Number Theory and Operations Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 2

If $x = -2$ , $y = 24$ , and $z = -4$ , what is the value of the following expression?

-2x^3 + \frac{y}{|z|}

-10

70

22

10

Show answer & explanation

Answer: 22

Answer

The correct answer is 22.

The correct value is found by accurately substituting the negative numbers into the expression. Applying the exponent to -2 yields -8. Multiplying -8 by the coefficient -2 gives 16. Finally, adding the result of 24 divided by the absolute value of -4 (which simplifies to 6) produces 22.

Step-by-Step Solution

1

Substitute the given values into the algebraic expression.

-2(-2)^3 + \frac{24}{|-4|}

Replacing variables with their numerical equivalents is the required first step before evaluating the expression.

2

Evaluate the absolute value in the denominator.

|-4| = 4

, so the fraction becomes

\frac{24}{4}

.

Absolute value symbols act as grouping symbols, and the absolute value of a negative number is positive.

3

Evaluate the exponent.

(-2)^3 = -8

, so the expression becomes

-2(-8) + \frac{24}{4}

.

According to the order of operations (PEMDAS), exponents must be evaluated before multiplication.

4

Perform the multiplication and division from left to right.

-2(-8) = 16

and

24 \div 4 = 6

. The expression simplifies to

16 + 6

.

Multiplication and division share the same priority and are performed before addition.

5

Perform the final addition.

16 + 6 = 22

Addition is the final operation remaining in the expression.

Key Concept

Order of Operations with Variable Substitution

Question 22Question →

Determine the final simplified value of the mathematical expression shown below:

-2^4 - 3[12 - 4(3 - 5)^2]

Which of the following represents the correct result?

-112

-4

28

76

Show answer & explanation

Answer: -4

Answer

The correct simplified value of the expression is -4.

The expression is accurately evaluated by following the standard order of operations. The innermost parentheses are solved to get -2, which is then squared to yield 4. Multiplying this by 4 inside the brackets gives 16, and subtracting this from 12 leaves -4 in the brackets. The leading exponent

-2^4

is -16. Multiplying the bracket by -3 gives 12, and adding this to -16 yields a final result of -4.

Step-by-Step Solution

1

Evaluate the expression inside the innermost parentheses:

(3 - 5)

.

(3 - 5) = -2

According to the order of operations (PEMDAS), parentheses must be resolved first.

2

Apply the exponents to the respective bases:

(-2)^2

and

-2^4

.

(-2)^2 = 4

and

-2^4 = -16

Exponents are evaluated next. Note that

-2^4

lacks parentheses around the negative sign, meaning it is interpreted as

-(2^4) = -16

.

3

Substitute the exponent results back into the bracket and multiply:

4(4)

.

The bracket becomes

[12 - 4(4)]

, which simplifies to

[12 - 16]

.

Inside the brackets, multiplication must be performed before subtraction.

4

Complete the subtraction inside the brackets:

[12 - 16]

.

[12 - 16] = -4

Finish evaluating the operations grouped by the brackets.

5

Multiply the bracket's result by the coefficient outside:

-3(-4)

.

-3 \times -4 = 12

Multiplication is performed before the final addition/subtraction.

6

Combine the first term with the multiplication result:

-16 + 12

.

-16 + 12 = -4

Addition and subtraction are the final steps in the order of operations.

Key Concept

Order of Operations (PEMDAS)

Estimated Time:45s

Question 23Question →

Determine the correct chronological sequence of steps required to evaluate the following mathematical expression according to the standard order of operations:

\frac{ -3 \times \left[ 4 - \left( 5 - 2^3 \right) \right] }{ -7 + 4 }

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence begins with the innermost exponent, resolves the nested grouping symbols outward, simplifies the numerator and denominator independently, and finishes with division.

The correct evaluation sequence strictly follows the order of operations (PEMDAS). It begins by isolating the innermost grouping symbols and addressing the exponent. Once the innermost parentheses are reduced to a single value, the surrounding brackets are resolved, carefully tracking negative signs. The large fraction bar acts as an invisible grouping symbol, requiring the entire numerator and denominator to be independently simplified before performing the final division.

Step-by-Step Solution

1

Identify innermost grouping symbols and evaluate exponents.

The expression

(5 - 2^3)

becomes

(5 - 8)

.

Parentheses and Exponents have the highest priority in the order of operations.

2

Complete operations inside the innermost parentheses.

The expression

(5 - 8)

evaluates to

-3

.

The innermost group must be resolved to a single number before moving outward to the next level of grouping.

3

Evaluate operations inside the outer square brackets.

The expression

[4 - (-3)]

becomes

[4 + 3]

, which simplifies to

7

.

Square brackets function as the next level of grouping and must be completely simplified before multiplying.

4

Simplify the remainder of the numerator and the denominator.

The numerator becomes

-3 \times 7 = -21

. The denominator becomes

-7 + 4 = -3

.

A fraction bar acts as a grouping symbol, meaning the top and bottom expressions must be fully evaluated independently before division occurs.

5

Perform the final division.

-21 \div -3 = 7

.

Division is the final operation combining the fully simplified numerator and denominator.

Key Concept

Order of Operations (PEMDAS) with nested grouping symbols, negative numbers, and fraction bars.

Question 24Question →

A supply store reduces the price of a generator by $20\%$ . During a clearance sale, this discounted price is reduced by an additional $25\%$ . If the final clearance price is $\$360$ , what was the original price of the generator?

\$540

\$600

\$800

\$900

Show answer & explanation

Answer: $\$600$

Answer

The original price of the generator was

\$600

.

The problem involves successive percentages. The first discount leaves

80\%

of the original price. The second discount removes

25\%

of that

80\%

, leaving

75\%

of

80\%

, which mathematically is

0.80 \times 0.75 = 0.60

. This means the final price is exactly

60\%

of the original price. Dividing the final price of

\$360

by

0.60

yields the correct original price of

\$600

.

Step-by-Step Solution

1

Represent the original price as a variable and apply the first discount.

If the original price is

x

, the price after a

20\%

discount is

x - 0.20x = 0.80x

.

A

20\%

reduction means the buyer pays

80\%

of the original price.

2

Apply the second discount to the newly reduced price.

The clearance price is

0.80x - (0.25 \times 0.80x) = 0.80x \times 0.75 = 0.60x

.

Successive discounts are applied to the previously discounted price, not the original price. Taking

25\%

off means keeping

75\%

of the current value.

3

Set up the equation with the given final price and solve for

x

.

0.60x = 360

, so

x = \frac{360}{0.60} = 600

.

The final represented percentage (

60\%

) equals the final dollar amount (

\$360

), allowing us to isolate the original price.

Key Concept

Successive Percentage Decreases

Question 25Question →

A military recruit needs to correctly answer $75\%$ of the questions on a 60-question training exam to pass. How many questions must the recruit answer correctly?

15

45

4.5

80

Show answer & explanation

Answer: 45

Answer

45

To find

75\%

of 60, first convert the percentage to a decimal (

0.75

) or a fraction (

\frac{3}{4}

). Then, multiply this value by the total number of questions (

60

). Calculating

60 \times 0.75

exactly yields 45.

Step-by-Step Solution

1

Convert the required percentage to a decimal or fraction.

75\% = 0.75

or

\frac{3}{4}

Percentages must be converted into their numerical equivalents before applying them in an arithmetic operation.

2

Multiply the total number of questions by the converted percentage.

60 \times 0.75 = 45

To find a specific percentage of a total amount, multiply the total by the percentage rate.

Key Concept

Calculating the percentage of a given number

Question 26Question →

What is the exact value of the following mathematical expression?

\frac{10 - (-2)}{3} + 4^2

Show answer & explanation

Answer: 20

Answer

The correct value is 20.

The expression is correctly evaluated by first simplifying the numerator grouping

10 - (-2)

to get

12

, dividing by

3

to obtain

4

, evaluating the exponent

4^2

to get

16

, and finally adding the results together to reach

20

.

Step-by-Step Solution

1

Simplify the grouping symbol formed by the numerator.

10 - (-2) = 12

The numerator of a fraction acts as a grouping symbol and must be evaluated first.

2

Divide the simplified numerator by the denominator.

\frac{12}{3} = 4

The fraction bar represents division, which should be completed before addition.

3

Evaluate the exponent.

4^2 = 16

According to the order of operations, exponents must be evaluated before addition.

4

Add the remaining values.

4 + 16 = 20

Addition is the final operation left in the expression.

Key Concept

Order of Operations with fractions and signed numbers

Estimated Time:45s

Question 27Question →

A military supply depot receives an initial stock of fuel. Over the first month, the stock is increased by $25\%$ . During the second month, the new stock is decreased by $x\%$ . If the final amount of fuel at the end of the second month is exactly $10\%$ less than the initial stock, what is the value of $x$ ?

28

30

35

72

Show answer & explanation

Answer: 28

Answer

The value of x is 28.

The correct answer is derived by setting up an algebraic equation for successive percentage changes. An initial increase of 25% results in 1.25 times the original amount. To reach a final amount that is 10% less than the original (0.90 times the original), the intermediate amount must be multiplied by 0.72 (since 0.90 / 1.25 = 0.72). A multiplier of 0.72 means that 72% of the amount remains, representing a 28% decrease.

Step-by-Step Solution

1

Define a variable for the initial stock.

Let the initial stock be S.

Establishing a variable allows for clear algebraic translation of the successive changes.

2

Express the amount of fuel after the first month.

S + 0.25S = 1.25S

A 25% increase means multiplying the initial base by 1.25.

3

Express the amount of fuel after the second month.

1.25S \times (1 - \frac{x}{100})

The x% decrease must be applied to the new intermediate base of 1.25S, not the original base.

4

Set up an equation based on the final given condition.

1.25S \times (1 - \frac{x}{100}) = 0.90S

The problem states the final amount is exactly 10% less than the initial stock (which is 0.90S).

5

Solve the equation for the remaining percentage multiplier.

1 - \frac{x}{100} = \frac{0.90}{1.25} = \frac{90}{125} = \frac{18}{25} = 0.72

Dividing both sides by 1.25S isolates the multiplier associated with the decrease.

6

Calculate the value of x.

\frac{x}{100} = 1 - 0.72 = 0.28, so x = 28

Subtracting the remaining percentage (72%) from 100% yields the percentage decrease.

Key Concept

Successive Percentage Changes and Base Identification

Question 28Question →

A supplier sells aircraft parts. A specific component is marked up by $60\%$ from its original manufacturing cost to establish a catalog price. Later, the supplier applies a $25\%$ discount to the catalog price for a bulk order. If the discounted bulk price of the component is $\$300$ , what was its original manufacturing cost?

\$150

\$195

\$250

\$360

Show answer & explanation

Answer: \$250

Answer

The original manufacturing cost was \$250.

To find the original cost, the successive percentage changes must be expressed as multipliers. A

60\%

markup is a multiplier of

1.60

, and a

25\%

discount is a multiplier of

0.75

. Multiplying these together gives a net multiplier of

1.20

(

1.60 \times 0.75 = 1.20

). This means the final bulk price is

120\%

of the original cost. Since the final price is

\$300

, dividing this by

1.20

yields the original manufacturing cost of

\$250

.

Step-by-Step Solution

1

Translate the 60% markup into a decimal multiplier.

Catalog Price =

1.60 \times

Cost.

A 60% increase means the new price is 100% + 60% = 160% of the original cost.

2

Translate the 25% discount into a decimal multiplier.

Bulk Price =

0.75 \times

Catalog Price.

A 25% decrease means the customer pays 100% - 25% = 75% of the catalog price.

3

Combine the multipliers to find the net percentage effect on the original cost.

1.60 \times 0.75 = 1.20

. The net price is 120% of the original cost.

Successive percentage changes are multiplicative, not additive.

4

Set up the equation with the given final price and solve for the original cost (

C

).

1.20C = 300

, which solves to

C = 300 \div 1.20 = 250

.

Dividing the final amount by the net multiplier isolates the starting amount.

Key Concept

Successive Percentage Changes and Reverse Percentages

Question 29Question →

Arrange the following calculation steps in the exact chronological sequence they must be performed relative to each other when evaluating the mathematical expression below using the standard order of operations:

\frac{-2[(-3)^2 - 4(5 - 7)^3]}{|-9| - (8 - 1)} - 9

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence starts with the innermost exponents and grouping symbols, resolves the bracket, evaluates the full numerator and denominator, divides, and finally subtracts 9.

The correct order strictly follows the standard order of operations (PEMDAS). Operations within grouping symbols (innermost parentheses first, then brackets) must be evaluated before external operations. Exponents are applied immediately after their base is simplified. The fraction bar acts as a major grouping symbol, requiring the numerator and denominator to be fully evaluated before dividing. The final subtraction occurs last.

Step-by-Step Solution

1

Simplify the innermost parentheses and absolute value

$5 - 7 = -2$ and $8 - 1 = 7$ and $|-9| = 9$

Parentheses and absolute value brackets (P in PEMDAS) are evaluated first.

2

Evaluate the exponents

$(-3)^2 = 9$ and $(-2)^3 = -8$

Exponents (E in PEMDAS) follow parentheses.

3

Perform multiplication inside the square brackets

4 \times (-8) = -32

Multiplication (M in PEMDAS) occurs before subtraction inside the grouping symbols.

4

Perform subtraction inside the square brackets

$9 - (-32) = 41$

Subtraction is the final operation needed to simplify the inside of the square bracket.

5

Multiply the bracket by the external coefficient

-2 \times 41 = -82

This completes the evaluation of the numerator.

6

Simplify the denominator

$9 - 7 = 2$

The denominator must be evaluated to a single number before division can occur.

7

Divide the numerator by the denominator

-82 \div 2 = -41

The fraction bar represents division of the evaluated top and bottom parts.

8

Perform the final subtraction

$-41 - 9 = -50$

Subtraction outside the fraction is the last operational step.

Key Concept

Order of Operations with Nested Grouping Symbols and Fractions

Question 30Question →

What is the exact numerical value of the following mathematical expression?

|-18| - 7^2 + \frac{45}{-9}

Show answer & explanation

Answer: -36

Answer

The correct numerical value is -36.

To evaluate the expression, follow the order of operations strictly. First, resolve the absolute value: |-18| becomes 18. Next, evaluate the exponent: 7^2 becomes 49 (note that the negative sign is separate from the squared base). Then, perform the division: 45 divided by -9 equals -5. Finally, perform the remaining subtraction and addition from left to right: 18 - 49 equals -31, and -31 + (-5) yields the final answer of -36.

Step-by-Step Solution

1

Evaluate the absolute value.

|-18| = 18

Absolute value acts as a grouping symbol and must be processed first, returning the non-negative distance from zero.

2

Evaluate the exponential term.

7^2 = 49

Exponents are evaluated before multiplication, division, addition, and subtraction. The negative sign is outside the base, so only the 7 is squared.

3

Evaluate the division.

\frac{45}{-9} = -5

Division and multiplication are evaluated before addition and subtraction.

4

Evaluate addition and subtraction from left to right.

18 - 49 + (-5) = -36

Addition and subtraction hold the same precedence and are solved strictly from left to right as they appear.

Key Concept

Order of Operations with signed numbers, absolute value, and exponents

Question 31Question →

Calculate the final sale price and fill in the blank with the correct numerical value.

Fill in the blanks below

A hardware store is selling a power drill that originally costs

120

dollars. During a weekend sale, the drill is discounted by

20\%

. The final sale price of the drill is dollars.

Show answer & explanation

Answer

96

The correct final price is

96

, which is found by taking

20\%

of

120

to determine the discount amount (

24

), and then subtracting that discount from the original price (

120 - 24 = 96

).

Step-by-Step Solution

1

Calculate the amount of the discount.

0.20 \times 120 = 24

To find the discount amount, multiply the original price by the discount percentage expressed as a decimal.

2

Subtract the discount from the original price to find the final sale price.

120 - 24 = 96

The final sale price is the original price minus the amount of the discount.

Key Concept

Calculating a percentage decrease to find a final price

Alternative Method

You can calculate the final price directly in one step. Since the drill is discounted by

20\%

, you are paying

100\% - 20\% = 80\%

of the original price. Multiply

0.80 \times 120 = 96

.

Estimated Time:45s

Question 32Question →

What is the evaluated result of the arithmetic expression below?

(-3)^2 - 4 \times (-2)

-10

-1

1

17

Show answer & explanation

Answer: 17

Answer

17

Following the proper order of operations, the exponent is evaluated first to yield 9. Next, multiplying 4 by -2 gives -8. Finally, subtracting -8 from 9 results in 9 + 8, which equals 17.

Step-by-Step Solution

1

Evaluate the exponent

(-3)^2

.

9

According to the order of operations, exponents must be resolved before multiplication and subtraction.

2

Perform the multiplication

4 \times (-2)

.

-8

Multiplication takes precedence over subtraction.

3

Subtract the result of the multiplication from the result of the exponent.

17

Subtracting a negative number is equivalent to adding its positive counterpart (

9 - (-8) = 9 + 8 = 17

).

Key Concept

Order of Operations with Integers

Question 33Question →

An engineer uses the formula below to calculate a specific stress factor in a structural component:

p^3 - \frac{4pq}{r - q} + (r - p)^2

If the component's parameters are measured as $p = -2$ , $q = -1$ , and $r = 3$ , what is the calculated stress factor?

-35

15

-9

31

Show answer & explanation

Answer: 15

Answer

15

The correct answer is found by strictly following the order of operations (PEMDAS) while properly applying integer sign rules during substitution. First, evaluating grouping symbols yields a denominator of 4 and an exponent base of 5. Second, resolving the exponents gives -8 and 25. Third, calculating the middle fraction evaluates to 2. Finally, computing -8 - 2 + 25 from left to right results in 15.

Step-by-Step Solution

1

Substitute the given values into the expression.

(-2)^3 - \frac{4(-2)(-1)}{3 - (-1)} + (3 - (-2))^2

Variable substitution must be done precisely, maintaining all negative signs within parentheses to avoid sign errors.

2

Simplify the expressions acting as grouping symbols (the denominator and the base of the final exponent).

(-2)^3 - \frac{4(-2)(-1)}{4} + (5)^2

According to the order of operations, operations inside parentheses and implied grouping symbols like fraction bars must be completed first.

3

Evaluate the exponents.

-8 - \frac{4(-2)(-1)}{4} + 25

Exponents are the next priority in the order of operations. Note that an odd exponent preserves the negative sign of its base:

(-2)^3 = -8

.

4

Perform the multiplication in the numerator, then the division.

-8 - \frac{8}{4} + 25 = -8 - 2 + 25

Multiplication and division are evaluated from left to right before addition and subtraction.

5

Perform the remaining addition and subtraction from left to right.

-10 + 25 = 15

Addition and subtraction have equal priority and must be evaluated strictly from left to right.

Key Concept

Evaluating Complex Expressions using Order of Operations and Negative Numbers

Question 34Question →

A base exchange marks up the wholesale price of a standard watch by $20\%$ to determine its retail price. If the retail price of the watch is $\$150$ , what was the original wholesale price and the markup amount? Complete the statement below with your calculated values.

Fill in the blanks below

The original wholesale price of the watch was dollars, and the exact amount of the markup was dollars.

Show answer & explanation

Answer

The original wholesale price was 125 dollars, and the exact amount of the markup was 25 dollars.

Because the watch is marked up by

20\%

, the

\$150

retail price represents

120\%

of the wholesale price. Dividing

150

by

1.20

yields the correct wholesale price of

125

. The markup is the difference between the retail and wholesale prices (

150 - 125 = 25

).

Step-by-Step Solution

1

Identify the mathematical relationship between the wholesale price, the markup percentage, and the retail price.

\text{Wholesale Price} + 20\% \text{ of Wholesale Price} = \text{Retail Price}

The markup is calculated based on the original wholesale price, not the final retail price.

2

Set up an algebraic equation where

w

represents the wholesale price.

w + 0.20w = \$150

, which simplifies to

1.20w = \$150

Translating the percentage relationship into a solvable equation allows us to find the unknown base amount.

3

Solve for the wholesale price (

w

).

w = \frac{150}{1.20} = 125

Dividing the retail price by

1.20

isolates

w

, giving the original wholesale price of

\$125

.

4

Calculate the exact markup amount by finding the difference between the retail and wholesale prices.

\$150 - \$125 = \$25

The markup amount is the portion added to the wholesale price to reach the retail price.

Key Concept

Reverse percentage calculation to find an original base amount.

Question 35Question →

An engineer is using the formula $P = 2a^3 - b[a - (c - b)]^2$ to calculate the tolerance of a specific component. If the parameters are measured as $a = -2$ , $b = 3$ , and $c = -1$ , what is the calculated value of $P$ ?

-28

-52

4

-124

Show answer & explanation

Answer: -28

Answer

The calculated value of P is -28.

The correct answer is derived by substituting the values and strictly following PEMDAS: resolving the innermost parentheses to get -4, resolving the brackets to get 2, cubing -2 to get -8 and squaring 2 to get 4, performing the multiplications to yield -16 and 12, and subtracting to reach the final value of -28.

Step-by-Step Solution

1

Substitute the given values into the formula.

P = 2(-2)^3 - 3[-2 - (-1 - 3)]^2

Variable substitution must be done carefully, keeping negative numbers in parentheses to avoid sign drops.

2

Evaluate the innermost grouping symbols first.

(-1 - 3) = -4

According to the order of operations, innermost parentheses must be simplified before outer brackets.

3

Simplify the expression inside the brackets by handling the double negative.

[-2 - (-4)] = [-2 + 4] = 2

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

4

Apply the exponents to their respective bases.

(-2)^3 = -8

and

2^2 = 4

Exponents must be evaluated before multiplication. A negative base raised to an odd power remains negative.

5

Perform the multiplication operations.

2(-8) = -16

and

3(4) = 12

Multiplication precedes subtraction in the order of operations.

6

Perform the final subtraction.

-16 - 12 = -28

Subtraction is the final step after all grouping, exponents, and multiplication are resolved.

Key Concept

Order of Operations with Variable Substitution and Signed Integers

Estimated Time:45s

Question 36Question →

A drainage system is clearing a flooded trench. During the first hour of operation, the system removes $30\%$ of the initial volume of water. During the second hour, it removes $40\%$ of the water that remained after the first hour. If there are exactly $2,100$ gallons of water still left in the trench after the second hour, calculate the initial volume of water and the exact amount of water removed during the second hour.

Fill in the blanks below

The initial volume of water in the trench was gallons, and the system removed exactly gallons during the second hour.

Show answer & explanation

Answer

The initial volume was 5,000 gallons, and 1,400 gallons were removed during the second hour.

The problem requires tracking successive percentage changes relative to an unknown original base. Because

30\%

was removed initially,

70\%

of the original volume remained. During the second hour,

40\%

of that

70\%

was removed, which equates to removing an additional

28\%

of the original volume. Subtracting

28\%

from the

70\%

leaves

42\%

of the original volume. We are told this

42\%

equals

2,100

gallons. By dividing

2,100

by

0.42

, we find the initial volume was

5,000

gallons. The water removed during the second hour is

28\%

of

5,000

, which equals

1,400

gallons.

Step-by-Step Solution

1

Calculate the percentage of the initial volume remaining after the first hour.

100\% - 30\% = 70\%

of the initial volume remains.

The second hour's removal rate is applied to the remaining water, so establishing this new base is required.

2

Determine the percentage of the initial volume that is removed during the second hour.

40\%

of

70\%

is

0.40 \times 0.70 = 0.28

, or

28\%

of the initial volume.

This converts the second hour's removal rate into a percentage relative to the overall initial volume.

3

Calculate the final percentage of the initial volume remaining after the second hour.

70\% - 28\% = 42\%

of the initial volume remains.

Subtracting the second hour's removed percentage from the first hour's remaining percentage yields the final remaining proportion.

4

Set up an equation using the final remaining volume to find the initial volume (

V

).

0.42V = 2,100

, which solves to

V = \frac{2,100}{0.42} = 5,000

gallons.

This algebraic equation equates the final percentage representation to the actual known volume to isolate the initial total.

5

Calculate the exact volume removed during the second hour.

0.28 \times 5,000 = 1,400

gallons.

Applying the percentage removed during the second hour to the solved initial volume provides the specific amount removed.

Key Concept

Successive Percentages and Reverse Base Calculations

Question 37Question →

Calculate the base production cost and the net percentage increase based on the successive price changes described below.

Fill in the blanks below

A technology firm sets the retail price of an industrial 3D printer by applying a

125\%

markup to its base production cost. During a trade show, the retail price is discounted by

40\%

. Partners who place orders at the show receive an additional

20\%

discount on the trade show price.

If a partner pays

\$5,400

for the printer, the base production cost is

\$\{\{blank_1\}\}

, and the firm's final selling price represents an overall increase of

\{\{blank_2\}\}\%

over the base production cost.

Show answer & explanation

Answer

The base production cost is $5,000, and the final selling price represents an 8% overall increase.

To solve problems with successive percentage changes, express each change as a multiplier. A 125% markup is a multiplier of 2.25. A 40% discount is a multiplier of 0.60. A 20% discount is a multiplier of 0.80. Multiplying these sequentially (2.25 × 0.60 × 0.80) yields a final compound multiplier of 1.08. This means the final price is 1.08 times the base cost, which represents an 8% overall increase. Setting 1.08C equal to the final price of

5,400 allows us to solve for C by dividing 5,400 by 1.08, giving a base cost of

5,000.

Step-by-Step Solution

1

Express the initial retail price in terms of the base production cost (C).

Retail Price = C + 1.25C = 2.25C

A 125% markup means adding 125% of the cost to the original 100% cost, resulting in a multiplier of 2.25.

2

Apply the first discount (40%) to the retail price.

Trade Show Price = 2.25C × (1 - 0.40) = 2.25C × 0.60 = 1.35C

A 40% discount means the buyer pays 60% of the current price. Discounts are multiplicative.

3

Apply the second discount (20%) to the trade show price.

Final Partner Price = 1.35C × (1 - 0.20) = 1.35C × 0.80 = 1.08C

Successive discounts apply to the most recently discounted amount, not the original price.

4

Set up an equation for the final price and solve for the base cost (C).

1.08C = 5400 → C = 5400 / 1.08 = 5000

This reverse-calculates the initial unknown variable using the final compound multiplier and the given final amount.

5

Determine the net percentage increase from the final compound multiplier.

Multiplier is 1.08, which represents 100% + 8%. Thus, the net increase is 8%.

The coefficient of C directly indicates what percentage the final price is of the original cost.

Key Concept

Successive Percentage Adjustments and Reverse Calculation

Estimated Time:3m 0s

Question 38Question →

A numerical value decreases from $80$ to $64$ . Which of the following represents the percentage of decrease?

25\%

80\%

20\%

0.2\%

Show answer & explanation

Answer: $20\%$

Answer

The correct answer is

20\%

.

The percentage decrease is found by dividing the absolute change by the original amount. The value decreased by 16 (from 80 to 64). Dividing 16 by the original value of 80 gives 0.20, which converts to 20%.

Step-by-Step Solution

1

Calculate the absolute decrease in the value.

80 - 64 = 16

To find the percentage decrease, the total amount of change must first be determined.

2

Divide the change by the original starting value.

\frac{16}{80} = 0.20

Percentage change is always based on the initial value, not the final value.

3

Convert the resulting decimal to a percentage.

0.20 \times 100\% = 20\%

Multiplying by 100 converts the decimal proportion into a percentage format.

Key Concept

Percentage Decrease

Estimated Time:45s

Question 39Question →

The greatest common factor (GCF) of two positive integers, $x$ and $y$ , is $15$ , and their least common multiple (LCM) is $3,150$ . If $x = 150$ , which of the following represents the prime factorization of $y$ ?

3^2 \times 5 \times 7

2 \times 3 \times 5 \times 7

3 \times 7

2 \times 3^2 \times 5^2 \times 7

Show answer & explanation

Answer: $3^2 \times 5 \times 7$

Answer

The prime factorization of y is

3^2 \times 5 \times 7

.

The correct answer can be found by comparing prime exponents. The exponent of a prime in the GCF is the minimum of its exponents in

x

and

y

, while in the LCM it is the maximum. Given

x = 2^1 \times 3^1 \times 5^2

,

\text{GCF} = 2^0 \times 3^1 \times 5^1

, and

\text{LCM} = 2^1 \times 3^2 \times 5^2 \times 7^1

. For prime 2, the min is 0, so

y

must have

2^0

. For prime 3, the max is 2, so

y

must have

3^2

. For prime 5, the min is 1, so

y

must have

5^1

. For prime 7, the max is 1, so

y

must have

7^1

. Combining these yields

3^2 \times 5 \times 7

.

Step-by-Step Solution

1

Find the prime factorizations of

x

, the GCF, and the LCM.

x = 150 = 2^1 \times 3^1 \times 5^2

\text{GCF} = 15 = 2^0 \times 3^1 \times 5^1

\text{LCM} = 3,150 = 2^1 \times 3^2 \times 5^2 \times 7^1

Comparing the exponents of prime factors between the numbers, their GCF, and their LCM allows us to deduce the exact prime factorization of

y

.

2

Determine the necessary exponent for the prime factor

2

in

y

.

The exponent for

2

in

y

must be

0

.

The GCF exponent is the minimum between

x

and

y

. Since the GCF has

2^0

and

x

has

2^1

,

y

must provide the minimum of

0

, meaning

y

does not contain a factor of

2

.

3

Determine the necessary exponent for the prime factor

3

in

y

.

The exponent for

3

in

y

must be

2

.

The LCM exponent is the maximum between

x

and

y

. Since the LCM has

3^2

and

x

has

3^1

,

y

must provide the maximum of

2

, so it has

3^2

.

4

Determine the necessary exponents for the prime factors

5

and

7

in

y

.

The exponent for

5

is

1

, and the exponent for

7

is

1

.

For

5

: The GCF has

5^1

and

x

has

5^2

, so

y

must provide the minimum

1

. For

7

: The LCM has

7^1

and

x

has

7^0

, so

y

must provide the maximum

1

.

5

Combine the deduced prime factors to form the final prime factorization of

y

.

y = 2^0 \times 3^2 \times 5^1 \times 7^1 = 3^2 \times 5 \times 7

Multiplying the prime powers gives the complete prime factorization of

y

.

Key Concept

Using prime factorizations to determine relationships between numbers, their Greatest Common Factor (GCF), and their Least Common Multiple (LCM).

Alternative Method

Instead of using the prime factorization minimum/maximum rules, you can use the mathematical property that the product of two positive integers is equal to the product of their GCF and LCM. Therefore,

x \times y = \text{GCF} \times \text{LCM}

. Substituting the given values:

150 \times y = 15 \times 3,150

. This simplifies to

150y = 47,250

. Solving for

y

gives

y = 315

. Finding the prime factorization of

315

yields

3^2 \times 5 \times 7

.

Estimated Time:2m 0s

Question 40Question →

Consider the mathematical expression below:

-4^2 - \frac{36 \div (-3) \times 2}{|1 - 4|}

What is the exact evaluated value of this expression?

Show answer & explanation

Answer: -8

Answer

-8

The correct answer is derived by carefully following the order of operations (PEMDAS/BODMAS). First, the denominator's absolute value evaluates to 3. Second, evaluating the numerator strictly from left to right gives

36 \div (-3) = -12

, and

-12 \times 2 = -24

. The entire fraction therefore simplifies to

-24 \div 3 = -8

. Next, the exponent

-4^2

evaluates to

-16

because the negative sign is not inside parentheses. Finally, subtracting the fraction's value yields

-16 - (-8)

, which simplifies to

-16 + 8 = -8

.

Step-by-Step Solution

1

Evaluate the absolute value grouping in the denominator.

|1 - 4| = |-3| = 3

Absolute value bars act as grouping symbols and must be evaluated first. Absolute value always returns a non-negative distance.

2

Evaluate the numerator from left to right.

36 \div (-3) \times 2 = -12 \times 2 = -24

Multiplication and division have equal precedence, so they must be evaluated strictly from left to right.

3

Simplify the fraction by dividing the numerator by the denominator.

\frac{-24}{3} = -8

The fraction bar acts as a grouping symbol separating the numerator and denominator.

4

Evaluate the exponent term.

-4^2 = -16

Without parentheses around the base, the exponent applies only to the 4, not the negative sign.

5

Perform the final subtraction.

-16 - (-8) = -16 + 8 = -8

Subtracting a negative number is equivalent to adding its positive value.

Key Concept

Applying the order of operations (PEMDAS), specifically evaluating multiplication/division from left to right, processing absolute values, and understanding how exponents apply to negative bases without parentheses.

Number Theory and Operations

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