ascending order:

4, 6, 7, 8, 15, 20, 24

. Since there are 7 data points, the median is the fourth value, which is 8.

3

Calculate the positive difference between the median and the mean.

Positive difference = 4

Subtract the median from the mean (or vice versa) and take the absolute value:

|8 - 12| = 4

.

Key Concept

Calculating and comparing the mean and median of a dataset, including the necessity of sorting data before finding the median.

Question 230Question →

A web designer is creating a layout with three sections: a header, a main content area, and a sidebar. The ratio of the area of the header to the area of the main content area is $1:3$ . The ratio of the area of the main content area to the area of the sidebar is $4:3$ . If the total area of the layout is $2,000$ square pixels, what is the area, in square pixels, of the sidebar?

320

720

960

1,000

1,500

Show answer & explanation

Answer: 720

Answer

720

The correct answer is 720 square pixels. To find this, combine the ratios

H:M = 1:3

(which scales to

4:12

) and

M:S = 4:3

(which scales to

12:9

) using their common term

M

. This yields a joint ratio of

H:M:S = 4:12:9

. Let the areas be

4x

,

12x

, and

9x

. The sum is

25x = 2,000

, which gives

x = 80

. The sidebar area is

9x = 9(80) = 720

square pixels.

Step-by-Step Solution

1

Express the ratios with a common term.

The ratio of the header to the main content is

H:M = 1:3

, which is equivalent to

4:12

. The ratio of the main content to the sidebar is

M:S = 4:3

, which is equivalent to

12:9

. Combining these gives a joint ratio of

H:M:S = 4:12:9

.

Finding a common term for the main content area (

M

) allows us to express all three sections in a single ratio.

2

Set up an equation for the total area in terms of a multiplier

x

.

4x + 12x + 9x = 2,000 \Rightarrow 25x = 2,000

The sum of the individual areas must equal the total area of

2,000

square pixels.

3

Solve for

x

.

x = 80

This determines the value of one part of the ratio.

4

Calculate the area of the sidebar.

9 \times 80 = 720

The sidebar's area is represented by

9x

, so we multiply

9

by

80

.

Key Concept

Combining multiple ratios through a common term to solve multi-part ratio problems.

Question 231Question →

A commuter train travels at a constant speed to complete a trip of $D$ miles in $t$ hours. Due to track maintenance, the train's speed is decreased by 15 miles per hour for a 20-minute portion of the trip. Which of the following expressions represents the distance, in miles, the train traveled during this 20-minute portion of the trip?

\frac{D}{t} - 5

\frac{D}{3t} - 15

\frac{D}{3t} - 5

\frac{D}{3t} - 45

\frac{3D}{t} - 5

Show answer & explanation

Answer: $\frac{D}{3t} - 5$

Answer

The expression

\frac{D}{3t} - 5

represents the distance traveled during the 20-minute portion of the trip.

The train's original speed is

\frac{D}{t}

miles per hour. During the maintenance segment, its speed is reduced by 15 miles per hour, giving a speed of

\frac{D}{t} - 15

miles per hour. To find the distance traveled over a 20-minute interval (which represents

\frac{20}{60} = \frac{1}{3}

of an hour), the reduced speed is multiplied by the elapsed time. Applying the distributive property to

\left(\frac{D}{t} - 15\right) \times \frac{1}{3}

results in

\frac{D}{3t} - 5

.

Step-by-Step Solution

1

Determine the train's original speed.

Original speed is

\frac{D}{t}

miles per hour.

Speed is defined as distance divided by time.

2

Express the reduced speed during the track maintenance portion.

Reduced speed is

\frac{D}{t} - 15

miles per hour.

The speed was decreased by 15 miles per hour from the original speed.

3

Convert the duration of the maintenance portion from minutes to hours.

20 minutes is equal to

\frac{20}{60} = \frac{1}{3}

hour.

Since the speed is in miles per hour, the time must be converted to hours to ensure unit consistency.

4

Calculate the distance traveled during the 20-minute portion.

Distance is

\left(\frac{D}{t} - 15\right) \times \frac{1}{3} = \frac{D}{3t} - 5

miles.

Distance is equal to the product of speed and time. The distributive property must be applied to both terms in the speed expression.

Key Concept

Translating a verbal rate and time relationship into a simplified algebraic expression using the distributive property and unit conversion.

Alternative Method

Alternatively, you can assign plug-in values to make the problem concrete. Let the total distance

D = 120

miles and the total time

t = 2

hours. This gives an original speed of

60

miles per hour. The reduced speed is

60 - 15 = 45

miles per hour. Traveling at

45

miles per hour for

20

minutes (

\frac{1}{3}

hour) results in a distance of

45 \times \frac{1}{3} = 15

miles. Substituting

D = 120

and

t = 2

into the correct expression gives

\frac{120}{3(2)} - 5 = \frac{120}{6} - 5 = 20 - 5 = 15

miles, which matches our result.

Estimated Time:1m 30s

Question 232Question →

A courier uses three modes of transportation to deliver packages: a bicycle, an electric scooter, and a delivery van. The ratio of the average speed of the electric scooter to the average speed of the bicycle is $3:2$ , and the ratio of the average speed of the delivery van to the average speed of the electric scooter is $5:2$ . If it takes the courier $60$ minutes to complete a delivery route using the bicycle, how many minutes would it take the courier to complete the exact same delivery route using the delivery van, assuming all average speeds remain constant?

16

24

36

100

225

Show answer & explanation

Answer: 16

Answer

16 minutes

Because travel time is inversely proportional to speed when distance is constant, the time required decreases as speed increases. With a scooter-to-bicycle speed ratio of

3:2

, the scooter takes

\frac{2}{3}

of the bicycle's time:

60 \times \frac{2}{3} = 40

minutes. With a van-to-scooter speed ratio of

5:2

, the van takes

\frac{2}{5}

of the scooter's time:

40 \times \frac{2}{5} = 16

minutes.

Step-by-Step Solution

1

Determine the relationship between the speed of the bicycle and the speed of the electric scooter.

v_{\text{scooter}} = \frac{3}{2}v_{\text{bicycle}}

The given ratio of the average speed of the scooter to the bicycle is

3:2

.

2

Calculate the time it would take to complete the route using the electric scooter.

40

minutes

Since speed and travel time are inversely proportional for a constant distance, the scooter's time is

60 \times \frac{2}{3} = 40

minutes.

3

Determine the relationship between the speed of the electric scooter and the delivery van.

v_{\text{van}} = \frac{5}{2}v_{\text{scooter}}

The given ratio of the average speed of the van to the scooter is

5:2

.

4

Calculate the time it would take to complete the route using the delivery van.

16

minutes

Since speed and travel time are inversely proportional, the van's time is

40 \times \frac{2}{5} = 16

minutes.

Key Concept

Inverse proportionality in rate-time-distance relationships and compounding multiple ratios

Question 233Question →

The diameter of a human red blood cell is approximately $7.0 \times 10^{-6}$ meters, and the diameter of a typical influenza virus is approximately $1.4 \times 10^{-7}$ meters. How many times larger is the diameter of the red blood cell than the diameter of the influenza virus?

Show answer & explanation

Answer: 50

Answer

50

To determine how many times larger the diameter of the red blood cell is than the diameter of the influenza virus, divide the larger diameter by the smaller diameter:

\frac{7.0 \times 10^{-6}}{1.4 \times 10^{-7}}

. Dividing the coefficients gives

\frac{7.0}{1.4} = 5

. Dividing the powers of ten using the quotient rule for exponents gives

\frac{10^{-6}}{10^{-7}} = 10^{-6 - (-7)} = 10^1

. Multiplying these results gives

5 \times 10^1 = 50

.

Step-by-Step Solution

1

Set up the ratio of the larger diameter to the smaller diameter

\frac{7.0 \times 10^{-6}}{1.4 \times 10^{-7}}

To find how many times larger one quantity is than another, divide the larger quantity by the smaller quantity.

2

Divide the decimal coefficients

5.0

Dividing

7.0

by

1.4

simplifies the numerical coefficient.

3

Divide the exponential terms using exponent properties

10^1

Using the quotient rule for exponents,

\frac{10^a}{10^b} = 10^{a - b}

, so

10^{-6 - (-7)} = 10^1

.

4

Combine and simplify the final value

50

Multiplying the coefficient by the simplified power of ten yields

5.0 \times 10 = 50

.

Key Concept

Division of numbers in scientific notation using properties of exponents

Estimated Time:1m 30s

Question 234Question →

A geology student collected a group of rocks and recorded their weights, in grams, in the table below. One frequency, $f$ , is missing.

Weight (grams)	Frequency
25	5
10	4
20	$f$
30	2
15	6

If the mean weight of the rocks in the group is exactly $18.75$ grams, what is the median weight, in grams, of the rocks?

3

10

15

17.5

20

Show answer & explanation

Answer: 17.5

Answer

17.5

The correct answer is 17.5. By setting up the weighted mean equation, we find that the missing frequency

f

is 3. This means there are 20 rocks in total. When sorted in ascending order, the weights are four 10s, six 15s, three 20s, five 25s, and two 30s. The 10th rock weighs 15 grams and the 11th rock weighs 20 grams. The median is the average of these two middle values: (15 + 20) / 2 = 17.5 grams.

Step-by-Step Solution

1

Set up an equation for the mean weight using the frequencies and weights from the table.

The sum of the weights is

25(5) + 10(4) + 20(f) + 30(2) + 15(6) = 315 + 20f

. The total number of rocks is

5 + 4 + f + 2 + 6 = 17 + f

. The mean is

\frac{315 + 20f}{17 + f} = 18.75

.

To find the missing frequency

f

using the given mean.

2

Solve the equation for

f

.

315 + 20f = 18.75(17 + f) \implies 315 + 20f = 318.75 + 18.75f \implies 1.25f = 3.75 \implies f = 3

.

To determine the number of rocks that weigh 20 grams.

3

Find the total number of rocks and identify the median position.

The total number of rocks is

17 + 3 = 20

. Since the total number is even, the median is the average of the 10th and 11th values when the weights are sorted in ascending order.

The median of an even number of data points is the average of the two middle values.

4

List the sorted weights and find the 10th and 11th values.

The sorted weights are: four 10s, six 15s (making 10 rocks total), followed by three 20s. Thus, the 10th rock weighs 15 grams and the 11th rock weighs 20 grams.

To find the values of the two middle rocks.

5

Calculate the average of the 10th and 11th values.

Median =

\frac{15 + 20}{2} = 17.5

grams.

To find the median weight.

Key Concept

Calculating the median from a frequency distribution table after finding a missing frequency using the mean.

Estimated Time:2m 30s

Question 235Question →

A scientist monitors the temperature of two research chambers. Chamber A is kept at $-5^\circ\text{C}$ and Chamber B is kept at $7^\circ\text{C}$ . The scientist sets a third chamber, Chamber C, to a temperature of $T^\circ\text{C}$ such that the distance between $T$ and the temperature of Chamber A on the Celsius scale is exactly $3$ times the distance between $T$ and the temperature of Chamber B. If the temperature of Chamber C is warmer than Chamber A but colder than Chamber B, what is the value of $T$ ?

Show answer & explanation

Answer: 4

Answer

The correct temperature value of Chamber C is 4.

The temperature of Chamber C must be

4^\circ\text{C}

because the distance from

4

to

-5

is

|4 - (-5)| = 9

and the distance from

4

to

7

is

|4 - 7| = 3

. The distance of

9

is exactly

3

times the distance of

3

. Furthermore,

4

lies between

-5

and

7

, satisfying the condition that Chamber C is warmer than Chamber A but colder than Chamber B.

Step-by-Step Solution

1

Represent the distances on the number line using absolute value expressions.

The distance to Chamber A is

|T - (-5)| = |T + 5|

and the distance to Chamber B is

|T - 7|

.

Distance between two points

x

and

y

on a number line is represented by

|x - y|

.

2

Set up the algebraic equation reflecting the relationship between the distances.

|T + 5| = 3|T - 7|

The problem states the distance to Chamber A is exactly 3 times the distance to Chamber B.

3

Solve the absolute value equation by considering both positive and negative cases.

Case 1:

T + 5 = 3(T - 7) \Rightarrow T = 13

. Case 2:

T + 5 = -3(T - 7) \Rightarrow T = 4

.

The equation

|x| = |y|

implies

x = y

or

x = -y

.

4

Verify which solution satisfies the temperature boundary condition.

Since Chamber C must be warmer than

-5^\circ\text{C}

but colder than

7^\circ\text{C}

, the only valid value is

T = 4

.

The value

T = 13

is warmer than both chambers and does not lie between them.

Key Concept

Using absolute value to represent distance on a number line and solving absolute value equations with boundary conditions.

Alternative Method

Alternatively, visualize this on a number line. The total distance between Chamber A (

-5

) and Chamber B (

7

) is

12

units. Since Chamber C lies between them and the distance from C to A is

3

times the distance from C to B, we can divide the

12

-unit interval into

3 + 1 = 4

equal parts. Each part is

12 \div 4 = 3

units. Chamber C is located

1

part away from Chamber B (towards Chamber A), which places it at

7 - 3 = 4

.

Estimated Time:1m 30s

Question 236Question →

A group of friends agrees to share the total cost, $d$ dollars, of renting a cabin equally. There are 6 friends in the group, and each friend's share of the cost is $145. Which of the following equations can be solved to find$ d$?

d + 6 = 145

d - 6 = 145

6d = 145

\frac{d}{6} = 145

\frac{6}{d} = 145

Show answer & explanation

Answer: $\frac{d}{6} = 145$

Answer

The equation

\frac{d}{6} = 145

can be solved to find the total cost.

Because the total cost of renting the cabin,

d

dollars, is split equally among 6 friends, each friend's individual share is represented by the total cost divided by 6, which is written as

\frac{d}{6}

. Since each friend's share is given as

145, this expression is set equal to 145, resulting in the equation

\frac{d}{6} = 145$.

Step-by-Step Solution

1

Determine the relationship between the total cost, the number of friends sharing the cost, and each friend's individual share.

Since the cost is shared equally, the individual share is the total cost,

d

, divided by the number of friends, 6.

This translates the sharing scenario into the algebraic expression

\frac{d}{6}

.

2

Set the algebraic expression equal to the known individual share value.

\frac{d}{6} = 145

The problem states that each friend's share is $145, so the expression representing each share must equal 145.

Key Concept

Translating real-world sharing scenarios into one-step equations involving division.

Question 237Question →

To calibrate a temperature sensor, a technician uses the formula $T_c = T_r - e$ , where $T_c$ is the calibrated temperature, $T_r$ is the raw sensor reading, and $e$ is the calibration error of the sensor. For a particular sensor, the calibration error is $-2.4^\circ\text{C}$ . If the calibrated temperature is measured to be $18.6^\circ\text{C}$ , which of the following equations can be solved to find the raw sensor reading, $T_r$ , in degrees Celsius?

18.6 = T_r - 2.4

18.6 = T_r + 2.4

18.6 = 2.4 - T_r

18.6 = -2.4 - T_r

18.6 = \\frac{T_r}{-2.4}

Show answer & explanation

Answer: $18.6 = T_r + 2.4$

Answer

The equation

18.6 = T_r + 2.4

To find the correct equation, substitute the given values into the formula

T_c = T_r - e

. Substituting

T_c = 18.6

and

e = -2.4

yields

18.6 = T_r - (-2.4)

. Subtracting a negative number is equivalent to addition, which simplifies the equation to

18.6 = T_r + 2.4

.

Step-by-Step Solution

1

Identify the given variables and their values from the problem statement.

The calibrated temperature

T_c = 18.6

and the calibration error

e = -2.4

.

Knowing the values allows for direct substitution into the formula.

2

Substitute the identified values into the calibration formula

T_c = T_r - e

.

The equation becomes

18.6 = T_r - (-2.4)

.

This establishes the relationship between the raw reading

T_r

and the known values.

3

Simplify the subtraction of the negative number on the right side of the equation.

The equation simplifies to

18.6 = T_r + 2.4

.

Subtracting a negative number is mathematically equivalent to adding its positive counterpart.

Key Concept

Substituting signed values into algebraic expressions and simplifying one-step linear equations.

Question 238Question →

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 239Question →

A gymnast competed in 5 events during a state meet, earning scores of $14.2$ , $13.8$ , $14.5$ , $13.5$ , and $14.0$ . She has one event remaining. If her goal is to have an average (arithmetic mean) score of exactly $14.1$ for all 6 events, what score must she earn on her final event?

14.0

14.1

14.5

14.6

15.1

Show answer & explanation

Answer: 14.6

Answer

14.6

To find the score needed on the final event, we first calculate the total sum of the scores from the first 5 events:

14.2 + 13.8 + 14.5 + 13.5 + 14.0 = 70.0

. Next, we find the total sum required for all 6 events to average exactly

14.1

:

14.1 \times 6 = 84.6

. Finally, we subtract the sum of the first 5 events from this target total:

84.6 - 70.0 = 14.6

. Therefore, the required score on the final event is 14.6.

Step-by-Step Solution

1

Calculate the sum of the scores from the first 5 events.

14.2 + 13.8 + 14.5 + 13.5 + 14.0 = 70.0

To find the missing score, we need to compare the current total score to the target total score.

2

Calculate the target total score for all 6 events to achieve an average of

14.1

.

14.1 \times 6 = 84.6

Since the average is the sum divided by the number of events, the total sum is the average multiplied by 6.

3

Subtract the sum of the first 5 events from the target total sum to find the required final score.

84.6 - 70.0 = 14.6

The difference between the required total sum and the current sum represents the score needed on the final event.

Key Concept

Calculating a missing value in a data set to achieve a target arithmetic mean.

Question 240Question →

A logistics company distributes $T$ tons of cargo among a fleet of trucks. A standard truck has a maximum capacity of $c$ tons, and a heavy-duty truck has a maximum capacity of twice that of a standard truck. The fleet consists of $x$ standard trucks and $y$ heavy-duty trucks, all loaded to their maximum capacities. If the ratio of standard trucks to heavy-duty trucks in the fleet is exactly $3:1$ , which of the following equations correctly expresses the capacity of a standard truck, $c$ , in terms of $T$ and $y$ ?

c = \frac{T}{4y}

c = \frac{3T}{7y}

c = \frac{T}{5y}

c = \frac{T}{2y + 3}

c = T - 5y

Show answer & explanation

Answer: $c = \frac{T}{5y}$

Answer

The equation

c = \frac{T}{5y}

correctly expresses the capacity of a standard truck.

The total cargo

T

is the sum of the capacities of all trucks. Since there are

x

standard trucks carrying

c

tons each and

y

heavy-duty trucks carrying

2c

tons each, the equation is

T = cx + 2cy

. The

3:1

ratio of standard trucks to heavy-duty trucks means

x = 3y

. Substituting this relationship into the cargo equation gives

T = c(3y) + 2cy = 5cy

. To isolate the capacity of a standard truck,

c

, divide both sides by

5y

to get

c = \frac{T}{5y}

.

Step-by-Step Solution

1

Write the equation for the total cargo weight by summing the capacities of all trucks in the fleet.

T = cx + 2cy

The total cargo

T

is distributed among

x

standard trucks, each carrying

c

tons, and

y

heavy-duty trucks, each carrying

2c

tons.

2

Translate the ratio of standard trucks to heavy-duty trucks into an algebraic relationship.

x = 3y

The ratio of standard trucks (

x

) to heavy-duty trucks (

y

) is

3:1

, meaning there are 3 times as many standard trucks as heavy-duty trucks.

3

Substitute the ratio relationship

x = 3y

into the total cargo equation.

T = c(3y) + 2cy

Replacing

x

with

3y

eliminates the variable

x

and expresses

T

in terms of

c

and

y

.

4

Combine like terms to simplify the expression.

T = 5cy

Adding

3cy

and

2cy

yields

5cy

.

5

Isolate the variable

c

by dividing both sides of the equation by

5y

.

c = \frac{T}{5y}

Division is the inverse operation of multiplication, allowing us to solve the one-step equation

T = (5y)c

for

c

.

Key Concept

Translating verbal descriptions into algebraic equations and solving one-step literal equations

Estimated Time:2m 0s

Pre-Algebra

Answer

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