Pre-Algebra Practice Questions — ACT — Page 2

Question 21Question →

For all real numbers $a$ , $b$ , and $c$ , let the operation $\oplus$ be defined by $a \oplus b = ab + 1$ . Which of the following statements must be true?

I. $a \oplus b = b \oplus a$
II. $a \oplus (b \oplus c) = (a \oplus b) \oplus c$
III. $a \oplus (b + c) = (a \oplus b) + (a \oplus c)$

I only

II only

I and II only

I and III only

I, II, and III

Show answer & explanation

Answer: I only

Answer

The correct option is the one stating that only Statement I must be true.

The correct option is the one stating that only Statement I must be true. This is because multiplication of real numbers is commutative, making the custom operation commutative. Statement II is false because the operation is not associative, and Statement III is false because the operation does not distribute over addition.

Step-by-Step Solution

1

Evaluate Statement I (

a \oplus b = b \oplus a

) by substituting the definition of the operation.

a \oplus b = ab + 1

and

b \oplus a = ba + 1

. Since multiplication of real numbers is commutative,

ab = ba

, which means

ab + 1 = ba + 1

. Thus, Statement I is true for all real numbers.

To test the commutative property of the defined operation.

2

Evaluate Statement II (

a \oplus (b \oplus c) = (a \oplus b) \oplus c

) by applying the operation's definition to both sides.

Left side:

a \oplus (b \oplus c) = a \oplus (bc + 1) = a(bc + 1) + 1 = abc + a + 1

. Right side:

(a \oplus b) \oplus c = (ab + 1) \oplus c = (ab + 1)c + 1 = abc + c + 1

. Since

abc + a + 1

is not equal to

abc + c + 1

for all real numbers (for example, if

a = 1

,

b = 1

, and

c = 2

, then

4 \neq 5

), Statement II is not always true.

To test the associative property of the defined operation.

3

Evaluate Statement III (

a \oplus (b + c) = (a \oplus b) + (a \oplus c)

) using the operation's definition.

Left side:

a \oplus (b + c) = a(b + c) + 1 = ab + ac + 1

. Right side:

(a \oplus b) + (a \oplus c) = (ab + 1) + (ac + 1) = ab + ac + 2

. Since

ab + ac + 1 \neq ab + ac + 2

for all real numbers, Statement III is false.

To test whether the operation distributes over addition.

Key Concept

Identifying properties of operations (commutativity, associativity, and distributivity) on real numbers.

Alternative Method

To quickly disprove Statements II and III, you can substitute simple counterexamples. For Statement II, let

a=1, b=1, c=2

:

1 \oplus (1 \oplus 2) = 1 \oplus 3 = 4

, while

(1 \oplus 1) \oplus 2 = 2 \oplus 2 = 5

. Since

4 \neq 5

, Statement II is false. For Statement III, let

a=1, b=1, c=1

:

1 \oplus (1+1) = 1 \oplus 2 = 3

, while

(1 \oplus 1) + (1 \oplus 1) = 2 + 2 = 4

. Since

3 \neq 4

, Statement III is false.

Estimated Time:2m 0s

Question 22Question →

Evaluate the four 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

Expression X, Expression W, Expression Z, Expression Y

Evaluating all four expressions using the proper order of operations (PEMDAS) yields the values: Expression X is

-16

, Expression W is

-10

, Expression Z is

6

, and Expression Y is

7

. Ordering these values from least to greatest results in the sequence Expression X, Expression W, Expression Z, Expression Y.

Step-by-Step Solution

1

Evaluate Expression X:

-(-3)^2 - | -6 \times 2 + 5 |

-16

First, evaluate the exponent:

(-3)^2 = 9

, which makes the first term

-9

. Next, calculate the expression inside the absolute value bars:

-6 \times 2 + 5 = -12 + 5 = -7

. The absolute value of

-7

is

7

. Subtracting this value gives

-9 - 7 = -16

.

2

Evaluate Expression W:

-2^4 - 3(5 - 7)^2 \div (-2)

-10

First, evaluate the exponent:

-2^4 = -(2^4) = -16

. Then, evaluate inside the parentheses:

5 - 7 = -2

, and square it:

(-2)^2 = 4

. Next, perform multiplication and division from left to right:

-3(4) \div (-2) = -12 \div (-2) = 6

. Adding this to

-16

gives

-16 + 6 = -10

.

3

Evaluate Expression Z:

\frac{-18 \div (-3) \times (-2)}{-4 + 2^3 \div 4}

6

In the numerator, perform multiplication and division from left to right:

-18 \div (-3) = 6

, and then

6 \times (-2) = -12

. In the denominator, evaluate the exponent:

2^3 = 8

, divide:

8 \div 4 = 2

, and add:

-4 + 2 = -2

. Finally, divide the numerator by the denominator:

\frac{-12}{-2} = 6

.

4

Evaluate Expression Y:

4 - 3 \times [8 - (2 - 5)^2]

7

Evaluate the innermost parentheses first:

2 - 5 = -3

, and square it:

(-3)^2 = 9

. Simplify inside the brackets:

8 - 9 = -1

. Multiply:

-3 \times (-1) = 3

. Add to the remaining term:

4 + 3 = 7

.

5

Compare the evaluated values to find the correct ordering.

Expression X (

-16

) < Expression W (

-10

) < Expression Z (

6

) < Expression Y (

7

)

Comparing the values

-16

,

-10

,

6

, and

7

from least to greatest establishes the correct sequence.

Key Concept

Order of operations dictates evaluating parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). Negation outside exponents and absolute values must be treated with care.

Question 23Question →

What is the value of the expression $15 - 3 \times 2^3 + 10$ ?

-191

-19

7

1

106

Show answer & explanation

Answer: $1$

Answer

The value of the expression is

1

.

Evaluating the exponent first gives

2^3 = 8

. Next, multiplying 3 by 8 yields 24, resulting in the expression

15 - 24 + 10

. Finally, performing addition and subtraction from left to right gives

15 - 24 = -9

, and

-9 + 10 = 1

. Therefore, the value of the expression is

1

.

Step-by-Step Solution

1

Evaluate the exponential expression.

2^3 = 8

According to the order of operations, exponents must be evaluated before multiplication, division, addition, and subtraction.

2

Perform the multiplication.

3 \times 8 = 24

Multiplication has higher priority than addition and subtraction.

3

Perform addition and subtraction from left to right.

15 - 24 + 10 = -9 + 10 = 1

Addition and subtraction have equal priority and must be performed in the order they appear from left to right.

Key Concept

Order of Operations (PEMDAS)

Question 24Question →

What is the greatest common factor of $36$ and $54$ ?

Show answer & explanation

Answer: 18

Answer

The greatest common factor of

36

and

54

is

18

.

The greatest common factor of

36

and

54

is

18

because it is the largest integer that divides both numbers evenly (

36 \div 18 = 2

and

54 \div 18 = 3

).

Step-by-Step Solution

1

Find the prime factorization of

36

and

54

.

36 = 2^2 \times 3^2

and

54 = 2 \times 3^3

Prime factorization decomposes each number into its basic prime building blocks.

2

Identify the common prime factors with their lowest exponents.

The common prime factors are

2

with an exponent of

1

(

2^1

) and

3

with an exponent of

2

(

3^2

).

The greatest common factor consists of the product of the lowest powers of the shared prime factors.

3

Multiply the common factors raised to their lowest exponents.

2^1 \times 3^2 = 2 \times 9 = 18

Calculating this product gives the greatest common factor.

Key Concept

The greatest common factor (GCF) of two integers is the largest integer that divides both numbers without leaving a remainder. It is calculated by taking the product of the lowest powers of all common prime factors.

Alternative Method

List all factors of both numbers: Factors of

36

are

1, 2, 3, 4, 6, 9, 12, 18, 36

. Factors of

54

are

1, 2, 3, 6, 9, 18, 27, 54

. The largest number present in both lists is

18

.

Estimated Time:45s

Question 25Question →

A red light flashes every 8 seconds, and a blue light flashes every 12 seconds. If they both flash at the same instant, how many seconds will pass before they next flash at the same instant?

4

20

24

48

96

Show answer & explanation

Answer: 24

Answer

24 seconds

The correct answer is 24 seconds. To find when both lights will flash at the same time again, we need to find the least common multiple (LCM) of their individual flashing intervals, 8 seconds and 12 seconds. Listing the multiples of each number:
Multiples of 8: 8, 16, 24, 32, 40, 48...
Multiples of 12: 12, 24, 36, 48...
The smallest number that appears in both lists is 24. Thus, the lights will flash together next after 24 seconds.

Step-by-Step Solution

1

Identify the mathematical concept needed to solve the problem.

The problem requires finding the least common multiple (LCM) of the two flashing intervals, 8 and 12.

The lights will next flash together at the smallest positive integer that is a multiple of both 8 and 12.

2

Find the prime factorization of each number.

8 = 2^3

and

12 = 2^2 \times 3

.

Prime factorization allows us to construct the LCM by taking the highest power of each prime factor present in either number.

3

Calculate the LCM from the prime factors.

\text{LCM}(8, 12) = 2^3 \times 3 = 8 \times 3 = 24

.

We take the highest power of 2, which is

2^3

, and the highest power of 3, which is

3^1

.

Key Concept

Least Common Multiple

Estimated Time:45s

Question 26Question →

For all non-zero real numbers $x$ and $y$ , let the operation $\star$ be defined by $x \star y = \frac{x}{y} - \frac{y}{x}$ . Which of the following equations must be true for all non-zero real numbers $a$ , $b$ , and $c$ ?

a \star b = b \star a

a \star (b \star c) = (a \star b) \star c

a \star (b + c) = (a \star b) + (a \star c)

a \star b = -(b \star a)

(a + b) \star c = (a \star c) + (b \star c)

Show answer & explanation

Answer: $a \star b = -(b \star a)$

Answer

The equation stating that the operation on two numbers is equal to the negative of the operation on those numbers in reverse order, which is

a \star b = -(b \star a)

Applying the definition of the custom operation, the expression for

b \star a

is

\frac{b}{a} - \frac{a}{b}

. Factoring out

-1

from this expression yields

-\left(\frac{a}{b} - \frac{b}{a}\right)

, which is equivalent to

-(a \star b)

. Multiplying both sides by

-1

results in the equation

a \star b = -(b \star a)

, which must be true for all non-zero real numbers

a

and

b

.

Step-by-Step Solution

1

Evaluate the expression for

b \star a

using the custom operation definition.

b \star a = \frac{b}{a} - \frac{a}{b}

To analyze the result of reversing the order of operands.

2

Factor out a negative sign from the right-hand side of the evaluated expression.

b \star a = -\left(-\frac{b}{a} + \frac{a}{b}\right) = -\left(\frac{a}{b} - \frac{b}{a}\right)

To align the expression's terms with the definition of

a \star b

.

3

Substitute the definition of

a \star b

back into the expression.

b \star a = -(a \star b)

To establish the relationship between the two permutations of the operation.

4

Negate both sides of the equation to find the equivalent representation.

a \star b = -(b \star a)

To match the final equation with the options.

Key Concept

Algebraic properties of custom operations and order of operations

Alternative Method

Instead of algebraic proof, substitute small, distinct non-zero integers (e.g.,

a = 2

,

b = 1

,

c = 3

) to evaluate the expressions in each option, enabling the elimination of equations that do not hold.

Estimated Time:1m 30s

Question 27Question →

For all non-zero real numbers $x$ , $y$ , and $z$ , which of the following mathematical equations must be true?

x - (y + z) = x - y + z

x(y - z) = xy - z

\frac{z}{x + y} = \frac{z}{x} + \frac{z}{y}

\frac{x + y}{z} = \frac{x}{z} + \frac{y}{z}

x \div (y \div z) = (x \div y) \div z

Show answer & explanation

Answer: $\frac{x + y}{z} = \frac{x}{z} + \frac{y}{z}$

Answer

The equation

\frac{x + y}{z} = \frac{x}{z} + \frac{y}{z}

must be true.

The equation stating that a sum in the numerator divided by a single term in the denominator can be split into two separate fractions is correct because division distributes over addition from the right side. This can be verified by rewriting the division as multiplication by the reciprocal:

\frac{x + y}{z} = (x + y) \cdot \frac{1}{z} = x \cdot \frac{1}{z} + y \cdot \frac{1}{z} = \frac{x}{z} + \frac{y}{z}

.

Step-by-Step Solution

1

Analyze the properties of division and addition in the equation

\frac{x + y}{z} = \frac{x}{z} + \frac{y}{z}

.

The division of a sum by a number can be rewritten as the multiplication of the sum by the reciprocal of that number:

(x + y) \cdot \frac{1}{z}

.

To apply the distributive property of multiplication over addition.

2

Distribute the term

\frac{1}{z}

to both

x

and

y

.

(x + y) \cdot \frac{1}{z} = x \cdot \frac{1}{z} + y \cdot \frac{1}{z} = \frac{x}{z} + \frac{y}{z}

.

The distributive property states that

a(b + c) = ab + ac

for any real numbers

a

,

b

, and

c

.

3

Confirm that this identity holds for all non-zero real numbers

x

,

y

, and

z

.

Since

z \neq 0

, the denominators are defined, and the relation

\frac{x + y}{z} = \frac{x}{z} + \frac{y}{z}

is always true.

To establish that the statement must be true under the given conditions.

Key Concept

Distributive Property of Real Numbers

Alternative Method

We can plug in simple numbers (e.g.,

x = 2

,

y = 3

,

z = 4

) to test each option and eliminate those that do not produce a true statement.

Estimated Time:1m 0s

Question 28Question →

Let the values of three mathematical expressions be represented by $A$ , $B$ , and $C$ , where:

A = -3^2 - ( -2 - 4 )^2 \div 3 \times 2

B = |-5 - (-3)| - 2^4 \div (-2)^3 \times (4 - 7)

C = \frac{4 - 3 \times 2^2}{2} - ( -1 )^5 \times 3

Arrange the expressions $A$ , $B$ , and $C$ in order from least to greatest numerical value.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

Expression A, Expression B, Expression C

Evaluating each expression using correct PEMDAS order of operations yields

A = -33

,

B = -4

, and

C = -1

. Comparing these values gives the order from least to greatest as Expression A, followed by Expression B, followed by Expression C.

Step-by-Step Solution

1

Evaluate Expression A by applying order of operations.

A = -33

Simplify the grouping

(-2 - 4) = -6

. Then apply exponents:

-3^2 = -9

and

(-6)^2 = 36

. Perform division and multiplication from left to right:

36 \div 3 \times 2 = 12 \times 2 = 24

. Subtract to find

-9 - 24 = -33

.

2

Evaluate Expression B by applying order of operations.

B = -4

Simplify grouping terms:

|-5 - (-3)| = 2

and

(4 - 7) = -3

. Evaluate exponents:

2^4 = 16

and

(-2)^3 = -8

. Perform division and multiplication from left to right:

16 \div (-8) \times (-3) = -2 \times (-3) = 6

. Subtract:

2 - 6 = -4

.

3

Evaluate Expression C by applying order of operations.

C = -1

Simplify the numerator:

4 - 3 \times 2^2 = 4 - 12 = -8

, so the fraction is

-4

. Evaluate the exponent:

(-1)^5 = -1

. Multiply:

-1 \times 3 = -3

. Subtract:

-4 - (-3) = -1

.

4

Compare the resulting values to determine the correct order from least to greatest.

Expression A (

-33

) < Expression B (

-4

) < Expression C (

-1

)

Comparing negative integers,

-33

is the smallest value and

-1

is the largest value.

Key Concept

Order of Operations and Number Properties

Question 29Question →

What is the value of the expression $(-2)^4 - 3 \times \frac{|12 - 18|}{-2} + 5 \times (3 - 5)^3$ ?

Show answer & explanation

Answer: -15

Answer

The correct answer is -15.

Evaluating the expression according to order of operations yields -15. First, simplify inside the grouping symbols to obtain

(3 - 5) = -2

and

|12 - 18| = 6

. Next, evaluate exponents to obtain

(-2)^4 = 16

and

(-2)^3 = -8

. After this, multiply and divide from left to right to obtain

-3 \times \frac{6}{-2} = 9

and

5 \times (-8) = -40

. Finally, add and subtract from left to right to obtain

16 + 9 - 40 = -15

.

Step-by-Step Solution

1

Evaluate expressions inside grouping symbols.

The grouping symbols evaluate to

(3 - 5) = -2

and

|12 - 18| = |-6| = 6

.

Operations inside parentheses and absolute value bars are prioritized first in the order of operations.

2

Evaluate the exponential terms.

(-2)^4 = 16

and

(-2)^3 = -8

.

Exponents must be calculated after grouping symbols and before multiplication or division.

3

Evaluate multiplication and division from left to right.

-3 \times \frac{6}{-2} = 9

and

5 \times (-8) = -40

.

Multiplication and division are equal in precedence and must be executed in order from left to right.

4

Perform addition and subtraction from left to right.

16 + 9 - 40 = -15

.

Addition and subtraction are the final operations performed, in order from left to right.

Key Concept

Order of operations with exponents, absolute values, and signed numbers.

Question 30Question →

What is the value of the expression $3 \times (4^2 - 10) - 5 \times |-2|$ ?

Show answer & explanation

Answer: 8

Answer

The correct value of the expression is 8.

Evaluating the expression following the order of operations (parentheses, exponents, multiplication/division, and addition/subtraction) results in 8.

Step-by-Step Solution

1

Evaluate the exponent inside the parentheses

4^2 = 16

Exponents must be evaluated before basic arithmetic operations inside parentheses.

2

Subtract the numbers within the parentheses

16 - 10 = 6

Operations enclosed in parentheses are prioritized first.

3

Evaluate the absolute value term

|-2| = 2

The absolute value of a negative number is its positive distance from zero.

4

Perform the multiplications from left to right

3 \times 6 = 18

and

5 \times 2 = 10

Multiplications are performed before addition and subtraction.

5

Perform the final subtraction

18 - 10 = 8

Subtraction is performed last according to the order of operations.

Key Concept

Order of Operations and Number Properties

Question 31Question →

A baker has $12$ chocolate cupcakes and $15$ vanilla cupcakes. She wants to package them into boxes such that each box has the same number of chocolate cupcakes and the same number of vanilla cupcakes, with no cupcakes left over. What is the greatest number of boxes she can make?

3

5

27

60

180

Show answer & explanation

Answer: $3$

Answer

The greatest number of boxes she can make is

3

.

The greatest number of boxes that can be made is found by calculating the greatest common factor (GCF) of the two cupcake quantities. The factors of

12

are

1, 2, 3, 4, 6,

and

12

. The factors of

15

are

1, 3, 5,

and

15

. The greatest factor common to both lists is

3

. This allows the baker to make

3

boxes, each containing

4

chocolate cupcakes and

5

vanilla cupcakes.

Step-by-Step Solution

1

Identify that the problem requires finding the greatest common factor (GCF) of

12

and

15

to distribute the cupcakes evenly into the maximum number of boxes.

The problem is recognized as a greatest common factor (GCF) calculation.

We need the largest integer that divides both

12

and

15

with no remainder.

2

List the factors of

12

and

15

.

Factors of

12

are

1, 2, 3, 4, 6, 12

. Factors of

15

are

1, 3, 5, 15

.

Listing the factors helps identify all shared divisors.

3

Find the greatest common factor shared by both lists.

The common factors are

1

and

3

, so the greatest common factor is

3

.

The greatest common factor is the largest number of identical groups we can form.

Key Concept

Factors, Multiples, and Prime Factorization

Estimated Time:45s

Question 32Question →

What is the value of the expression $-3 \times (4 - 7)^2 + \frac{24}{2^3 - 4}$ ?

87

33

-15

-21

-45

Show answer & explanation

Answer: $-21$

Answer

The correct answer is

-21

.

Following the order of operations, we first compute the term inside the parentheses:

4 - 7 = -3

. Next, we evaluate the exponents:

(-3)^2 = 9

and

2^3 = 8

. This simplifies the expression to

-3 \times 9 + \frac{24}{8 - 4}

. Simplifying the denominator gives

8 - 4 = 4

. We then perform the multiplication and division:

-3 \times 9 = -27

and

\frac{24}{4} = 6

. Adding these results together yields

-27 + 6 = -21

.

Step-by-Step Solution

1

Evaluate the subtraction inside the parentheses.

4 - 7 = -3

According to the order of operations, grouping symbols such as parentheses must be evaluated first.

2

Evaluate the exponential terms in the expression.

(-3)^2 = 9

and

2^3 = 8

Exponents must be evaluated after parentheses and before multiplication, division, addition, or subtraction.

3

Simplify the denominator of the fraction.

8 - 4 = 4

The denominator must be fully simplified before dividing the numerator by it.

4

Perform multiplication and division from left to right.

-3 \times 9 = -27

and

\frac{24}{4} = 6

Multiplication and division have higher priority than addition and subtraction.

5

Perform the final addition.

-27 + 6 = -21

Addition and subtraction are performed last.

Key Concept

Order of Operations (PEMDAS)

Question 33Question →

For a positive integer $n$ , the greatest common factor of $n$ and $120$ is $15$ , and the least common multiple of $n$ and $120$ is $360$ . What is the value of $n$ ?

5

24

45

90

180

Show answer & explanation

Answer: 45

Answer

45

The correct answer is 45. The relationship between two positive integers

a

and

b

and their GCF and LCM is

a \times b = \text{GCF}(a, b) \times \text{LCM}(a, b)

. Substituting

a = n

,

b = 120

,

\text{GCF} = 15

, and

\text{LCM} = 360

gives

n \times 120 = 15 \times 360

. Dividing both sides of the equation by 120 yields

n = 45

.

Step-by-Step Solution

1

Identify the given values and the mathematical relationship between two positive integers, their greatest common factor (GCF), and their least common multiple (LCM).

Given

a = n

,

b = 120

,

\text{GCF}(n, 120) = 15

, and

\text{LCM}(n, 120) = 360

. The relationship is

a \times b = \text{GCF}(a, b) \times \text{LCM}(a, b)

.

This formula provides a direct algebraic path to solve for the unknown integer when the other parameters are known.

2

Substitute the given values into the relationship formula.

n \times 120 = 15 \times 360

To set up an equation containing only the unknown variable

n

.

3

Isolate and solve for

n

.

n = \frac{15 \times 360}{120} = 15 \times 3 = 45

To find the specific integer value of

n

.

Key Concept

The product of two positive integers is equal to the product of their greatest common factor (GCF) and their least common multiple (LCM).

Alternative Method

Another way to solve this is to write the prime factorizations. Since

120 = 2^3 \times 3 \times 5

, and their GCF is

15 = 3 \times 5

,

n

must contain

3 \times 5

but cannot contain any factor of 2. Since their LCM is

360 = 2^3 \times 3^2 \times 5

,

n

must contain

3^2

because 120 only has

3^1

. Thus, the prime factorization of

n

is

3^2 \times 5 = 45

.

Estimated Time:1m 0s

Question 34Question →

What is the value of the expression below?

\frac{-2^4 - |-12 + (-2)^3|}{(-1)^5 \times 3^2 - (-2)^3} \div \left( \frac{1}{3} - 1 \right)

-24

-6

6

-54

54

Show answer & explanation

Answer: $-54$

Answer

-54

Evaluating the numerator gives

-36

, and the denominator simplifies to

-1

, resulting in a fraction of

36

. Dividing

36

by

(1/3 - 1) = -2/3

is performed by multiplying

36

by the reciprocal

-3/2

, which yields the final result of

-54

.

Step-by-Step Solution

1

Evaluate the numerator of the first fraction.

Numerator =

-36

First, evaluate the exponent inside the absolute value:

(-2)^3 = -8

. Next, evaluate the expression inside the absolute value:

-12 + (-8) = -20

. Taking the absolute value gives

|-20| = 20

. Separately, evaluate the exponent

-2^4 = -(2^4) = -16

because exponentiation has priority over negation. Finally, subtract the absolute value term from this to get

-16 - 20 = -36

.

2

Evaluate the denominator of the first fraction.

Denominator =

-1

Evaluate the exponents first:

(-1)^5 = -1

,

3^2 = 9

, and

(-2)^3 = -8

. Then, perform the multiplication:

-1 \times 9 = -9

. Finally, perform the subtraction:

-9 - (-8) = -9 + 8 = -1

.

3

Simplify the first fraction.

Fraction =

36

Divide the numerator by the denominator:

\frac{-36}{-1} = 36

.

4

Evaluate the expression inside the divisor parentheses.

Divisor =

-\frac{2}{3}

Perform the fraction subtraction:

\frac{1}{3} - 1 = \frac{1}{3} - \frac{3}{3} = -\frac{2}{3}

.

5

Divide the simplified fraction by the divisor.

Final Value =

-54

To divide by a fraction, multiply by its reciprocal:

36 \div \left(-\frac{2}{3}\right) = 36 \times \left(-\frac{3}{2}\right) = 18 \times (-3) = -54

.

Key Concept

Order of Operations (PEMDAS) and Number Properties

Question 35Question →

What is the sum of the distinct prime factors of $30$ ?

Show answer & explanation

Answer: 10

Answer

The sum of the distinct prime factors of

30

is

10

.

The prime factorization of

30

is

2 \times 3 \times 5

. The distinct prime factors are

2

,

3

, and

5

, and their sum is

10

.

Step-by-Step Solution

1

Find the prime factorization of

30

.

30 = 2 \times 3 \times 5

To break the composite number

30

down into its prime components.

2

Identify the distinct prime factors.

The distinct prime factors are

2

,

3

, and

5

.

Only prime numbers that divide

30

should be included in the sum.

3

Add the distinct prime factors.

2 + 3 + 5 = 10

To calculate the final sum as requested by the question.

Key Concept

Prime factorization is the process of factoring a composite number into a product of prime numbers. The distinct prime factors of a number are the unique prime numbers that divide it.

Question 36Question →

On a standard number line, point $A$ is located at $-7$ and point $B$ is located at $5$ . What is the distance between point $A$ and point $B$ ?

Show answer & explanation

Answer: 12

Answer

The distance between the two points is 12.

The distance between two points on a number line is the absolute value of their difference. Calculating

|-7 - 5|

yields

|-12|

, which is

12

. Alternatively, subtracting the lesser coordinate from the greater coordinate gives

5 - (-7) = 5 + 7 = 12

.

Step-by-Step Solution

1

Identify the coordinates of the two points on the number line.

The coordinates are

-7

for point

A

and

5

for point

B

.

These coordinates represent the positions from which we need to find the distance.

2

Apply the absolute value formula for the distance between two points,

|a - b|

.

The expression is

|-7 - 5|

.

Distance on a number line is always non-negative and is defined by the absolute value of the difference between the two coordinates.

3

Perform the subtraction and find the absolute value.

|-12| = 12

.

Subtracting

5

from

-7

gives

-12

, and the absolute value of

-12

is

12

.

Key Concept

The distance between two points

a

and

b

on a number line is given by

|a - b|

.

Estimated Time:45s

Question 37Question →

An algebra student is simplifying the expression $a(b - c) - d(e - f)$ by applying the distributive property. Instead of distributing correctly, the student incorrectly writes the expansion as $a(b - c) - de - df$ . If $a = 2$ , $b = 3$ , $c = 5$ , $d = -3$ , $e = 4$ , and $f = -2$ , what is the absolute difference between the student's incorrect result and the correct value of the expression?

Show answer & explanation

Answer: 12

Answer

The correct answer is 12.

The correct expression evaluates to 14, and the incorrect expression evaluates to 2. The absolute difference between them is

|14 - 2| = 12

.

Step-by-Step Solution

1

Evaluate the correct expression

a(b - c) - d(e - f)

using the given values.

14

To find the mathematically correct value of the expression.

2

Evaluate the incorrect expression

a(b - c) - de - df

using the same values.

2

To find the value resulting from the student's distribution error.

3

Find the absolute difference between the correct value and the incorrect value.

12

To determine the error magnitude as requested by the question.

Key Concept

Distributive Property and Order of Operations

Question 38Question →

A student is adding the fractions $\frac{5}{12}$ and $\frac{7}{18}$ . What is the least common denominator they should use to add these fractions?

6

30

36

72

216

Show answer & explanation

Answer: 36

Answer

36

To find the least common denominator of the two fractions, find the least common multiple (LCM) of the denominators, 12 and 18. The prime factorization of 12 is

2^2 \times 3

, and the prime factorization of 18 is

2 \times 3^2

. The LCM is found by taking the highest power of each prime factor that appears in either factorization:

2^2 \times 3^2 = 4 \times 9 = 36

.

Step-by-Step Solution

1

Identify the denominators of the fractions to find their least common denominator.

The denominators are 12 and 18.

The least common denominator of two fractions is equal to the least common multiple (LCM) of their denominators.

2

Find the prime factorization of each denominator.

12 = 2^2 \times 3

and

18 = 2 \times 3^2

Prime factorization helps systematically determine the least common multiple.

3

Calculate the least common multiple by multiplying the highest power of each prime factor present.

2^2 \times 3^2 = 4 \times 9 = 36

This yields the smallest positive integer that is divisible by both 12 and 18.

Key Concept

Finding the least common denominator by calculating the least common multiple (LCM) of the denominators.

Estimated Time:45s

Question 39Question →

If $x$ , $y$ , and $z$ are integers such that $-5 \leq x < y < z \leq 5$ , what is the minimum possible value of the expression $(x - y)^2 - z(y - x) - |x| \cdot |y - z|$ ?

-1

-25

-39

-41

-49

Show answer & explanation

Answer: -49

Answer

The minimum possible value of the expression is

-49

.

The correct value is

-49

. By setting

d_1 = y - x \geq 1

and

d_2 = z - y \geq 1

, we can rewrite the expression as

d_1^2 - z d_1 - |x| d_2

. Substituting

z = x + d_1 + d_2

gives

-x d_1 - d_2(d_1 + |x|)

. To minimize this, we choose the smallest possible value for

x

, which is

-5

. This gives

5d_1 - d_2(d_1 + 5)

. Since

x = -5

and

z \leq 5

, we have

d_1 + d_2 \leq 10

. Substituting

d_2 = 10 - d_1

yields

5d_1 - (10 - d_1)(d_1 + 5) = d_1^2 - 50

. The minimum value of this quadratic expression for integer

d_1 \geq 1

occurs at

d_1 = 1

, giving

1^2 - 50 = -49

. This corresponds to

x = -5, y = -4, z = 5

. Evaluating the expression directly confirms this result:

(-5 - (-4))^2 - 5(-4 - (-5)) - |-5| \cdot |-4 - 5| = 1 - 5 - 45 = -49

.

Step-by-Step Solution

1

Define variables for the differences between the ordered integers.

Let

d_1 = y - x \geq 1

and

d_2 = z - y \geq 1

, which implies

y = x + d_1

and

z = x + d_1 + d_2

.

Using differences simplifies the inequality constraints and allows us to express the objective function in terms of positive integers

d_1

and

d_2

.

2

Substitute the differences into the given expression

(x - y)^2 - z(y - x) - |x| \cdot |y - z|

.

The expression becomes

d_1^2 - (x + d_1 + d_2)d_1 - |x|d_2 = -x d_1 - d_2(d_1 + |x|)

.

This rewrites the expression in terms of the initial variable

x

and the positive differences

d_1

and

d_2

.

3

Analyze how to minimize

-x d_1 - d_2(d_1 + |x|)

given the boundaries.

Setting

x = -5

makes the expression

5d_1 - d_2(d_1 + 5)

. Since

x = -5

and

z \leq 5

, we have

d_1 + d_2 \leq 10

.

Choosing the minimum value for

x

maximizes the positive coefficient of

d_1

and the term

|x|

in the negative product, leading to the smallest possible value.

4

Maximize

d_2

by setting

d_2 = 10 - d_1

and substitute it into the expression.

The expression becomes

5d_1 - (10 - d_1)(d_1 + 5) = d_1^2 - 50

.

Since

d_2

has a negative coefficient, maximizing

d_2

minimizes the overall expression.

5

Minimize

d_1^2 - 50

subject to

d_1 \geq 1

.

The minimum occurs at

d_1 = 1

, yielding

1^2 - 50 = -49

, which corresponds to

x = -5, y = -4, z = 5

.

Since

d_1^2

is strictly increasing for positive integers, the minimum value of the quadratic is achieved at the smallest boundary value of

d_1

.

Key Concept

Order of Operations and Number Properties

Question 40Question →

Given the algebraic expression $A = x - y \times z^2$ , evaluate the expression for each of the following sets of values for $(x, y, z)$ and arrange the sets in order of their resulting values of $A$ from least to greatest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order of the sets from least to greatest is:

(x, y, z) = (-5, 1, 2)

resulting in

-9

, followed by

(x, y, z) = (-2, 3, -1)

resulting in

-5

, then

(x, y, z) = (8, 4, 0)

resulting in

8

, and finally

(x, y, z) = (4, -2, 3)

resulting in

22

.

The correct order follows from evaluating the algebraic expression for each set of values according to the order of operations (PEMDAS): exponentiation first, then multiplication, and finally subtraction. This yields values of

-9

,

-5

,

8

, and

22

, which are ordered from least to greatest.

Step-by-Step Solution

1

Evaluate the expression

A = x - y \times z^2

for the set

(x, y, z) = (-5, 1, 2)

.

A = -9

According to the order of operations, evaluate the exponent first:

2^2 = 4

. Then perform multiplication:

1 \times 4 = 4

. Finally, perform subtraction:

-5 - 4 = -9

.

2

Evaluate the expression

A = x - y \times z^2

for the set

(x, y, z) = (-2, 3, -1)

.

A = -5

Evaluate the exponent first:

(-1)^2 = 1

. Next, multiply:

3 \times 1 = 3

. Finally, subtract:

-2 - 3 = -5

.

3

Evaluate the expression

A = x - y \times z^2

for the set

(x, y, z) = (8, 4, 0)

.

A = 8

Evaluate the exponent first:

0^2 = 0

. Next, multiply:

4 \times 0 = 0

. Finally, subtract:

8 - 0 = 8

.

4

Evaluate the expression

A = x - y \times z^2

for the set

(x, y, z) = (4, -2, 3)

.

A = 22

Evaluate the exponent first:

3^2 = 9

. Next, multiply:

-2 \times 9 = -18

. Finally, subtract:

4 - (-18) = 4 + 18 = 22

.

5

Arrange the resulting values from least to greatest.

-9 < -5 < 8 < 22

Comparing the four calculated values shows that

-9

is the least, followed by

-5

, then

8

, and

22

is the greatest.

Key Concept

Order of Operations (PEMDAS)

Alternative Method

To verify the calculations, substitute each set of coordinates carefully, using parentheses around negative values. Evaluate each term step-by-step: first compute

z^2

, then multiply by

y

, and finally subtract this result from

x

.

Estimated Time:1m 30s

Pre-Algebra

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