Fraction Operations Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB)

Question 1Question →

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 2Question →

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 3Question →

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 4Question →

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 5Question →

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 6Question →

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 7Question →

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 8Question →

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 9Question →

What is the value of the following expression?

\frac{3}{4} + \frac{1}{2} \div \frac{2}{3} - \frac{1}{8}

\frac{5}{8}

1 \frac{3}{4}

1 \frac{3}{8}

1 \frac{5}{8}

Show answer & explanation

Answer: $1 \frac{3}{8}$

Answer

The correct answer is

1 \frac{3}{8}

.

The correct answer is found by first performing the division operation (which becomes

3/4

), then finding a common denominator (8) to add and subtract the remaining fractions. This yields

11/8

, which simplifies to

1 \frac{3}{8}

.

Step-by-Step Solution

1

Evaluate the division portion of the expression first according to the order of operations.

\frac{1}{2} \div \frac{2}{3} = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}

The order of operations requires division to be performed before addition or subtraction.

2

Substitute the division result back into the original expression.

\frac{3}{4} + \frac{3}{4} - \frac{1}{8}

This updates the expression to contain only addition and subtraction operations.

3

Find the least common multiple (LCM) for the denominators to perform addition and subtraction.

The LCM of 4 and 8 is 8. Converting the fractions yields

\frac{6}{8} + \frac{6}{8} - \frac{1}{8}

.

Fractions must have a common denominator to be added or subtracted.

4

Perform the addition and subtraction sequentially from left to right.

\frac{6}{8} + \frac{6}{8} - \frac{1}{8} = \frac{12}{8} - \frac{1}{8} = \frac{11}{8}

Addition and subtraction carry the same precedence and are evaluated from left to right.

5

Convert the resulting improper fraction into a mixed number.

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

The answer choices are presented as mixed numbers, requiring the improper fraction to be converted.

Key Concept

Applying the order of operations (PEMDAS) to compound fraction arithmetic.

Question 10Question →

A carpenter is cutting a wooden board that is initially $12 \frac{1}{2}$ feet long. First, he cuts off a piece that is $3 \frac{3}{4}$ feet long and sets it aside. Then, he divides the remaining board into $5$ equal sections. What is the length of each of these $5$ sections?

11 \frac{3}{4}

feet

1

foot

1 \frac{3}{4}

feet

2 \frac{1}{20}

feet

Show answer & explanation

Answer: $1 \frac{3}{4}$ feet

Answer

Each of the

5

sections has a length of

1 \frac{3}{4}

feet.

The correct sequence of operations is to first determine the remaining length by subtracting

3 \frac{3}{4}

from

12 \frac{1}{2}

, yielding

8 \frac{3}{4}

feet. Then, dividing this remaining length by

5

results in

1 \frac{3}{4}

feet per section.

Step-by-Step Solution

1

Calculate the length of the remaining board by subtracting the cut piece from the total length.

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

The carpenter first removes a piece, leaving a smaller board to be divided.

2

Find a common denominator to perform the subtraction.

12 \frac{2}{4} - 3 \frac{3}{4}

Fractions must have the same denominator to be subtracted.

3

Borrow from the whole number to subtract the fractions.

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

feet

Since

\frac{2}{4}

is less than

\frac{3}{4}

, one whole unit must be converted into

\frac{4}{4}

.

4

Divide the remaining length by 5 to find the length of each section.

8 \frac{3}{4} \div 5

The remaining board is split into 5 equal parts.

5

Convert the mixed number to an improper fraction and multiply by the reciprocal of 5.

\frac{35}{4} \times \frac{1}{5} = \frac{7}{4} = 1 \frac{3}{4}

feet

Division by a whole number is equivalent to multiplying by its reciprocal.

Key Concept

Solving multi-step problems involving subtraction and division of mixed numbers with unlike denominators.

Question 11Question →

Arrange the following fractions and mixed numbers in ascending order (from least to greatest).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order is

2 \frac{7}{12}

,

2 \frac{5}{8}

,

\frac{8}{3}

, and

\frac{11}{4}

.

By converting all values to mixed numbers with a common denominator of 24, we find their fractional parts are

\frac{14}{24}

,

\frac{15}{24}

,

\frac{16}{24}

, and

\frac{18}{24}

. Because all whole numbers are 2, the order of the original fractions matches the ascending order of these fractional parts.

Step-by-Step Solution

1

Convert the improper fractions to mixed numbers to simplify comparison.

\frac{11}{4} = 2 \frac{3}{4}

and

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

. The other numbers are already mixed numbers:

2 \frac{5}{8}

and

2 \frac{7}{12}

.

Because all the whole numbers are 2, we only need to compare their fractional parts to determine the order.

2

Identify the least common denominator (LCD) for the fractional parts:

\frac{3}{4}

,

\frac{5}{8}

,

\frac{7}{12}

, and

\frac{2}{3}

.

The denominators are 4, 8, 12, and 3. The lowest number they all divide into evenly is 24.

A common denominator is required to accurately compare the sizes of fractions with unlike denominators.

3

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

\frac{3}{4} = \frac{18}{24}

,

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

,

\frac{7}{12} = \frac{14}{24}

, and

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

.

Multiplying both the numerator and denominator by the same value preserves the fraction's overall mathematical value.

4

Order the converted fractions from least to greatest based on their numerators.

\frac{14}{24} < \frac{15}{24} < \frac{16}{24} < \frac{18}{24}

.

When denominators are equal, a larger numerator represents a larger overall value.

5

Match the ordered fractions back to their original given forms.

2 \frac{7}{12}

(from

\frac{14}{24}

),

2 \frac{5}{8}

(from

\frac{15}{24}

),

\frac{8}{3}

(from

\frac{16}{24}

), and

\frac{11}{4}

(from

\frac{18}{24}

).

The final answer must present the exact numbers provided in the original stem.

Key Concept

Comparing and Ordering Fractions with Unlike Denominators

Question 12Question →

A carpenter has a wooden board that is $5 \frac{3}{4}$ inches wide. If he shaves off exactly $\frac{1}{8}$ of an inch from the width, what is the new width of the board?

5 \frac{5}{8}

inches

5 \frac{1}{2}

inches

5 \frac{7}{8}

inches

5 \frac{1}{4}

inches

Show answer & explanation

Answer: $5 \frac{5}{8}$ inches

Answer

The new width of the board is

5 \frac{5}{8}

inches.

To find the new width, subtract the shaved amount from the original width:

5 \frac{3}{4} - \frac{1}{8}

. First, find a common denominator, which is 8. Convert

\frac{3}{4}

to

\frac{6}{8}

. Subtracting

\frac{1}{8}

from

5 \frac{6}{8}

gives the correct answer of

5 \frac{5}{8}

inches.

Step-by-Step Solution

1

Set up the subtraction problem based on the context.

5 \frac{3}{4} - \frac{1}{8}

The phrase 'shaves off' indicates subtraction from the original width.

2

Find a common denominator for the fractions

\frac{3}{4}

and

\frac{1}{8}

.

The common denominator is

8

.

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

3

Convert

\frac{3}{4}

to an equivalent fraction with a denominator of

8

.

\frac{3 \times 2}{4 \times 2} = \frac{6}{8}

Multiplying both the numerator and denominator by

2

keeps the fraction's value the same while converting the denominator.

4

Subtract the fractions.

5 \frac{6}{8} - \frac{1}{8} = 5 \frac{5}{8}

Subtract the numerators while keeping the common denominator the same.

Key Concept

Subtracting mixed numbers with unlike denominators

Question 13Question →

A mechanical engineer is determining the load factor for a support beam. The calculation requires evaluating the equation below.

\text{Load Factor} = \left( 5 \frac{5}{8} - 2 \frac{3}{4} \right) \div \left( 1 \frac{1}{3} \times \frac{3}{8} \right)

What is the final load factor? Express your answer as an exact decimal.

Show answer & explanation

Answer: 5.75

Answer

The final load factor is exactly 5.75.

The expression evaluates to exactly 5.75. This is found by first converting all mixed numbers to improper fractions, calculating the values inside the parentheses (

\frac{23}{8}

and

\frac{1}{2}

respectively), and then performing fraction division by multiplying by the reciprocal (

\frac{23}{8} \times 2 = \frac{23}{4} = 5.75

).

Step-by-Step Solution

1

Convert the mixed numbers in the expression to improper fractions.

5 \frac{5}{8} = \frac{45}{8}

,

2 \frac{3}{4} = \frac{11}{4}

, and

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

.

Fraction multiplication and division require working with improper fractions rather than mixed numbers.

2

Perform the subtraction inside the first set of parentheses:

\frac{45}{8} - \frac{11}{4}

.

Convert

\frac{11}{4}

to

\frac{22}{8}

. Then,

\frac{45}{8} - \frac{22}{8} = \frac{23}{8}

.

To subtract fractions, they must have a common denominator. The lowest common denominator for 8 and 4 is 8.

3

Perform the multiplication inside the second set of parentheses:

\frac{4}{3} \times \frac{3}{8}

.

Multiply the numerators and denominators:

\frac{4 \times 3}{3 \times 8} = \frac{12}{24}

. This simplifies to

\frac{1}{2}

.

Following the order of operations (PEMDAS), operations inside parentheses must be evaluated first.

4

Divide the result of the first parentheses by the result of the second parentheses.

\frac{23}{8} \div \frac{1}{2} = \frac{23}{8} \times \frac{2}{1} = \frac{46}{8}

.

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

5

Simplify the final fraction and convert it to a decimal format.

\frac{46}{8} = \frac{23}{4} = 5 \frac{3}{4}

. As a decimal,

5 \frac{3}{4} = 5.75

.

The question specifically requests the final load factor expressed as a decimal.

Key Concept

Compound Fraction Operations and Order of Operations

Question 14Question →

Arrange the following fractions and mixed numbers in ascending order (from smallest to largest).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order is

\frac{7}{8}

,

1 \frac{1}{4}

,

\frac{3}{2}

, and

1 \frac{3}{4}

.

The proper fraction

\frac{7}{8}

is the only value less than 1, placing it first. Converting

\frac{3}{2}

to

1 \frac{1}{2}

(or

1.5

) allows easy comparison with

1 \frac{1}{4}

(1.25) and

1 \frac{3}{4}

(1.75), confirming the correct ascending sequence.

Step-by-Step Solution

1

Identify values less than 1.

\frac{7}{8}

is the only proper fraction (less than 1), making it the smallest.

Proper fractions are always less than mixed numbers or improper fractions.

2

Convert the improper fraction to a mixed number to make comparison easier.

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

.

Matching the format of the other numbers allows for a direct comparison of the fractional parts.

3

Find a common denominator for the fractional parts of the values greater than 1 (

1 \frac{1}{4}

,

1 \frac{1}{2}

, and

1 \frac{3}{4}

).

The common denominator is 4.

1 \frac{1}{2}

becomes

1 \frac{2}{4}

.

Common denominators allow you to directly compare the numerators.

4

Order the mixed numbers by their fractional parts.

1 \frac{1}{4}

is smaller than

1 \frac{2}{4}

(which is

\frac{3}{2}

), and

1 \frac{2}{4}

is smaller than

1 \frac{3}{4}

.

Since the whole numbers are identical, the fractions determine the order.

Key Concept

Ordering mixed numbers and improper fractions

Question 15Question →

A supply clerk is organizing copper wire. A spool initially contains $15\frac{4}{5}$ meters of wire. If a technician uses $6\frac{1}{2}$ meters of wire from the spool, how many meters of wire remain? Express your answer as a decimal.

Show answer & explanation

Answer: 9.3

Answer

9.3

The initial length of the wire is

15\frac{4}{5}

meters and the used portion is

6\frac{1}{2}

meters. By converting these to decimals, you get

15.8

meters and

6.5

meters. Subtracting the used portion from the initial portion yields

15.8 - 6.5 = 9.3

meters. Alternatively, using a common denominator of 10 gives

15\frac{8}{10} - 6\frac{5}{10} = 9\frac{3}{10}

, which also equals 9.3.

Step-by-Step Solution

1

Convert the mixed numbers to decimals or find a common denominator.

15\frac{4}{5} = 15.8

and

6\frac{1}{2} = 6.5

Numbers must be in a uniform format to perform subtraction easily.

2

Subtract the length of wire used from the initial length.

15.8 - 6.5 = 9.3

The remaining length is the difference between the starting amount and the used amount.

Key Concept

Fraction Operations

Estimated Time:45s

Question 16Question →

An industrial bakery has a large container holding $10\frac{3}{8}$ gallons of vegetable oil. On Monday morning, a baker uses $3\frac{1}{2}$ gallons for a batch of muffins. In the afternoon, they use an additional $2\frac{3}{4}$ gallons for bread dough. How many gallons of vegetable oil remain in the container? (Enter your answer as a decimal.)

Show answer & explanation

Answer: 4.125

Answer

4.125

Converting all fractions to a common denominator of 8 yields

10\frac{3}{8}

total,

3\frac{4}{8}

used first, and

2\frac{6}{8}

used second. Total used is

6\frac{2}{8}

. Subtracting this from

10\frac{3}{8}

leaves

4\frac{1}{8}

. Converting

\frac{1}{8}

to a decimal yields 0.125, making the final answer 4.125.

Step-by-Step Solution

1

Find a common denominator for all the fractions involved.

The common denominator for 8, 2, and 4 is 8.

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

2

Convert the amounts of oil used into mixed numbers with the common denominator of 8.

3\frac{1}{2} = 3\frac{4}{8}

and

2\frac{3}{4} = 2\frac{6}{8}

.

This prepares the values for addition.

3

Add the amounts used together to find the total amount of oil removed.

3\frac{4}{8} + 2\frac{6}{8} = 5\frac{10}{8} = 6\frac{2}{8}

We must consolidate the used portions before subtracting from the total.

4

Subtract the total oil used from the original amount in the container.

10\frac{3}{8} - 6\frac{2}{8} = 4\frac{1}{8}

Subtracting the total used gives the remaining oil.

5

Convert the final mixed number to a decimal format.

4\frac{1}{8} = 4 + (1 \div 8) = 4.125

The question explicitly requires the answer to be entered as a decimal.

Key Concept

Adding and subtracting mixed numbers with unlike denominators

Question 17Question →

A mechanic is draining old oil from a vehicle into a waste oil catch pan that has a maximum capacity of $15$ quarts. The catch pan already contains $4\frac{2}{3}$ quarts of oil from a previous job. If the vehicle's engine holds $7\frac{3}{4}$ quarts of oil, and all of it is drained into the catch pan, how much more room, in quarts, is left in the catch pan?

3\frac{5}{12}

3\frac{2}{7}

2\frac{7}{12}

7\frac{1}{4}

Show answer & explanation

Answer: $2\frac{7}{12}$

Answer

The correct answer is

2\frac{7}{12}

.

To find the remaining capacity, first add the two amounts of oil using a common denominator of 12 (

4\frac{8}{12} + 7\frac{9}{12} = 11\frac{17}{12} = 12\frac{5}{12}

quarts). Then, subtract this total from the pan's capacity of 15. By borrowing, 15 becomes

14\frac{12}{12}

. Subtracting

12\frac{5}{12}

leaves

2\frac{7}{12}

quarts.

Step-by-Step Solution

1

Determine the total amount of oil that will be in the catch pan by adding the existing oil and the vehicle's oil.

4\frac{2}{3} + 7\frac{3}{4}

The pan's occupied space is the sum of what was already there and what is being added.

2

Find a common denominator to add the fractions.

The least common multiple of 3 and 4 is 12. Convert the fractions:

4\frac{8}{12} + 7\frac{9}{12}

.

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

3

Add the whole numbers and the numerators, then simplify.

4 + 7 = 11

and

\frac{8}{12} + \frac{9}{12} = \frac{17}{12}

. Since

\frac{17}{12} = 1\frac{5}{12}

, the total is

11 + 1\frac{5}{12} = 12\frac{5}{12}

quarts.

Combines the mixed numbers to find the total volume of oil.

4

Subtract the total volume of oil from the catch pan's maximum capacity.

15 - 12\frac{5}{12}

The remaining room is the difference between the total capacity and the occupied space.

5

Borrow 1 from the whole number 15 to perform the subtraction.

Rewrite

15

as

14\frac{12}{12}

. Subtract:

14\frac{12}{12} - 12\frac{5}{12} = (14 - 12) + (\frac{12}{12} - \frac{5}{12}) = 2\frac{7}{12}

quarts.

Borrowing allows subtraction of the fractional part.

Key Concept

Adding mixed numbers with unlike denominators and subtracting mixed numbers from whole numbers by borrowing.

Question 18Question →

What is the value of the following expression?

\frac{2}{3} - \frac{1}{4} \times 1\frac{1}{2} + \frac{5}{8}

1\frac{1}{4}

\frac{11}{12}

-\frac{1}{3}

\frac{3}{8}

Show answer & explanation

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

Answer

The correct value of the expression is

\frac{11}{12}

.

The correct answer is found by rigorously applying the order of operations. The multiplication

\frac{1}{4} \times 1\frac{1}{2}

must be done first. After converting

1\frac{1}{2}

to

\frac{3}{2}

, the product is

\frac{3}{8}

. Substituting this back gives

\frac{2}{3} - \frac{3}{8} + \frac{5}{8}

. By combining the terms with eighths, we get

-\frac{3}{8} + \frac{5}{8} = \frac{2}{8}

, or

\frac{1}{4}

. Adding this to the initial term requires a common denominator of 12:

\frac{8}{12} + \frac{3}{12} = \frac{11}{12}

.

Step-by-Step Solution

1

Convert the mixed number to an improper fraction.

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

Fraction multiplication requires all terms to be expressed as proper or improper fractions.

2

Perform the multiplication operation.

\frac{1}{4} \times \frac{3}{2} = \frac{3}{8}

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

3

Substitute the multiplication result back into the expression.

\frac{2}{3} - \frac{3}{8} + \frac{5}{8}

This updates the expression for the remaining addition and subtraction operations.

4

Combine the fractions that already share a common denominator.

-\frac{3}{8} + \frac{5}{8} = \frac{2}{8}

, which simplifies to

\frac{1}{4}

.

Grouping terms with the same denominator makes the calculation faster and simpler.

5

Find a common denominator for the remaining terms and add.

\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}

Fractions must have a common denominator (in this case, 12) before their numerators can be added.

Key Concept

Order of Operations with Fractions

Estimated Time:1m 0s

Question 19Question →

A portable generator's fuel tank holds $8\frac{1}{4}$ gallons of diesel. If the generator consumes fuel at a rate of $1\frac{2}{3}$ gallons per hour, how many gallons of fuel remain in the tank after exactly $2$ hours of continuous operation?

4\frac{11}{12}

13\frac{1}{6}

5\frac{1}{12}

1\frac{11}{12}

Show answer & explanation

Answer: $4\frac{11}{12}$

Answer

The correct answer is

4\frac{11}{12}

gallons.

The correct answer accurately calculates the total fuel used by multiplying the hourly consumption rate by the operating time (

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

). It then correctly subtracts this total from the initial tank capacity by finding a common denominator and borrowing appropriately (

8\frac{1}{4} - 3\frac{1}{3} = 8\frac{3}{12} - 3\frac{4}{12} = 7\frac{15}{12} - 3\frac{4}{12} = 4\frac{11}{12}

).

Step-by-Step Solution

1

Calculate the total amount of fuel consumed in 2 hours.

2 \times 1\frac{2}{3} = 2 \times \frac{5}{3} = \frac{10}{3} = 3\frac{1}{3}

gallons

Total consumption is the hourly rate multiplied by the number of hours operated.

2

Set up the subtraction to find the remaining fuel.

8\frac{1}{4} - 3\frac{1}{3}

Remaining fuel is the initial capacity minus the total consumed fuel.

3

Find a common denominator to subtract the mixed numbers.

The least common multiple of

4

and

3

is

12

. The expression becomes

8\frac{3}{12} - 3\frac{4}{12}

.

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

4

Borrow from the whole number to subtract the fractions.

8\frac{3}{12}

becomes

7\frac{15}{12}

. Then,

7\frac{15}{12} - 3\frac{4}{12} = 4\frac{11}{12}

gallons.

Since

\frac{3}{12}

is less than

\frac{4}{12}

, one whole unit must be converted into twelfths (

\frac{12}{12}

) and added to the fraction.

Key Concept

Fraction Operations

Question 20Question →

A hiker plans to complete a trail that is $15\frac{3}{4}$ miles long. If she hikes at a consistent average speed of $3\frac{1}{2}$ miles per hour, how many hours will it take her to complete the entire trail?

Show answer & explanation

Answer: 4.5

Answer

4.5

The correct answer is found by converting the total distance (

15\frac{3}{4}

) and the speed (

3\frac{1}{2}

) into improper fractions (

\frac{63}{4}

and

\frac{7}{2}

), and then dividing the distance by the speed. This translates to

\frac{63}{4} \times \frac{2}{7}

, which cross-simplifies perfectly to

\frac{9}{2}

or

4.5

hours.

Step-by-Step Solution

1

Convert the mixed numbers into improper fractions.

15\frac{3}{4} = \frac{63}{4}

and

3\frac{1}{2} = \frac{7}{2}

Fraction arithmetic requires mixed numbers to be in improper fraction form before dividing.

2

Set up the division equation to solve for time.

Time =

\frac{63}{4} \div \frac{7}{2}

Time is calculated by dividing total distance by the average speed.

3

Change the division to multiplication by using the reciprocal of the divisor.

\frac{63}{4} \times \frac{2}{7}

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

4

Multiply the fractions and simplify the result.

\frac{63 \times 2}{4 \times 7} = \frac{126}{28} = \frac{9}{2} = 4.5

Simplifying the fraction yields the exact number of hours.

Key Concept

Dividing mixed numbers by converting to improper fractions and multiplying by the reciprocal.

Estimated Time:1m 0s

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