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

Which of the following represents the correct evaluation of the mathematical expression given below?

-10 + 4 \times (6 - 8)

-18

12

-2

18

Show answer & explanation

Answer: -18

Answer

-18

The correct evaluation requires solving the parentheses first to obtain

-2

. Next, multiplying

4

by

-2

yields

-8

. Finally, adding

-8

to the initial

-10

results in

-18

.

Step-by-Step Solution

1

Evaluate the expression inside the parentheses.

6 - 8 = -2

According to the order of operations, operations enclosed in parentheses must be completed first.

2

Perform the multiplication.

4 \times (-2) = -8

Multiplication takes precedence over addition.

3

Perform the final addition.

-10 + (-8) = -18

Addition is the only remaining operation.

Key Concept

Order of Operations

Question 2Question →

A fuel tank on a generator is initially $\frac{7}{8}$ full. After the generator consumes $15$ gallons of fuel, the tank is exactly $\frac{1}{4}$ full. What is the total capacity of the fuel tank in gallons?

Show answer & explanation

Answer: 24

Answer

The total capacity of the fuel tank is 24 gallons.

The correct answer is derived by finding the difference in the fraction of the tank's capacity (

\frac{7}{8} - \frac{1}{4} = \frac{5}{8}

). Since this

\frac{5}{8}

portion corresponds to

15

gallons, the total capacity is found by dividing

15

by

\frac{5}{8}

, which yields

24

gallons.

Step-by-Step Solution

1

Find a common denominator for the given fractions.

\frac{1}{4} = \frac{2}{8}

To subtract the fractions, they must have the same denominator.

2

Calculate the fraction of the tank that was consumed by the generator.

\frac{7}{8} - \frac{2}{8} = \frac{5}{8}

The difference between the initial and final fuel levels represents the exact amount consumed.

3

Set up an algebraic equation linking the fraction consumed to the actual volume in gallons.

\frac{5}{8} \times \text{Capacity} = 15

We know that

\frac{5}{8}

of the total capacity corresponds directly to the

15

gallons used.

4

Solve the equation for the total capacity.

15 \times \frac{8}{5} = 24

Multiplying by the reciprocal of the fraction isolates the capacity variable.

Key Concept

Applying fraction subtraction with unlike denominators and solving a missing-value equation using a reciprocal.

Question 3Question →

Evaluate the following expression involving fractions:

-\frac{2}{3} + \frac{1}{4}

What is the value of the expression in simplest form?

-\frac{11}{12}

-\frac{1}{7}

-\frac{5}{12}

\frac{5}{12}

Show answer & explanation

Answer: $-\frac{5}{12}$

Answer

The correct value is

-\frac{5}{12}

.

The correct answer is found by first determining the least common denominator, which is 12. The expression is rewritten as

-\frac{8}{12} + \frac{3}{12}

. By combining the numerators (

-8 + 3

), we get

-5

, resulting in a final simplified fraction of

-\frac{5}{12}

.

Step-by-Step Solution

1

Find the least common denominator (LCD) for the fractions.

The LCD for 3 and 4 is 12.

Fractions must have the same denominator before they can be added or subtracted.

2

Rewrite both fractions as equivalent fractions with a denominator of 12.

-\frac{2}{3} = -\frac{8}{12}

and

\frac{1}{4} = \frac{3}{12}

.

Multiply the numerator and denominator of the first fraction by 4, and the second fraction by 3.

3

Add the numerators while keeping the denominator the same, applying integer sign rules.

\frac{-8 + 3}{12} = -\frac{5}{12}

.

Adding a positive 3 to a negative 8 results in a negative 5.

Key Concept

Adding fractions with unlike denominators and applying integer sign rules.

Question 4Question →

A drone is flying at an altitude of $45.5$ meters. It descends $12.75$ meters to take a photo, and then ascends $8.2$ meters. What is the new altitude of the drone?

40.95 meters

24.55 meters

33.57 meters

50.05 meters

Show answer & explanation

Answer: 40.95 meters

Answer

The new altitude of the drone is 40.95 meters.

The correct answer is found by sequentially subtracting the descent from the initial altitude (

45.5 - 12.75 = 32.75

) and then adding the ascent (

32.75 + 8.2 = 40.95

). Aligning the decimal points correctly ensures an accurate calculation.

Step-by-Step Solution

1

Identify the starting altitude and the mathematical operations for descending and ascending.

Starting altitude is

45.5

meters. Descending means subtracting (

12.75

), and ascending means adding (

8.2

).

Translating the real-world scenario into a mathematical expression ensures the correct operations are applied in the correct sequence.

2

Subtract the descent distance from the initial altitude.

45.5 - 12.75 = 32.75

This finds the drone's intermediate altitude after its initial descent.

3

Add the ascent distance to the intermediate altitude, making sure to align the decimal points.

32.75 + 8.20 = 40.95

This calculates the final altitude after the drone moves back up.

Key Concept

Decimal Operations

Question 5Question →

If $x = -3$ , what is the value of the expression $8 - 3|2x - 7|$ ?

-31

5

47

65

Show answer & explanation

Answer: -31

Answer

-31

The expression is evaluated using the correct order of operations. First, the value inside the absolute value is simplified by substituting -3, yielding -13. The absolute value of -13 is positive 13. Finally, multiplying 13 by 3 gives 39, and subtracting 39 from 8 results in -31.

Step-by-Step Solution

1

Substitute -3 for x in the expression.

8 - 3|2(-3) - 7|

To evaluate the expression for the specific variable value.

2

Multiply the numbers inside the absolute value.

8 - 3|-6 - 7|

Following the order of operations, calculations grouped inside absolute value bars must be simplified first.

3

Subtract the remaining numbers inside the absolute value.

8 - 3|-13|

Continuing to simplify the contents within the absolute value grouping.

4

Evaluate the absolute value of -13.

8 - 3(13)

The absolute value of any non-zero number is its positive distance from zero.

5

Multiply 3 by 13.

8 - 39

According to the order of operations (PEMDAS), multiplication must be performed before subtraction.

6

Subtract 39 from 8.

-31

This is the final operation to find the total value of the expression.

Key Concept

Evaluating absolute value expressions with order of operations

Question 6Question →

A carpenter is sorting four wooden planks based on their lengths to construct a custom frame. The lengths of the planks, in feet, are listed below. Sequence the plank lengths from shortest to longest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending sequence is

1\frac{1}{3}

feet,

1\frac{3}{8}

feet,

\frac{7}{5}

feet, and

\frac{17}{12}

feet.

To compare fractions with different denominators, they must be converted to a common denominator. The least common multiple of

3

,

8

,

5

, and

12

is

120

. Converting all values yields

\frac{160}{120}

,

\frac{165}{120}

,

\frac{168}{120}

, and

\frac{170}{120}

. Arranging these from smallest to largest gives the proper sequence of

1\frac{1}{3}

feet,

1\frac{3}{8}

feet,

\frac{7}{5}

feet, and

\frac{17}{12}

feet.

Step-by-Step Solution

1

Convert all mixed numbers to improper fractions.

1\frac{3}{8} = \frac{11}{8}

and

1\frac{1}{3} = \frac{4}{3}

.

Fractions must be in a consistent improper format to find a common denominator efficiently.

2

Find the least common multiple (LCM) for the denominators

8

,

12

,

3

, and

5

.

The LCM of

8

,

12

,

3

, and

5

is

120

.

A common denominator is required to directly compare the relative sizes of the fractions.

3

Convert each fraction to an equivalent fraction with a denominator of

120

.

\frac{4}{3} = \frac{160}{120}

,

\frac{11}{8} = \frac{165}{120}

,

\frac{7}{5} = \frac{168}{120}

, and

\frac{17}{12} = \frac{170}{120}

.

Normalizing the denominators isolates the variation to the numerators, making comparison straightforward.

4

Arrange the fractions by comparing their normalized numerators from smallest to largest.

160 < 165 < 168 < 170

, which corresponds to the original values

1\frac{1}{3} < 1\frac{3}{8} < \frac{7}{5} < \frac{17}{12}

.

The problem specifies sequencing the plank lengths from shortest to longest.

Key Concept

Ordering and Comparing Fractions

Alternative Method

Convert each fraction to its decimal equivalent:

1\frac{1}{3} \approx 1.333

,

1\frac{3}{8} = 1.375

,

\frac{7}{5} = 1.400

, and

\frac{17}{12} \approx 1.417

. The decimals can then be easily ordered from least to greatest.

Estimated Time:1m 0s

Question 7Question →

A quality control inspector is verifying the thickness of four different steel plates. Arrange the following plate thicknesses in order from thinnest to thickest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from thinnest to thickest is

1\frac{5}{16}

inches,

1\frac{11}{32}

inches,

1\frac{3}{8}

inches, and

\frac{23}{16}

inches.

To accurately compare these lengths, they must all be expressed with a common denominator. The lowest common denominator for 8, 16, and 32 is 32. Converting the values yields

1\frac{5}{16} = \frac{42}{32}

,

1\frac{11}{32} = \frac{43}{32}

,

1\frac{3}{8} = \frac{44}{32}

, and

\frac{23}{16} = \frac{46}{32}

. Ordering by the numerators from lowest to highest gives the sequence

1\frac{5}{16}

,

1\frac{11}{32}

,

1\frac{3}{8}

, and

\frac{23}{16}

.

Step-by-Step Solution

1

Convert all mixed numbers to improper fractions.

1\frac{3}{8} = \frac{11}{8}

;

1\frac{5}{16} = \frac{21}{16}

;

1\frac{11}{32} = \frac{43}{32}

. The fraction

\frac{23}{16}

is already in improper form.

Working with improper fractions simplifies the process of finding a common denominator.

2

Determine the lowest common denominator (LCD) for the fractions

\frac{11}{8}

,

\frac{23}{16}

,

\frac{21}{16}

, and

\frac{43}{32}

.

The LCD for 8, 16, and 32 is 32.

A common denominator is required to accurately compare the absolute sizes of the fractions.

3

Convert each fraction into an equivalent fraction with a denominator of 32.

\frac{11}{8} = \frac{44}{32}

;

\frac{23}{16} = \frac{46}{32}

;

\frac{21}{16} = \frac{42}{32}

;

\frac{43}{32}

remains

\frac{43}{32}

.

This normalization allows for direct comparison of the numerators to determine size.

4

Order the numerators from smallest to largest and map them back to their original forms.

The numerators in order are

42 < 43 < 44 < 46

. This corresponds to

\frac{42}{32}

(

1\frac{5}{16}

),

\frac{43}{32}

(

1\frac{11}{32}

),

\frac{44}{32}

(

1\frac{3}{8}

), and

\frac{46}{32}

(

\frac{23}{16}

).

When the denominators are identical, the magnitude of the fraction is directly proportional to its numerator.

Key Concept

Comparing and Ordering Fractions

Question 8Question →

A logistics specialist is organizing four replacement fuel hoses by length. The lengths are $\frac{17}{4}$ meters, $4 \frac{3}{8}$ meters, $\frac{14}{3}$ meters, and $4 \frac{5}{12}$ meters. Arrange these hose lengths in ascending order from shortest to longest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order is

\frac{17}{4}

meters,

4 \frac{3}{8}

meters,

4 \frac{5}{12}

meters, and

\frac{14}{3}

meters.

By converting all values to mixed numbers with a common denominator of 24, the fractional parts become

\frac{6}{24}

,

\frac{9}{24}

,

\frac{10}{24}

, and

\frac{16}{24}

. Ordering these from smallest to largest provides the correct sequence.

Step-by-Step Solution

1

Convert the improper fractions to mixed numbers for easier comparison.

\frac{17}{4} = 4 \frac{1}{4}

and

\frac{14}{3} = 4 \frac{2}{3}

.

Comparing values is simpler when whole numbers and fractional parts are separated.

2

Identify the fractional parts to be compared.

Since all lengths share a whole number of 4, we only need to compare the fractions:

\frac{1}{4}

,

\frac{3}{8}

,

\frac{5}{12}

, and

\frac{2}{3}

.

The whole numbers are identical, so the overall size depends entirely on the fractional parts.

3

Find a common denominator for the fractions 4, 8, 12, and 3.

The least common multiple (LCM) of the denominators 4, 8, 12, and 3 is 24.

A common denominator is required to directly and accurately compare the sizes of the numerators.

4

Convert each fraction to an equivalent fraction with a denominator of 24.

\frac{1}{4} = \frac{6}{24}

;

\frac{3}{8} = \frac{9}{24}

;

\frac{5}{12} = \frac{10}{24}

;

\frac{2}{3} = \frac{16}{24}

.

This standardizes the fractions so their relative sizes are visually clear.

5

Order the fractions from smallest to largest based on their numerators.

\frac{6}{24} < \frac{9}{24} < \frac{10}{24} < \frac{16}{24}

, which corresponds to

\frac{17}{4} < 4 \frac{3}{8} < 4 \frac{5}{12} < \frac{14}{3}

.

Ordering the standardized fractions reveals the correct ascending sequence for the original lengths.

Key Concept

Comparing and ordering mixed numbers and improper fractions by finding a common denominator.

Question 9Question →

Which of the following represents the correct evaluation of the mathematical expression below?

-1.5 - 2.4 \times (6.2 - 3.45)

-8.1

-10.725

-8.34

5.1

Show answer & explanation

Answer: -8.1

Answer

The correct answer is -8.1.

The correct evaluation requires following the exact order of operations. First, correctly align the decimals to subtract inside the parentheses (

6.20 - 3.45 = 2.75

). Second, multiply that result by

2.4

to get

6.6

. Finally, calculate

-1.5 - 6.6

, which correctly yields

-8.1

.

Step-by-Step Solution

1

Evaluate the expression inside the parentheses.

6.20 - 3.45 = 2.75

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

6.2

to properly align the decimals.

2

Perform the multiplication.

-2.4 \times 2.75 = -6.6

Multiplication takes precedence over addition and subtraction. The product of a negative and a positive number is negative.

3

Perform the final subtraction.

-1.5 - 6.6 = -8.1

Combine the remaining terms. Subtracting a positive number from a negative number moves further left on the number line.

Key Concept

Order of Operations with Decimals

Question 10Question →

Arrange the following mathematical expressions in ascending order, from the least value to the greatest value.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest is:

1.2 \times 0.6

, then

0.35 + 0.38

, then

4.2 - 3.45

, and finally

0.48 \div 0.6

.

Each arithmetic expression must be evaluated independently applying the correct decimal rules for that specific operation. The results are

0.72

,

0.73

,

0.75

, and

0.80

. Sorting these decimal values chronologically by magnitude yields the correct ascending sequence.

Step-by-Step Solution

1

Evaluate the multiplication expression:

1.2 \times 0.6

0.72

Multiply the whole numbers (

12 \times 6 = 72

) and count a total of two decimal places from the factors.

2

Evaluate the addition expression:

0.35 + 0.38

0.73

Align the decimal points vertically and add the hundredths, then the tenths.

3

Evaluate the subtraction expression:

4.2 - 3.45

0.75

Rewrite

4.2

as

4.20

to align the decimal points properly, then subtract.

4

Evaluate the division expression:

0.48 \div 0.6

0.80

Multiply both numbers by

10

to make the divisor a whole number (

4.8 \div 6

), then divide.

5

Order the evaluated results from least to greatest.

0.72 < 0.73 < 0.75 < 0.80

Comparing the values digit by digit from the tenths place to the hundredths place establishes the correct magnitude sequence.

Key Concept

Evaluating all four basic operations with decimals and comparing their relative magnitudes.

Question 11Question →

A carpenter is sorting four wooden dowels of different diameters. Order the dowel diameters from thinnest to thickest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from thinnest to thickest is

\frac{5}{8}

inch,

\frac{3}{4}

inch,

1 \frac{1}{4}

inches, and

\frac{3}{2}

inches.

By converting all fractions and mixed numbers to have a common denominator of

8

, we get

\frac{6}{8}

,

\frac{5}{8}

,

\frac{12}{8}

, and

\frac{10}{8}

. Ordering these by numerator from least to greatest gives

\frac{5}{8}

,

\frac{6}{8}

,

\frac{10}{8}

,

\frac{12}{8}

, which corresponds to the original values

\frac{5}{8}

,

\frac{3}{4}

,

1 \frac{1}{4}

,

\frac{3}{2}

.

Step-by-Step Solution

1

Identify a common denominator to compare all fractions easily.

The denominators are

4

,

8

,

2

, and

4

. The least common multiple for these is

8

.

Fractions must have the same denominator to compare their sizes accurately.

2

Convert each measurement to an equivalent fraction with a denominator of

8

.

\frac{3}{4} = \frac{6}{8}

,

\frac{5}{8} = \frac{5}{8}

,

\frac{3}{2} = \frac{12}{8}

, and

1 \frac{1}{4} = \frac{5}{4} = \frac{10}{8}

.

This allows for direct comparison of the numerators.

3

Order the converted fractions by comparing their numerators from smallest to largest.

Comparing the numerators (

5 < 6 < 10 < 12

), the order is

\frac{5}{8}

,

\frac{6}{8}

,

\frac{10}{8}

,

\frac{12}{8}

.

When denominators are equal, the fraction with the smaller numerator represents a smaller value.

4

Match the ordered fractions back to their original given forms.

The final order is

\frac{5}{8}

inch,

\frac{3}{4}

inch,

1 \frac{1}{4}

inches, and

\frac{3}{2}

inches.

The question asks to order the original measurements.

Key Concept

Comparing and Ordering Fractions

Alternative Method

Convert all fractions to decimals to compare them:

\frac{3}{4} = 0.75

,

\frac{5}{8} = 0.625

,

\frac{3}{2} = 1.5

, and

1 \frac{1}{4} = 1.25

. Then order the decimals:

0.625, 0.75, 1.25, 1.5

.

Estimated Time:45s

Question 12Question →

If $x$ and $y$ are both negative integers, the value of $x - y$ must also be a negative integer.

True

False

Show answer & explanation

Answer: False

Answer

False

The statement is false. When evaluating

x - y

with negative integers, the operation simplifies to adding a positive number. If the magnitude of

y

is larger than the magnitude of

x

, the result is positive. For instance, if

x = -1

and

y = -5

, then

-1 - (-5) = 4

.

Step-by-Step Solution

1

Analyze the expression

x - y

given that both variables represent negative integers.

Since

y

is negative, subtracting

y

is identical to adding the absolute value of

y

. The expression becomes

x + |y|

.

Understanding how double negatives simplify is essential for evaluating integer properties.

2

Test the statement using a counterexample where

y

is further left on the number line than

x

.

Assume

x = -3

and

y = -10

. Substitute these integers into the original expression:

-3 - (-10)

.

A universally phrased statement containing the word 'must' can be entirely disproven with just a single counterexample.

3

Evaluate the chosen counterexample.

-3 - (-10) = -3 + 10 = 7

. Because

7

is a positive integer, the expression does not always yield a negative result.

The positive outcome proves the initial statement is definitively false.

Key Concept

Integer Subtraction and Sign Rules

Estimated Time:35s

Question 13Question →

A high-efficiency water filter is designed to block microscopic particles that are $0.00072$ meters in diameter. How is this measurement expressed in scientific notation?

7.2 \times 10^{-3}

7.2 \times 10^{-4}

7.2 \times 10^4

72 \times 10^{-5}

Show answer & explanation

Answer: $7.2 \times 10^{-4}$

Answer

The correct measurement in scientific notation is

7.2 \times 10^{-4}

.

To convert

0.00072

to scientific notation, the decimal point must be moved

4

places to the right to create a valid coefficient of

7.2

. Because the original number is less than

1

, the exponent is negative, resulting in

7.2 \times 10^{-4}

.

Step-by-Step Solution

1

Identify the non-zero digits to create a coefficient that is greater than or equal to

1

and less than

10

.

The coefficient is

7.2

.

Scientific notation requires a single non-zero digit to the left of the decimal point.

2

Count the number of places the decimal point must move to change the original number (

0.00072

) into the coefficient (

7.2

).

The decimal point moves

4

places to the right.

The number of places moved determines the numerical value of the exponent.

3

Determine the sign of the exponent based on the original number's size.

The exponent is

-4

.

Because the original number (

0.00072

) is a decimal less than

1

, the exponent must be negative.

Key Concept

Converting small decimal numbers into scientific notation using negative exponents.

Question 14Question →

A painter has $\frac{3}{4}$ of a gallon of blue paint. If they use exactly $\frac{1}{3}$ of this paint to complete a small project, how many gallons of paint did they use? (Express your answer as a decimal.)

Show answer & explanation

Answer: 0.25

Answer

0.25

To find

\frac{1}{3}

of

\frac{3}{4}

, you must multiply the two fractions together. Calculating

\frac{3}{4} \times \frac{1}{3}

results in

\frac{3}{12}

. Simplifying by dividing the numerator and denominator by 3 gives

\frac{1}{4}

. Finally, converting

\frac{1}{4}

to a decimal yields 0.25.

Step-by-Step Solution

1

Identify the mathematical operation

Multiplication:

\frac{3}{4} \times \frac{1}{3}

The phrase '

\frac{1}{3}

of this paint' indicates finding a part of a part, which requires multiplication.

2

Multiply the fractions

\frac{3}{12}

Multiply the numerators together (

3 \times 1 = 3

) and the denominators together (

4 \times 3 = 12

).

3

Simplify the resulting fraction

\frac{1}{4}

Both

3

and

12

share a greatest common factor of

3

. Dividing both by

3

simplifies the fraction.

4

Convert the simplified fraction to a decimal

0.25

The question specifically requests the final answer to be expressed as a decimal.

Key Concept

Fraction multiplication and decimal conversion

Question 15Question →

A landscaper is combining two partial bags of fertilizer. The first bag contains $\frac{3}{4}$ of a pound, and the second bag contains $\frac{1}{6}$ of a pound. What is the total weight of the fertilizer when combined?

\frac{2}{5}

\frac{1}{8}

\frac{11}{12}

\frac{1}{6}

Show answer & explanation

Answer: $\frac{11}{12}$

Answer

The total weight is

\frac{11}{12}

of a pound.

To add fractions with different denominators, a common denominator must first be found. The least common multiple of

4

and

6

is

12

. Converting both fractions to have a denominator of

12

yields

\frac{9}{12}

and

\frac{2}{12}

. Adding the numerators while keeping the denominator the same gives the correct sum of

\frac{11}{12}

.

Step-by-Step Solution

1

Find a common denominator for the fractions

\frac{3}{4}

and

\frac{1}{6}

.

The least common multiple of

4

and

6

is

12

.

Fractions must have the same denominator before they can be added or subtracted.

2

Convert both fractions to have the common denominator of

12

.

\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}

and

\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}

.

This creates equivalent fractions that represent the same values but share a denominator.

3

Add the numerators together while keeping the denominator the same.

\frac{9}{12} + \frac{2}{12} = \frac{9 + 2}{12} = \frac{11}{12}

.

Once denominators are equal, only the fractional parts (numerators) are combined.

Key Concept

Adding fractions with unlike denominators

Alternative Method

You can also use the cross-multiplication method:

\frac{(3 \times 6) + (1 \times 4)}{4 \times 6} = \frac{18 + 4}{24} = \frac{22}{24}

. Then, simplify the fraction by dividing both the numerator and denominator by

2

to get

\frac{11}{12}

.

Estimated Time:45s

Question 16Question →

Evaluate the following expression:

12 - 2[3^2 - (5 - 8)] + 4

What is the value of the expression?

-8

4

-16

124

Show answer & explanation

Answer: -8

Answer

-8

Following the order of operations, the innermost parentheses (5 - 8) evaluate to -3. The exponent 3^2 evaluates to 9. The bracket becomes [9 - (-3)], which simplifies to 12. Multiplying this by 2 yields 24. Finally, evaluating 12 - 24 + 4 from left to right gives -12 + 4, which equals -8.

Step-by-Step Solution

1

Evaluate the innermost parentheses:

(5 - 8)

.

(5 - 8) = -3

According to the order of operations, innermost grouping symbols must be resolved first.

2

Evaluate the exponent inside the bracket:

3^2

.

3^2 = 9

Exponents are evaluated before subtraction inside the grouping symbols.

3

Simplify the expression inside the bracket:

[9 - (-3)]

.

9 - (-3) = 9 + 3 = 12

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

4

Perform the multiplication:

2[12]

.

2 \times 12 = 24

Multiplication takes precedence over addition and subtraction.

5

Perform addition and subtraction from left to right:

12 - 24 + 4

.

12 - 24 = -12

, then

-12 + 4 = -8

Addition and subtraction have the same precedence and must be evaluated strictly from left to right.

Key Concept

Order of Operations (PEMDAS/BODMAS) and integer arithmetic

Question 17Question →

What is the value of the following expression?

15 - 3 \times 4 + 2

5

50

1

-1

Show answer & explanation

Answer: 5

Answer

5

The correct answer is found by properly applying the order of operations. Multiplication takes precedence over subtraction and addition, so 3 * 4 is evaluated first to get 12. The expression then becomes 15 - 12 + 2. Because subtraction and addition have the same priority, they are evaluated from left to right. Subtracting 12 from 15 gives 3, and adding 2 to 3 gives the final result of 5.

Step-by-Step Solution

1

Perform the multiplication operation first.

3 \times 4 = 12

According to the order of operations (PEMDAS), multiplication must be completed before addition and subtraction.

2

Substitute the result back into the expression and subtract.

15 - 12 = 3

Addition and subtraction are of equal priority, so they are evaluated from left to right.

3

Perform the final addition operation.

3 + 2 = 5

This is the final remaining operation in the expression.

Key Concept

Order of Operations

Question 18Question →

Arrange the following calculation steps in the correct order to evaluate the mathematical expression according to the order of operations:

15 + (6 - 2) \times 3^2

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence is to first subtract 2 from 6 (resolving the parentheses), then square the number 3 (evaluating the exponent), multiply those two results together, and finally add 15.

The correct order strictly follows the PEMDAS sequence: Parentheses (

6 - 2

) must be resolved first, followed by Exponents (

3^2

). After those are evaluated, Multiplication (

4 \times 9

) is performed. Finally, Addition (

15 + 36

) is completed.

Step-by-Step Solution

1

Evaluate the operation inside the parentheses.

15 + (4) \times 3^2

Parentheses (P) have the highest priority in the order of operations.

2

Evaluate the exponent.

15 + 4 \times 9

Exponents (E) are evaluated after parentheses but before multiplication.

3

Perform the multiplication.

15 + 36

Multiplication (M) takes precedence over addition.

4

Perform the final addition.

51

Addition (A) is the final operation left in this expression.

Key Concept

Order of Operations

Question 19Question →

Given the variables $x = -2$ , $y = -3$ , and $z = 11$ , evaluate the algebraic expression below.

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

Which of the following represents the correctly calculated result?

-45

33

-15

-21

Show answer & explanation

Answer: 33

Answer

The correct result is 33.

The correct answer shows the result of correctly substituting the negative values, applying the order of operations by evaluating the innermost parentheses and exponents first, then multiplication, subtraction, and finally division and addition.

Step-by-Step Solution

1

Substitute the given values for x, y, and z into the expression.

\frac{(-2)^3 - 2(-3)(11 - (-2))}{|(-3)^2 - 11|} + (-2)

Accurate substitution, particularly retaining negative signs inside parentheses, is the required first step before performing arithmetic operations.

2

Evaluate the innermost grouping symbols and the exponents.

11 - (-2) = 13

,

(-2)^3 = -8

, and

(-3)^2 = 9

. The expression becomes

\frac{-8 - 2(-3)(13)}{|9 - 11|} - 2

.

Following the order of operations, computations inside parentheses and exponents must be resolved before multiplication.

3

Perform the multiplication in the numerator and the subtraction within the absolute value in the denominator.

Numerator multiplication:

2(-3)(13) = -6(13) = -78

. Absolute value:

|9 - 11| = |-2| = 2

. The expression is now

\frac{-8 - (-78)}{2} - 2

.

Multiplication precedes addition and subtraction. The absolute value acts as a grouping symbol that must be evaluated before division.

4

Simplify the numerator by resolving the subtraction of a negative number.

-8 - (-78) = -8 + 78 = 70

. The expression simplifies to

\frac{70}{2} - 2

.

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

5

Perform the division, then the final subtraction.

\frac{70}{2} = 35

, and

35 - 2 = 33

.

Division must be completed before the final addition or subtraction step.

Key Concept

Evaluating complex algebraic expressions with negative numbers using the correct order of operations.

Question 20Question →

Arrange the following mathematical operations in the exact sequence they must be performed to correctly evaluate the expression below:

40 - [2 \times (3 + 4)^2] \div 7

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence is: Add the numbers inside the innermost parentheses, apply the exponent of 2, multiply the squared result by 2, divide the value of the brackets by 7, and finally subtract the quotient from 40.

Evaluating the expression following the Order of Operations (PEMDAS) requires these steps in sequence:
1) Add (3 + 4) inside the parentheses.
2) Square that sum.
3) Multiply the squared value by 2 to clear the brackets.
4) Divide that result by 7.
5) Subtract the quotient from 40.

Step-by-Step Solution

1

Identify the innermost grouping symbols (parentheses) and perform the addition.

40 - [2 \times (7)^2] \div 7

Parentheses (P) are evaluated first in the order of operations.

2

Evaluate the exponent applied to the innermost group.

40 - [2 \times 49] \div 7

Exponents (E) are evaluated immediately after parentheses.

3

Complete the operations inside the brackets by performing multiplication.

40 - [98] \div 7

The brackets act as secondary grouping symbols and must be fully simplified to a single value.

4

Perform the division located outside the brackets.

40 - 14

Between division and subtraction, division (D) always takes precedence.

5

Perform the final subtraction.

26

Subtraction (S) is evaluated last.

Key Concept

Order of Operations (PEMDAS)

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