Pre-Algebra Practice Questions — ACT — Page 5

What is the integer value of the expression $-4^2 + | -15 \div 3 \times 2 | - (-3)^3$ ?

Show answer & explanation

Answer: 21

Answer

The correct answer is 21.

The correct answer is obtained by strictly following the order of operations (PEMDAS): first evaluating exponents (

-4^2 = -16

and

(-3)^3 = -27

), then performing the operations inside the absolute value from left to right (

-15 \div 3 = -5

followed by

-5 \times 2 = -10

, which simplifies to

|-10| = 10

), and finally executing addition and subtraction from left to right (

-16 + 10 - (-27) = 21

).

Step-by-Step Solution

1

Evaluate the exponent

-4^2

-16

According to the order of operations, the exponent is applied to the base 4 first, and then the negation is applied, yielding

-(4^2) = -16

.

2

Evaluate the absolute value expression

|-15 \div 3 \times 2|

10

Multiplication and division have the same precedence and must be performed from left to right: first

-15 \div 3 = -5

, then

-5 \times 2 = -10

. Taking the absolute value of

-10

yields

10

.

3

Evaluate the exponent

(-3)^3

-27

Since the negative sign is inside the parentheses, the base is

-3

, so

(-3)^3 = (-3) \times (-3) \times (-3) = -27

.

4

Combine the values and simplify

21

Substitute the evaluated parts into the original expression:

-16 + 10 - (-27)

. Simplify by performing addition and subtraction from left to right:

-16 + 10 = -6

, and

-6 - (-27) = -6 + 27 = 21

.

Key Concept

Order of operations (PEMDAS) and absolute value properties

Question 82Question →

An expression is given as $\sqrt{2^6 \cdot 3^4}$ . What is the value of this expression?

17

432

72

5,184

7,776

Show answer & explanation

Answer: 72

Answer

72

The correct answer of

72

is obtained by applying the rules of exponents under a radical. Evaluating

\sqrt{2^6 \cdot 3^4}

simplifies to

2^3 \cdot 3^2

, which equals

8 \cdot 9 = 72

.

Step-by-Step Solution

1

Express the square root as a fractional exponent of 1/2.

(2^6 \cdot 3^4)^{\frac{1}{2}}

The square root of any non-negative number

x

can be written as

x^{\frac{1}{2}}

.

2

Apply the power of a product rule

(ab)^n = a^n b^n

and power of a power rule

(a^m)^n = a^{mn}

to simplify the exponents.

2^{6 \cdot \frac{1}{2}} \cdot 3^{4 \cdot \frac{1}{2}} = 2^3 \cdot 3^2

Exponents are multiplied when raising a power to a power.

3

Evaluate the simplified exponential expressions and multiply the results.

8 \cdot 9 = 72

Evaluating

2^3 = 8

and

3^2 = 9

and calculating their product gives the final simplified value.

Key Concept

Simplifying expressions containing exponents and square roots

Question 83Question →

A security guard has two patrol routes. Route A takes 18 minutes to complete, and Route B takes 24 minutes to complete. The guard starts both routes at the same time at 6:00 PM. Assuming the guard works continuously without any breaks, how many times will both routes be completed at the same time between 6:01 PM and 11:59 PM on the same evening?

3

4

5

6

10

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. The least common multiple (LCM) of 18 and 24 is 72, which means both routes are completed at the same time every 72 minutes (1 hour and 12 minutes). Starting from 6:00 PM, the simultaneous completions occur at 7:12 PM, 8:24 PM, 9:36 PM, 10:48 PM, and 12:00 AM. Only the four times between 7:12 PM and 10:48 PM fall strictly between 6:01 PM and 11:59 PM.

Step-by-Step Solution

1

Find the Least Common Multiple (LCM) of the two cycle times, 18 minutes and 24 minutes.

LCM(18, 24) = 72 minutes

Both routes are completed at the same time at intervals that are common multiples of their individual durations. The first time they align after the start is the LCM.

2

List the times when both routes are completed simultaneously, starting after 6:00 PM.

7:12 PM, 8:24 PM, 9:36 PM, 10:48 PM, and 12:00 AM

Add multiples of 72 minutes (1 hour 12 minutes) to the starting time of 6:00 PM.

3

Determine which of the calculated times fall strictly within the window from 6:01 PM to 11:59 PM.

4 times (7:12 PM, 8:24 PM, 9:36 PM, and 10:48 PM)

The times 7:12 PM, 8:24 PM, 9:36 PM, and 10:48 PM are within the specified window. The completion at 12:00 AM is outside the window.

Key Concept

Least Common Multiple (LCM) in scheduling word problems

Question 84Question →

What is the value of the expression $5^3 - \sqrt{64}$ ?

Show answer & explanation

Answer: 117

Answer

The correct answer is

117

.

Evaluating

5^3

gives

125

since

5 \times 5 \times 5 = 125

. The square root of

64

is

8

because

8^2 = 64

. Subtracting

8

from

125

yields the correct result of

117

.

Step-by-Step Solution

1

Evaluate the exponential term

5^3

125

An exponent indicates how many times a base is multiplied by itself. Here,

5^3 = 5 \times 5 \times 5 = 125

.

2

Evaluate the radical term

\sqrt{64}

8

The square root of a number is the non-negative value that, when multiplied by itself, equals the original number. Since

8 \times 8 = 64

,

\sqrt{64} = 8

.

3

Subtract the evaluated terms

117

Subtract the value of the radical from the value of the exponent to simplify the expression:

125 - 8 = 117

.

Key Concept

Evaluating basic exponents and square roots

Question 85Question →

On a standard number line, the distance between two points, $A$ and $B$ , is $12$ units. If the coordinate of point $A$ is $-18$ , which of the following is a possible coordinate for point $B$ ?

-30

-12

-8

6

30

Show answer & explanation

Answer: $-30$

Answer

-30

The correct answer is

-30

because the distance between two points on a number line is found by taking the absolute value of the difference between their coordinates. Setting up the relation

|b - (-18)| = 12

simplifies to

|b + 18| = 12

. Solving this absolute value equation yields two possible locations:

b + 18 = 12

(which gives

b = -6

) and

b + 18 = -12

(which gives

b = -30

). Among the choices,

-30

is the only possible coordinate listed.

Step-by-Step Solution

1

Set up the absolute value equation representing the distance between point A and point B on the number line.

Let

b

be the coordinate of point

B

. The distance between

A

and

B

is given by

|-18 - b| = 12

.

On a standard number line, the distance between any two coordinates

x

and

y

is given by the absolute value of their difference,

|x - y|

.

2

Solve the absolute value equation for the two possible cases.

Case 1:

-18 - b = 12 \Rightarrow -b = 30 \Rightarrow b = -30

. Case 2:

-18 - b = -12 \Rightarrow -b = 6 \Rightarrow b = -6

.

An equation of the form

|x| = d

(where

d \geq 0

) splits into two distinct equations:

x = d

and

x = -d

.

3

Compare the possible solutions with the given options.

The value

-30

is one of the possible coordinates and is listed in the options.

To identify which of the two mathematically correct coordinates (-30 or -6) is provided as an answer choice.

Key Concept

The distance between two points on a number line with coordinates

x

and

y

is represented by the absolute value of their difference,

|x - y|

.

Estimated Time:45s

Question 86Question →

To complete a project, Alice works in shifts that last $\frac{3}{4}$ of an hour, and Bob works in shifts that last $\frac{5}{6}$ of an hour. If they both begin their first shift at the exact same time, after how many hours will they next begin a shift at the same time?

\frac{5}{8}

\frac{4}{5}

\frac{5}{4}

\frac{19}{12}

\frac{15}{2}

Show answer & explanation

Answer: $\frac{15}{2}$

Answer

The two workers will next begin a shift at the same time after

\frac{15}{2}

hours.

To find when both Alice and Bob will next start their shifts at the same time, we need to find the Least Common Multiple (LCM) of their shift durations,

\frac{3}{4}

and

\frac{5}{6}

hours. One way to do this is to convert the durations into minutes:

\frac{3}{4} \times 60 = 45

minutes, and

\frac{5}{6} \times 60 = 50

minutes. The prime factorizations are

45 = 3^2 \times 5

and

50 = 2 \times 5^2

. The LCM of

45

and

50

is

2 \times 3^2 \times 5^2 = 450

minutes. Converting this back to hours gives

\frac{450}{60} = \frac{15}{2}

hours. Alternatively, using the formula for the LCM of fractions:

\text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{GCD}(b, d)}

, we get

\frac{\text{LCM}(3, 5)}{\text{GCD}(4, 6)} = \frac{15}{2}

hours.

Step-by-Step Solution

1

Identify the mathematical concept needed to solve the simultaneous starting times.

The next simultaneous starting time is the Least Common Multiple (LCM) of the two shift durations,

\frac{3}{4}

and

\frac{5}{6}

hours.

Because both shifts start at multiples of their respective durations, the first time they start together again must be a common multiple of both intervals.

2

Calculate the LCM of the fractions

\frac{3}{4}

and

\frac{5}{6}

.

Using the fraction LCM formula,

\text{LCM}(\frac{a}{b}, \frac{c}{d}) = \frac{\text{LCM}(a, c)}{\text{GCD}(b, d)}

, we find

\text{LCM}(3, 5) = 15

and

\text{GCD}(4, 6) = 2

, yielding

\frac{15}{2}

hours.

This formula correctly scales the numerator multiples while accounting for the common division in the denominators.

Key Concept

Finding the Least Common Multiple (LCM) of fractional values or using unit conversion to find the LCM of integers.

Alternative Method

Convert the fractional hours into minutes first. Alice's shifts are

\frac{3}{4}

of an hour, which is

45

minutes. Bob's shifts are

\frac{5}{6}

of an hour, which is

50

minutes. Find the LCM of

45

and

50

by listing their multiples or using their prime factorizations:

45 = 3^2 \times 5

and

50 = 2 \times 5^2

, so their LCM is

2 \times 3^2 \times 5^2 = 450

minutes. Finally, convert

450

minutes back to hours by dividing by

60

, which simplifies to

\frac{450}{60} = \frac{15}{2}

hours.

Estimated Time:2m 0s

Question 87Question →

For all non-zero real numbers $x$ and $y$ , which of the following is equivalent to the expression below?

x - y[x - (y - x)] - y^2

2x^2 - 3xy

x - 2xy - 2y^2

x - 2xy

x

x - 2y^2

Show answer & explanation

Answer: $x - 2xy$

Answer

The correct answer is

x - 2xy

.

The correct answer is

x - 2xy

. We begin by simplifying the innermost grouping symbols first. The expression inside the parentheses is

x - (y - x) = x - y + x = 2x - y

. Substituting this back into the expression gives

x - y[2x - y] - y^2

. Next, we distribute the factor of

-y

to the terms inside the brackets, yielding

-y(2x) - y(-y) = -2xy + y^2

. The expression then becomes

x - 2xy + y^2 - y^2

. Finally, combining the like terms

+y^2

and

-y^2

cancels them out, resulting in the simplified expression

x - 2xy

.

Step-by-Step Solution

1

Simplify the expression inside the innermost parentheses:

x - (y - x)

.

2x - y

Distribute the negative sign to the terms inside the parentheses:

-(y - x) = -y + x

. Combining like terms gives

x + x - y = 2x - y

.

2

Substitute the simplified expression back into the main expression:

x - y[2x - y] - y^2

.

x - y[2x - y] - y^2

This updates the expression to show the next level of grouping symbols to resolve.

3

Distribute the

-y

to the terms inside the brackets:

-y[2x - y]

.

-2xy + y^2

Apply the distributive property:

-y \times 2x = -2xy

, and

-y \times (-y) = y^2

.

4

Combine the distributed terms back into the expression and simplify:

x - 2xy + y^2 - y^2

.

x - 2xy

The

+y^2

and

-y^2

terms cancel each other out, leaving the simplified expression

x - 2xy

.

Key Concept

Applying the order of operations (PEMDAS) to evaluate expressions with nested grouping symbols, and correctly distributing positive and negative signs.

Estimated Time:1m 30s

Question 88Question →

Arrange the following mathematical expressions in order of their simplified values from least to greatest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order of the expressions from least to greatest simplified value is:

-3 - (4 - 2)

, then

-3^2 + 4 \times 2

, then

(-3)^2 - 4 \times 2

, and finally

-(3 - 4 \times 2)

.

Evaluating each expression using the correct order of operations yields

-5

for

-3 - (4 - 2)

,

-1

for

-3^2 + 4 \times 2

,

1

for

(-3)^2 - 4 \times 2

, and

5

for

-(3 - 4 \times 2)

. Ordering these results from least to greatest yields the sequence:

-3 - (4 - 2)

, followed by

-3^2 + 4 \times 2

, then

(-3)^2 - 4 \times 2

, and finally

-(3 - 4 \times 2)

.

Step-by-Step Solution

1

Simplify the expression

-3 - (4 - 2)

.

-3 - (4 - 2) = -3 - 2 = -5

Perform the operation inside the parentheses first, then subtract the result from the leading term.

2

Simplify the expression

-3^2 + 4 \times 2

.

-3^2 + 4 \times 2 = -9 + 8 = -1

Evaluate the exponent first, noting that the negative sign is outside the base (

-3^2 = -9

), then multiply, and finally add the terms.

3

Simplify the expression

(-3)^2 - 4 \times 2

.

(-3)^2 - 4 \times 2 = 9 - 8 = 1

Evaluate the term inside parentheses raised to the power first (

(-3)^2 = 9

), then perform multiplication, and subtract.

4

Simplify the expression

-(3 - 4 \times 2)

.

-(3 - 4 \times 2) = -(3 - 8) = -(-5) = 5

Inside the parentheses, perform multiplication before subtraction, then apply the negative sign outside the parentheses.

5

Compare the simplified values of the four expressions.

-5 < -1 < 1 < 5

Arrange the resulting values from least to greatest to determine the correct sequence.

Key Concept

Applying the correct order of operations (PEMDAS/GEMS) to evaluate numerical expressions, specifically distinguishing between

-a^2

and

(-a)^2

.

Question 89Question →

Arrange the following mathematical values in order from least to greatest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest is

4 \times 10^{-2}

,

2^{-3}

,

\sqrt{0.09}

, and

\frac{3^2}{2 \times 10^1}

.

By converting each value to a decimal, we find

4 \times 10^{-2} = 0.04

,

2^{-3} = 0.125

,

\sqrt{0.09} = 0.3

, and

\frac{3^2}{2 \times 10^1} = 0.45

. Ordering these decimals from smallest to largest yields the correct sequence.

Step-by-Step Solution

1

Convert the scientific notation expression

4 \times 10^{-2}

to its decimal form.

4 \times 10^{-2} = 0.04

Multiplying by

10^{-2}

is equivalent to moving the decimal point of the coefficient

4

two places to the left.

2

Convert the negative exponent expression

2^{-3}

to a fraction and then to a decimal.

2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125

A negative exponent indicates the reciprocal of the base raised to the positive power.

3

Evaluate the square root of the decimal

\sqrt{0.09}

.

\sqrt{0.09} = 0.3

The square root of a decimal less than

1

results in a larger decimal value because

0.3 \times 0.3 = 0.09

.

4

Evaluate the exponential fraction expression

\frac{3^2}{2 \times 10^1}

.

\frac{3^2}{2 \times 10^1} = \frac{9}{20} = 0.45

Simplify the numerator to

9

and the denominator to

20

, then divide to find the decimal equivalent.

5

Compare the evaluated decimal values to determine the correct order from least to greatest.

0.04 < 0.125 < 0.3 < 0.45

, which correspond to

4 \times 10^{-2} < 2^{-3} < \sqrt{0.09} < \frac{3^2}{2 \times 10^1}

.

Comparing decimal place values shows that

0.04

is the smallest and

0.45

is the largest.

Key Concept

Converting and comparing exponential, radical, and scientific notation expressions to a common decimal format.

Question 90Question →

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

20

24

28

52

72

Show answer & explanation

Answer: 28

Answer

28

The correct answer is

28

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

5x

,

6x

, and

3x

, respectively. Adding

8

ounces of blue and

8

ounces of white paint gives new amounts of

5x + 8

for blue and

3x + 8

for white. The amount of yellow paint remains

6x

. The new ratio of yellow to white paint is

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

. Cross-multiplying yields

30x = 18x + 48

, which simplifies to

12x = 48

, giving

x = 4

. Substituting

x = 4

into the expression for the new amount of blue paint gives

5(4) + 8 = 28

ounces.

Step-by-Step Solution

1

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

x

.

Blue paint =

5x

, yellow paint =

6x

, and white paint =

3x

.

Since the ratio of blue to yellow to white paint is

5:6:3

, the actual amounts are multiples of these ratio parts.

2

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

New blue paint =

5x + 8

, new yellow paint =

6x

, and new white paint =

3x + 8

.

The artist adds

8\text{ ounces}

of blue and

8\text{ ounces}

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

3

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

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

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

6:5

.

4

Solve the equation for the multiplier

x

.

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

.

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

5

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

x = 4

.

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

.

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

5x + 8

.

Key Concept

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

Question 91Question →

An astronomical unit (AU) is a unit of length equal to approximately $1.50 \times 10^8$ kilometers. A research probe travels through space at a constant speed of $2.50 \times 10^4$ meters per second. Which of the following is closest to the number of hours it will take the probe to travel a distance of $5.00$ AU?

8.33 \times 10^0

8.33 \times 10^3

8.33 \times 10^4

3.00 \times 10^7

1.08 \times 10^{11}

Show answer & explanation

Answer: $8.33 \times 10^3$

Answer

The correct answer is

8.33 \times 10^3

hours.

To find the travel time in hours, the distance of

5.00

AU is first converted to meters:

5.00 \times 1.50 \times 10^8\text{ km} \times 10^3\text{ m/km} = 7.50 \times 10^{11}\text{ m}

. Dividing this distance by the speed of

2.50 \times 10^4\text{ m/s}

yields a time of

3.00 \times 10^7\text{ seconds}

. Dividing this time by the

3,600\text{ seconds}

in an hour results in

\frac{3.00 \times 10^7}{3,600} \approx 8,333.33\text{ hours}

, which is written in standard scientific notation as

8.33 \times 10^3\text{ hours}

.

Step-by-Step Solution

1

Convert the total distance from astronomical units (AU) to meters.

Distance =

7.50 \times 10^{11}

meters

Since

1\text{ AU} \approx 1.50 \times 10^8\text{ km}

and

1\text{ km} = 10^3\text{ m}

, then

1\text{ AU} \approx 1.50 \times 10^{11}\text{ m}

. Therefore, a distance of

5.00\text{ AU} = 5.00 \times (1.50 \times 10^{11}\text{ m}) = 7.50 \times 10^{11}\text{ m}

.

2

Calculate the travel time in seconds.

Time =

3.00 \times 10^7

seconds

Dividing the total distance in meters by the speed in meters per second gives:

\frac{7.50 \times 10^{11}\text{ m}}{2.50 \times 10^4\text{ m/s}} = 3.00 \times 10^7\text{ seconds}

.

3

Convert the travel time from seconds to hours.

Time

\approx 8.33 \times 10^3

hours

Since

1\text{ hour} = 3,600\text{ seconds}

(which is

3.60 \times 10^3\text{ seconds}

), we divide the time in seconds by

3,600

:

\frac{3.00 \times 10^7}{3.60 \times 10^3} \approx 0.833 \times 10^4 = 8.33 \times 10^3\text{ hours}

.

Key Concept

Applying division and unit conversion with numbers written in scientific notation.

Estimated Time:2m 0s

Question 92Question →

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

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

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

Drag items to arrange them in the correct order

Show answer & explanation

Answer

Jamie (

35\%

), Kayla (

\frac{3}{8}

), Leo (

0.4

), and Mia (

\frac{1}{2}

)

Converting all values to decimals yields

35\% = 0.35

,

\frac{3}{8} = 0.375

,

0.4 = 0.40

, and

\frac{1}{2} = 0.50

. Comparing these decimals gives

0.35 < 0.375 < 0.40 < 0.50

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

Step-by-Step Solution

1

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

Jamie:

35\% = 0.35

Kayla:

\frac{3}{8} = 0.375

Leo:

0.4 = 0.40

Mia:

\frac{1}{2} = 0.50

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

2

Compare the decimal values from least to greatest.

0.35 < 0.375 < 0.40 < 0.50

This establishes the relative sizes of the portions.

3

Match each decimal back to the student and order them.

Jamie (

35\%

), Kayla (

\frac{3}{8}

), Leo (

0.4

), Mia (

\frac{1}{2}

)

This provides the final ordered sequence.

Key Concept

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

Question 93Question →

What is the value of the expression $2^3 \cdot 2^2 - \sqrt{81}$ ?

-40

1

23

55

-49

Show answer & explanation

Answer: 23

Answer

23

To evaluate the expression, follow the order of operations. First, simplify the exponential multiplication:

2^3 \cdot 2^2 = 2^{3+2} = 2^5 = 32

. Next, evaluate the square root:

\sqrt{81} = 9

. Finally, perform the subtraction:

32 - 9 = 23

. Thus, the correct value is 23.

Step-by-Step Solution

1

Simplify the exponential term

2^3 \cdot 2^2

2^5 = 32

When multiplying exponential terms with the same base, add their exponents:

3 + 2 = 5

.

2

Evaluate the square root term

\sqrt{81}

9

The square root of 81 is the positive number that, when multiplied by itself, equals 81, which is 9.

3

Perform the subtraction:

32 - 9

23

Subtract 9 from 32 to find the final value of the expression.

Key Concept

Simplifying numerical expressions using laws of exponents, square roots, and order of operations

Question 94Question →

If $a = 3$ and $b = 2$ , what is the value of the expression $\sqrt{a^4 - b^5}$ ?

Show answer & explanation

Answer: 7

Answer

The value of the expression is 7.

Evaluating the exponents yields

3^4 = 81

and

2^5 = 32

. Substituting these values under the radical gives

\sqrt{81 - 32} = \sqrt{49}

. Evaluating the square root of 49 gives the correct value of 7.

Step-by-Step Solution

1

Evaluate the terms with exponents using the given values

a = 3

and

b = 2

.

a^4 = 3^4 = 81

and

b^5 = 2^5 = 32

Exponent operations must be evaluated before performing subtraction or taking the root.

2

Substitute the evaluated terms back into the radical expression and subtract.

\sqrt{81 - 32} = \sqrt{49}

Simplifying the expression under the radical is necessary before taking the square root.

3

Find the square root of the simplified value.

\sqrt{49} = 7

Evaluating the square root of 49 completes the simplification of the expression.

Key Concept

Evaluating expressions involving exponents, subtraction, and square roots

Question 95Question →

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

Show answer & explanation

Answer: 15

Answer

The correct answer is 15.

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

Step-by-Step Solution

1

Convert the fraction of students playing sports to a decimal

0.60

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

\frac{3}{5} = 0.60

.

2

Multiply the sports-participating portion by the track team percentage

0.15

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

25\% = 0.25

) and multiply:

0.60 \times 0.25 = 0.15

.

3

Convert the final decimal back to a percentage

15\%

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

0.15 \times 100 = 15\%

.

Key Concept

Fractions, Decimals, and Percentages

Estimated Time:45s

Question 96Question →

The prime factorization of a positive integer $N$ is of the form $2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are non-negative integers. The number $N$ has exactly $12$ positive factors. If $N$ is a multiple of $4$ but is not divisible by $3$ , what is the smallest possible value of $N$ ?

Show answer & explanation

Answer: 160

Answer

The smallest possible value of

N

is 160.

The correct value is 160 because it satisfies all the conditions: its prime factorization has only 2 and 5 (no 3, so not divisible by 3), the exponent of 2 is 5 (which is greater than or equal to 2, so it is a multiple of 4), the number of factors is

(5+1)(1+1) = 12

, and it is the smallest such integer.

Step-by-Step Solution

1

Set up the factor counting formula using the prime factorization.

The number of positive factors of

N = 2^a \times 3^b \times 5^c

is

(a+1)(b+1)(c+1) = 12

.

The number of positive factors of any integer is found by adding one to each exponent in its prime factorization and multiplying the results.

2

Apply the given divisibility conditions to simplify the exponents.

Since

N

is not divisible by

3

,

b = 0

. Since

N

is a multiple of

4

,

a \ge 2

. The equation simplifies to

(a+1)(c+1) = 12

with

a+1 \ge 3

.

A number is not divisible by a prime if its exponent in the prime factorization is 0. A number is a multiple of 4 if the exponent of 2 in its prime factorization is at least 2.

3

List all possible pairs of

(a+1, c+1)

that multiply to 12 where

a+1 \ge 3

, and compute the corresponding values of

N

.

Case 1:

a+1 = 3 \implies a = 2, c = 3 \implies N = 2^2 \times 5^3 = 500

. Case 2:

a+1 = 4 \implies a = 3, c = 2 \implies N = 2^3 \times 5^2 = 200

. Case 3:

a+1 = 6 \implies a = 5, c = 1 \implies N = 2^5 \times 5^1 = 160

. Case 4:

a+1 = 12 \implies a = 11, c = 0 \implies N = 2^{11} \times 5^0 = 2048

.

Testing all possible divisor pairs of 12 that satisfy the constraint on

a

allows us to find all possible candidate values for

N

.

4

Determine the smallest value of

N

from the candidate values.

The smallest candidate is 160.

Comparing the values 500, 200, 160, and 2048 shows that 160 is the minimum.

Key Concept

Factors, Multiples, and Prime Factorization

Question 97Question →

A student simplifies the expression $7 + 3(x + 4) - 7$ in three steps as shown below:

Step 1: $7 - 7 + 3(x + 4)$
Step 2: $0 + 3(x + 4)$
Step 3: $3x + 12$

Which of the following lists the mathematical properties of real numbers that justify Step 1 and Step 3, respectively?

Commutative property of addition; distributive property

Associative property of addition; distributive property

Commutative property of addition; associative property of multiplication

Associative property of addition; commutative property of multiplication

Distributive property; commutative property of addition

Show answer & explanation

Answer: Commutative property of addition; distributive property

Answer

Commutative property of addition; distributive property

In Step 1, the student reorders the terms from

7 + 3(x+4) - 7

to

7 - 7 + 3(x+4)

. Since changing the order of elements being added is governed by the commutative property of addition, this property justifies Step 1. In Step 3, the expression

3(x + 4)

is expanded to

3x + 12

by multiplying

3

by both

x

and

4

. This operation is justified by the distributive property. Combining these, the correct pair of properties is the commutative property of addition and the distributive property.

Step-by-Step Solution

1

Analyze the transition from the initial expression to Step 1:

7 + 3(x + 4) - 7 \rightarrow 7 - 7 + 3(x + 4)

The terms are rearranged such that the constant term

-7

is moved next to the initial term

7

.

The commutative property of addition states that the order in which real numbers are added does not affect the sum (

a + b = b + a

).

2

Analyze the transition from Step 2 to Step 3:

3(x + 4) \rightarrow 3x + 12

The coefficient

3

is multiplied by both terms inside the parentheses (

x

and

4

).

The distributive property states that multiplying a sum by a number is equivalent to multiplying each term in the sum individually (

a(b + c) = ab + ac

).

Key Concept

Identifying commutative and distributive properties of real numbers.

Estimated Time:1m 0s

Question 98Question →

Three distinct integers, $a$ , $b$ , and $c$ , lie on a standard number line such that $a < b < c$ . The distance between $a$ and $b$ is $3$ times the distance between $b$ and $c$ . If $|a| = 15$ , $|c| = 7$ , and $b < 0$ , what is the value of $b$ ?

Show answer & explanation

Answer: -9

Answer

The value of

b

is

-9

.

The correct answer is

-9

. By interpreting

|a| = 15

and

|c| = 7

with the constraint

a < b < c

, we find

a = -15

. Testing the possible values for

c

, when

c = -7

, we set up the distance equation

b - (-15) = 3(-7 - b)

, which simplifies to

b + 15 = -21 - 3b

, giving

4b = -36

and

b = -9

. This satisfies all constraints, including

b

being a negative integer.

Step-by-Step Solution

1

Find the possible coordinates for

a

and

c

based on their absolute values.

a \in \{-15, 15\}

and

c \in \{-7, 7\}

.

The absolute value of a number represents its distance from zero, so

|x| = d \implies x = \pm d

.

2

Use the ordering condition

a < b < c

to eliminate invalid combinations.

a = -15

and

c \in \{-7, 7\}

.

Since

a

must be less than

c

,

a

cannot be

15

because both possible values of

c

(

-7

and

7

) are less than

15

.

3

Set up an equation representing the distance relationship on the number line.

b + 15 = 3(c - b) \implies 4b = 3c - 15

.

For points on a number line ordered

a < b < c

, the distance between

a

and

b

is

b - a

, and the distance between

b

and

c

is

c - b

.

4

Substitute each possible value of

c

and solve for

b

to find the one that yields a negative integer.

For

c = -7

,

b = -9

.

If

c = 7

,

b = 1.5

, which is not an integer. If

c = -7

,

b = -9

, which is a negative integer, satisfying all given conditions.

Key Concept

Using absolute values and relative order to determine integer positions and distances on a number line.

Question 99Question →

Evaluate the mathematical expressions below and arrange them in order from least to greatest value.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest is

\sqrt{25}

,

2^3

, and

1.2 \times 10^1

.

Evaluating each expression gives

\sqrt{25} = 5

,

2^3 = 8

, and

1.2 \times 10^1 = 12

. Comparing these values gives

5 < 8 < 12

, which yields the sequence:

\sqrt{25}

,

2^3

, and

1.2 \times 10^1

.

Step-by-Step Solution

1

Evaluate the square root expression.

\sqrt{25} = 5

Identify the non-negative number that, when multiplied by itself, equals

25

.

2

Evaluate the exponential expression.

2^3 = 8

Multiply the base,

2

, by itself three times (

2 \times 2 \times 2

).

3

Evaluate the expression in scientific notation.

1.2 \times 10^1 = 12

Multiply

1.2

by

10

to shift the decimal point one spot to the right.

4

Order the resulting values from least to greatest.

5 < 8 < 12

, which corresponds to the expressions:

\sqrt{25}

,

2^3

, and

1.2 \times 10^1

.

Compare the numbers

5

,

8

, and

12

to establish their relative sizes.

Key Concept

Evaluating basic exponents, square roots, and scientific notation, then ordering their values.

Question 100Question →

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

16

39

44

49

77

Show answer & explanation

Answer: 77

Answer

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

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

Step-by-Step Solution

1

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

28 \div 4 = 7

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

2

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

7 \times 7 = 49

dogs

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

3

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

28 + 49 = 77

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

Key Concept

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

Alternative Method

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

Estimated Time:1m 0s

Pre-Algebra

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