Number Theory and Operations Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 7

Question 121Question →

Calculate the value of each decimal expression below. Arrange the results in ascending order (from smallest to largest).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from smallest to largest is:

1.5 \times 2

, followed by

1.2 + 2.4

, then

8.4 \div 2

, and finally

5.8 - 1.3

.

Evaluating each mathematical expression individually yields 4.5, 3.6, 4.2, and 3.0 respectively. Arranging these numerical results in ascending order gives 3.0, 3.6, 4.2, and 4.5. This directly maps to the expressions:

1.5 \times 2

, then

1.2 + 2.4

, then

8.4 \div 2

, and finally

5.8 - 1.3

.

Step-by-Step Solution

1

Calculate the value of the first expression.

5.8 - 1.3 = 4.5

To determine the numerical value for comparison.

2

Calculate the value of the second expression.

1.2 + 2.4 = 3.6

To determine the numerical value for comparison.

3

Calculate the value of the third expression.

8.4 \div 2 = 4.2

To determine the numerical value for comparison.

4

Calculate the value of the fourth expression.

1.5 \times 2 = 3.0

To determine the numerical value for comparison.

5

Compare the four calculated values and order them from smallest to largest.

3.0 < 3.6 < 4.2 < 4.5

To establish the final ascending sequence required by the question.

Key Concept

Performing basic operations (addition, subtraction, multiplication, division) with decimals and comparing their magnitudes.

Estimated Time:45s

Question 122Question →

A sporting goods store stocks basketballs and footballs in a ratio of $3:4$ . After receiving a new shipment of $10$ additional basketballs, the ratio of basketballs to footballs changes to $5:6$ . How many footballs are currently in stock?

20

120

100

12

Show answer & explanation

Answer: 120

Answer

There are 120 footballs currently in stock.

By scaling the initial ratio (

3:4

) to

9:12

and the new ratio (

5:6

) to

10:12

, we can see that the

10

added basketballs represent exactly

1

part (

10 - 9 = 1

). Since the footballs represent

12

parts in this scaled system, there are

12 \times 10 = 120

footballs in stock.

Step-by-Step Solution

1

Identify the quantity that remains unchanged.

The number of footballs does not change when the new shipment of basketballs arrives.

This provides an anchor point to compare the before and after ratios.

2

Find a common multiple for the football ratio parts.

The initial ratio has

4

parts for footballs and the new ratio has

6

parts. The least common multiple (LCM) of

4

and

6

is

12

.

Creating a common denominator for the unchanging item allows direct comparison of the changed item's parts.

3

Scale both ratios so the football parts equal

12

.

Initial ratio: Multiply

3:4

by

3

to get

9:12

. New ratio: Multiply

5:6

by

2

to get

10:12

.

This standardizes the 'size' of one part across both ratios.

4

Determine the change in basketball parts.

The scaled basketball parts increased from

9

to

10

, a difference of exactly

1

part.

This

1

-part increase corresponds directly to the physical items added.

5

Calculate the actual number of footballs.

Since

1

part equals the

10

basketballs added, the

12

parts for footballs equal

12 \times 10 = 120

footballs.

This yields the final answer by multiplying the scaled parts by the unit value.

Key Concept

Scaling Ratios and Proportions

Alternative Method

Using algebra, let the initial basketballs be

3x

and footballs be

4x

. After adding

10

basketballs, the new ratio yields the equation

\frac{3x + 10}{4x} = \frac{5}{6}

. Cross-multiplying gives

6(3x + 10) = 5(4x)

, which simplifies to

18x + 60 = 20x

. Solving for

x

gives

2x = 60

, so

x = 30

. The number of footballs is

4x = 4(30) = 120

.

Estimated Time:1m 0s

Question 123Question →

A supply convoy is distributing medical kits and rations to various outposts in a ratio of $2:7$ . If a particular outpost receives $56$ rations, how many medical kits should they receive?

Show answer & explanation

Answer: 16; 16 kits; 16 medical kits; sixteen

Answer

16

The correct amount is found by maintaining the ratio of

2

medical kits for every

7

rations. Since

56

is

8

times as large as

7

(

56 \div 7 = 8

), the number of medical kits must also be

8

times as large as

2

. Multiplying the kit ratio component (

2

) by this scale factor (

8

) gives

16

medical kits.

Step-by-Step Solution

1

Identify the given ratio and the known quantity.

Ratio of medical kits to rations is

2:7

. The known quantity is

56

rations.

Extracting the correct values and their relationship is required to set up the proportion.

2

Set up a proportion.

\frac{2}{7} = \frac{x}{56}

The ratio of kits to rations (

2/7

) must equal the ratio of actual kits (

x

) to actual rations (

56

).

3

Solve for the unknown value

x

.

x = \frac{2 \times 56}{7} = 16

Multiplying both sides by

56

isolates

x

, revealing the required number of medical kits.

Key Concept

Solving direct proportions using a scale factor or cross-multiplication

Alternative Method

Find the scale factor by dividing the actual number of rations by the ratio number for rations:

56 \div 7 = 8

. Then multiply the ratio number for medical kits by this scale factor:

2 \times 8 = 16

.

Estimated Time:45s

Question 124Question →

A communications technician is configuring signal transponders. The synchronization cycle for each transponder is determined by the Least Common Multiple (LCM) of its two input frequencies (measured in Hz). Arrange the following pairs of input frequencies in ascending order based on their resulting synchronization cycle (smallest LCM to largest LCM).

Drag items to arrange them in the correct order

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Answer

The correct order from smallest to largest LCM is: 12 Hz and 18 Hz (LCM=36), 16 Hz and 24 Hz (LCM=48), 10 Hz and 25 Hz (LCM=50), 8 Hz and 14 Hz (LCM=56), followed by 15 Hz and 20 Hz (LCM=60).

The correct sequence is determined by finding the LCM of each frequency pair and listing them in ascending order. The calculated LCMs are 36, 48, 50, 56, and 60 respectively.

Step-by-Step Solution

1

Calculate the Least Common Multiple (LCM) for the first pair,

12\text{ Hz}

and

18\text{ Hz}

.

The LCM of

12

and

18

is

36

.

The multiples of

18

are

18, 36, 54

. Since

36

is divisible by

12

(

12 \times 3 = 36

), it is the lowest common multiple.

2

Calculate the LCM for

16\text{ Hz}

and

24\text{ Hz}

.

The LCM of

16

and

24

is

48

.

The multiples of

24

are

24, 48

. Since

48

is divisible by

16

(

16 \times 3 = 48

), it is the lowest common multiple.

3

Calculate the LCM for

10\text{ Hz}

and

25\text{ Hz}

.

The LCM of

10

and

25

is

50

.

The multiples of

25

are

25, 50

. Since

50

is divisible by

10

(

10 \times 5 = 50

), it is the lowest common multiple.

4

Calculate the LCM for

8\text{ Hz}

and

14\text{ Hz}

.

The LCM of

8

and

14

is

56

.

Using prime factorizations:

8 = 2^3

and

14 = 2 \times 7

. The LCM takes the highest powers:

2^3 \times 7 = 8 \times 7 = 56

.

5

Calculate the LCM for

15\text{ Hz}

and

20\text{ Hz}

.

The LCM of

15

and

20

is

60

.

Using prime factorizations:

15 = 3 \times 5

and

20 = 2^2 \times 5

. The LCM takes the highest powers:

2^2 \times 3 \times 5 = 60

.

6

Arrange the pairs in ascending order based on their calculated LCMs.

36 < 48 < 50 < 56 < 60

The stem requires ordering the frequencies from the smallest resulting synchronization cycle (smallest LCM) to the largest.

Key Concept

Calculating the Least Common Multiple (LCM) to synchronize multiple periodic events.

Question 125Question →

An experimental particle travels a distance of $4.5 \times 10^5$ meters in $5.0 \times 10^{-3}$ seconds. What is the average speed of the particle in meters per second, expressed in proper scientific notation?

9.0 \times 10^7

9.0 \times 10^1

9.0 \times 10^9

9.0 \times 10^8

Show answer & explanation

Answer: $9.0 \times 10^7$

Answer

The correct speed is

9.0 \times 10^7

meters per second.

To divide numbers in scientific notation, divide the coefficients and subtract the exponents. Dividing

4.5

by

5.0

yields

0.9

. Subtracting the exponent

-3

from

5

yields

8

(since

5 - (-3) = 5 + 3 = 8

). This gives an intermediate result of

0.9 \times 10^8

. To convert this into proper scientific notation, the coefficient must be between

1

and

10

. Moving the decimal point one place to the right converts

0.9

to

9.0

, which requires decreasing the exponent by

1

, resulting in the final correct answer of

9.0 \times 10^7

.

Step-by-Step Solution

1

Divide the coefficients given in the problem:

4.5 \div 5.0

.

0.9

When dividing numbers in scientific notation, the coefficients are divided separately from the powers of 10.

2

Divide the powers of 10 by subtracting the denominator's exponent from the numerator's exponent:

10^5 \div 10^{-3}

.

10^{5 - (-3)} = 10^8

According to the laws of exponents, division of like bases requires subtracting the exponent of the divisor from the exponent of the dividend.

3

Combine the newly calculated coefficient and power of 10.

0.9 \times 10^8

This represents the mathematical result before applying proper formatting rules.

4

Normalize the answer by moving the decimal point one place to the right to make the coefficient between 1 and 10, and subtract 1 from the exponent to balance the shift.

9.0 \times 10^7

Proper scientific notation requires the coefficient to be greater than or equal to 1 and less than 10.

Key Concept

Scientific Notation Division and Normalization

Question 126Question →

A logistics coordinator is packing identical emergency relief kits. They have $180$ bottles of water, $252$ energy bars, and $324$ bandages. If every kit must contain the exact same combination of supplies with no items left over, what is the greatest number of kits that can be assembled?

18

36

72

11,340

Show answer & explanation

Answer: 36

Answer

The greatest number of kits that can be assembled is 36.

The scenario describes dividing three sets of items into identical groups with no remainder, which means finding the Greatest Common Factor (GCF). Breaking down the numbers into prime factors yields:

180 = 2^2 \times 3^2 \times 5

,

252 = 2^2 \times 3^2 \times 7

, and

324 = 2^2 \times 3^4

. The factors shared by all three are

2^2

and

3^2

. Multiplying these together (

4 \times 9

) gives the GCF of

36

. Therefore, a maximum of

36

kits can be made, each containing exactly

5

water bottles,

7

energy bars, and

9

bandages.

Step-by-Step Solution

1

Identify the mathematical operation required by the scenario.

We need to find the Greatest Common Factor (GCF) of 180, 252, and 324.

The problem asks for the maximum number of identical kits that can be made by dividing all supplies without any leftovers, which aligns with the definition of the GCF.

2

Determine the prime factorization for each quantity.

180 = 2^2 \times 3^2 \times 5

252 = 2^2 \times 3^2 \times 7

324 = 2^2 \times 3^4

Prime factorization is a reliable method to find all common prime factors among multiple large numbers.

3

Identify the common prime factors and their lowest exponents.

The common prime bases are 2 and 3. Their lowest exponents across all factorizations are

2^2

and

3^2

.

The GCF is the product of the lowest power of each prime factor that appears in all of the numbers.

4

Calculate the final GCF by multiplying the common prime factors.

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

Multiplying the lowest common prime factors gives the greatest numerical value that divides evenly into all three supply quantities.

Key Concept

Greatest Common Factor (GCF) via Prime Factorization

Alternative Method

Instead of prime factorization, you can check the factors of the common difference. The difference between the quantities is

72

(

252 - 180 = 72

and

324 - 252 = 72

). The factors of

72

are

72, 36, 24, 18, 12, 9, 8, 6, 4, 3, 2, 1

. Checking these against

180

from largest to smallest,

72

does not divide

180

, but

36

does (

180 \div 36 = 5

). Since

36

divides all three numbers, it is the GCF.

Estimated Time:1m 30s

Question 127Question →

A tactical transport vehicle's fuel efficiency is evaluated under two conditions. On paved roads, the ratio of miles driven to gallons of fuel consumed is $15:1$ . In off-road conditions, the vehicle's fuel efficiency decreases by $20\%$ . How many gallons of fuel are required to drive $60\text{ miles}$ in off-road conditions?

4.8

3.2

5

0.2

Show answer & explanation

Answer: $5$

Answer

The correct amount of fuel required is

5

gallons.

The original fuel efficiency is 15 miles per gallon. A 20% decrease reduces this rate by 3 mpg (0.20 × 15), resulting in a new off-road efficiency of 12 mpg. To find the required fuel for a 60-mile trip, divide the total distance by the new unit rate: 60 / 12 = 5 gallons.

Step-by-Step Solution

1

Identify the original fuel efficiency from the given ratio.

The ratio

15:1

means the vehicle gets

15\text{ miles per gallon}

(mpg) on paved roads.

Converting the ratio into a unit rate makes it easier to apply the percentage change.

2

Calculate the off-road fuel efficiency by applying the

20\%

decrease.

15 - (0.20 \times 15) = 15 - 3 = 12\text{ mpg}

.

The problem states the efficiency decreases by

20\%

in off-road conditions.

3

Set up a proportion to find the gallons required for the

60\text{-mile}

trip.

\frac{12\text{ miles}}{1\text{ gallon}} = \frac{60\text{ miles}}{x\text{ gallons}}

Fuel consumption is a direct proportion between miles driven and gallons used at a constant rate.

4

Solve for

x

.

12x = 60 \implies x = 5\text{ gallons}

.

Isolating the variable gives the final required amount of fuel.

Key Concept

Applying percentage changes to unit rates and setting up proportions to solve for an unknown quantity.

Alternative Method

Calculate the fuel needed for paved roads first (

60 / 15 = 4\text{ gallons}

). Recognize that since the new efficiency is

80\%

(or

\frac{4}{5}

) of the original, the fuel needed will be the inverse ratio (

\frac{5}{4}

) of the original fuel amount. Multiply

4\text{ gallons} \times \frac{5}{4} = 5\text{ gallons}

.

Estimated Time:1m 0s

Question 128Question →

A rectangular microchip has a length of $8.0 \times 10^{-3}$ meters and a width of $4.5 \times 10^{-3}$ meters. Calculate the area of the microchip in square meters and express the result in proper scientific notation as $a \times 10^n$ (where $1 \le a < 10$ ). What is the value of the coefficient $a$ ?

Show answer & explanation

Answer: 3.6

Answer

The coefficient

a

is 3.6.

The correct answer is derived by multiplying the coefficients (8.0 × 4.5 = 36), adding the exponents (-3 + -3 = -6), and normalizing the preliminary result of 36 × 10^-6 by moving the decimal left by one place. This creates a proper coefficient of 3.6 and changes the exponent to -5.

Step-by-Step Solution

1

Set up the multiplication for area

Area =

(8.0 \times 10^{-3}) \times (4.5 \times 10^{-3})

The area of a rectangle is found by multiplying its length by its width.

2

Multiply the coefficients

8.0 \times 4.5 = 36

When multiplying terms in scientific notation, the numerical coefficients are multiplied together.

3

Multiply the powers of 10

10^{-3} \times 10^{-3} = 10^{-6}

When multiplying identical bases, their exponents are added:

-3 + (-3) = -6

.

4

Combine into preliminary result

36 \times 10^{-6}

Bring the multiplied coefficient and combined power of 10 together.

5

Normalize to proper scientific notation

3.6 \times 10^{-5}

Proper scientific notation requires the coefficient

a

to satisfy

1 \le a < 10

. Moving the decimal one place left on 36 increases the exponent by 1 (

-6 + 1 = -5

).

6

Extract the coefficient

a

3.6

The question specifically asks for the value of the coefficient

a

in the final

a \times 10^n

format.

Key Concept

Multiplying and Normalizing Values in Scientific Notation

Question 129Question →

A plumber has a copper pipe that is $14.5$ inches long. She cuts off a piece that is $3.75$ inches long. What is the length of the remaining pipe in inches?

Show answer & explanation

Answer: 10.75

Answer

10.75

Subtracting the cut piece (

3.75

inches) from the total length (

14.5

inches) gives the remaining length. Aligning the decimals properly as

14.50 - 3.75

results in exactly

10.75

.

Step-by-Step Solution

1

Identify the operation needed

Subtraction:

14.5 - 3.75

Cutting a piece off means reducing the total length of the pipe.

2

Align the decimals for subtraction

14.50 - 3.75

To subtract decimals correctly, align the place values by lining up the decimal points. Add a placeholder zero to

14.5

to make it

14.50

.

3

Perform the subtraction

10.75

Subtracting

3.75

from

14.50

yields

10.75

.

Key Concept

Decimal Subtraction and Alignment

Estimated Time:45s

Question 130Question →

Determine the final value of each arithmetic expression below. What is the correct sequence when the expressions are arranged from smallest to largest result?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from smallest to largest is:

0.18 + 0.42

,

3.5 \div 5

,

2.1 - 1.3

, and

0.15 \times 6

.

To arrange the sequence properly, first calculate their values:

0.18 + 0.42 = 0.6

,

3.5 \div 5 = 0.7

,

2.1 - 1.3 = 0.8

, and

0.15 \times 6 = 0.9

. By comparing these results side by side, the accurate ascending order is

0.6

,

0.7

,

0.8

, and

0.9

.

Step-by-Step Solution

1

Calculate the sum for the fourth expression.

0.18 + 0.42 = 0.60

, which is

0.6

.

Adding the hundredths and tenths places properly yields

0.6

.

2

Calculate the quotient for the third expression.

3.5 \div 5 = 0.7

.

Since

35 \div 5 = 7

, dividing

3.5

by

5

gives

0.7

.

3

Calculate the difference for the first expression.

2.1 - 1.3 = 0.8

.

Borrowing from the ones place allows subtracting

1.3

from

2.1

to get

0.8

.

4

Calculate the product for the second expression.

0.15 \times 6 = 0.90

, which is

0.9

.

Multiplying

15

by

6

gives

90

. Since the original factors have a total of two decimal places, the result is

0.90

.

5

Order the evaluated results from smallest to largest.

0.6 < 0.7 < 0.8 < 0.9

.

Comparing the tenths digit of each calculated value provides the final arrangement.

Key Concept

Performing basic arithmetic operations with decimals and comparing their magnitudes.

Question 131Question →

An integer $k$ is defined as the Least Common Multiple (LCM) of $12$ and $28$ . What is the Greatest Common Factor (GCF) of $k$ and $40$ ?

2

4

40

840

Show answer & explanation

Answer: 4

Answer

The Greatest Common Factor of

k

and

40

is 4.

The problem requires two distinct mathematical steps. First, calculate the Least Common Multiple of

12

and

28

. Using prime factorizations (

12 = 2^2 \times 3

and

28 = 2^2 \times 7

), the LCM is the product of the highest powers of all prime factors:

2^2 \times 3 \times 7 = 84

. Therefore,

k = 84

. The second step asks for the Greatest Common Factor of

k

(

84

) and

40

. The prime factorization of

40

is

2^3 \times 5

. The only shared prime factor between

84

and

40

is

2

, and the lowest power they share is

2^2

. Thus, the GCF is

4

.

Step-by-Step Solution

1

Find the prime factorizations of 12 and 28.

12 = 2^2 \times 3

and

28 = 2^2 \times 7

.

Prime factorization is the most reliable method for finding the Least Common Multiple.

2

Calculate

k

, which is the LCM of 12 and 28.

k = 2^2 \times 3 \times 7 = 4 \times 21 = 84

.

The LCM is found by multiplying the highest power of every prime factor present in either number.

3

Find the prime factorization of 40.

40 = 2^3 \times 5

.

This is needed to compare against the prime factors of

k

(

84

) to find the Greatest Common Factor.

4

Calculate the GCF of 84 and 40.

Comparing

84

(

2^2 \times 3 \times 7

) and

40

(

2^3 \times 5

), the only shared prime base is 2. The lowest exponent for this base is 2. Thus, the GCF is

2^2 = 4

.

The GCF is the product of the lowest powers of all common prime factors shared by both integers.

Key Concept

Applying prime factorization to sequentially determine Least Common Multiples (LCM) and Greatest Common Factors (GCF).

Question 132Question →

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 133Question →

A biomedical researcher is cataloging the diameters of four microscopic specimens. To properly calibrate the imaging equipment, arrange the specimens in descending order based on their diameter, from the largest to the smallest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct descending order from largest to smallest is Specimen Beta, Specimen Delta, Specimen Gamma, and Specimen Alpha.

By converting all measurements into standard scientific notation (

a \times 10^n

), their magnitudes can be compared directly. Specimen Beta has the largest exponent (

-5

). The remaining three specimens all reduce to an exponent of

-6

. Ordering those three by their coefficients (

9.0

,

8.5

, and

6.5

) yields the final sequence.

Step-by-Step Solution

1

Convert Specimen Beta from standard decimal form to scientific notation.

0.000072

meters becomes

7.2 \times 10^{-5}

meters.

A uniform mathematical format is necessary to accurately compare the magnitudes of all specimens.

2

Normalize the coefficient for Specimen Gamma.

850 \times 10^{-8}

becomes

8.5 \times 10^{-6}

.

Proper scientific notation requires the coefficient to be greater than or equal to 1 and less than 10. Moving the decimal two places left increases the exponent by 2.

3

Normalize the coefficient for Specimen Delta.

0.9 \times 10^{-5}

becomes

9.0 \times 10^{-6}

.

Moving the decimal one place right to create a valid coefficient decreases the exponent by 1.

4

Compare the exponents first, then the coefficients to determine the descending order.

7.2 \times 10^{-5} > 9.0 \times 10^{-6} > 8.5 \times 10^{-6} > 6.5 \times 10^{-6}

.

A larger exponent (

-5

vs

-6

) dictates a larger overall number. For values sharing the identical

-6

exponent, the descending order is determined directly by their coefficients (

9.0 > 8.5 > 6.5

).

Key Concept

Normalizing Coefficients and Comparing Magnitudes in Scientific Notation

Estimated Time:1m 30s

Question 134Question →

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 135Question →

Given $x = -4$ and $y = 3$ , what is the exact value of the expression $3|x - y| - |y^2 - x^2|$ ?

Show answer & explanation

Answer: 14

Answer

14

By correctly substituting the variables, following the order of operations, and applying the definition of absolute value, the expression simplifies to

3(7) - 7

, which evaluates precisely to

14

.

Step-by-Step Solution

1

Substitute the given values for

x

and

y

into the expression.

3|-4 - 3| - |(3)^2 - (-4)^2|

Variable substitution is the first step in evaluating an algebraic expression.

2

Simplify the expression inside the first absolute value bars.

|-4 - 3| = |-7|

Absolute value bars act as grouping symbols, so operations inside them must be performed first.

3

Evaluate the exponents inside the second absolute value bars.

|(3)^2 - (-4)^2| = |9 - 16|

Following the order of operations, exponents are evaluated before subtraction.

4

Simplify the expression inside the second absolute value bars.

|9 - 16| = |-7|

Complete the subtraction inside the grouping symbols.

5

Apply the absolute value function to both simplified terms.

3(7) - 7

The absolute value of any non-zero real number is its positive counterpart:

|-7| = 7

.

6

Perform the final multiplication and subtraction.

21 - 7 = 14

Multiplication is performed before subtraction according to the standard order of operations.

Key Concept

Evaluating algebraic expressions involving absolute value, exponents, and signed numbers.

Estimated Time:1m 30s

Question 136Question →

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 137Question →

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 138Question →

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

Question 139Question →

If $y = -3$ , which of the following represents all possible values for $x$ in the equation $-2 - 2|x - y| = -16$ ?

10

and

-4

1

and

-7

4

and

-10

4

Show answer & explanation

Answer: $4$ and $-10$

Answer

The possible values for

x

are

4

and

-10

.

The correct answer accurately substitutes the negative value, isolates the absolute value term to yield

|x + 3| = 7

, and evaluates both the positive and negative branches to find the two valid solutions for

x

.

Step-by-Step Solution

1

Substitute the value

y = -3

into the original equation.

-2 - 2|x - (-3)| = -16 \Rightarrow -2 - 2|x + 3| = -16

To reduce the equation to a single variable so it can be solved.

2

Begin isolating the absolute value expression by adding

2

to both sides.

-2|x + 3| = -16 + 2 \Rightarrow -2|x + 3| = -14

To undo the constant subtraction outside the absolute value, following reverse order of operations.

3

Divide both sides of the equation by

-2

.

|x + 3| = 7

To completely isolate the absolute value expression.

4

Set up two separate equations for the positive and negative pathways.

x + 3 = 7

and

x + 3 = -7

The expression inside the absolute value can evaluate to either

7

or

-7

and still have an absolute distance of

7

from zero.

5

Solve both resulting linear equations for

x

.

x = 4

and

x = -10

To find all possible values that satisfy the original equation.

Key Concept

Solving linear absolute value equations requires accurately substituting variables, strictly following the order of operations to isolate the absolute value, and creating two distinct pathways (positive and negative) to find all solutions.

Estimated Time:1m 30s

Question 140Question →

A plumber is connecting three sections of PVC pipe to bridge a gap that is $12\frac{1}{4}$ feet wide. The first section of pipe measures $4\frac{2}{3}$ feet in length, and the second section measures $3\frac{3}{4}$ feet. Assuming there is no overlap between the connected sections, what must be the length of the third section of pipe to exactly bridge the gap?

3\frac{5}{6}

feet

4\frac{1}{6}

feet

4\frac{5}{6}

feet

20\frac{2}{3}

feet

Show answer & explanation

Answer: $3\frac{5}{6}$ feet

Answer

The third section of pipe must be

3\frac{5}{6}

feet long.

The correct answer accurately adds the two pipe lengths by finding a common denominator of 12, resulting in

8\frac{5}{12}

. It then subtracts this sum from the total gap width of

12\frac{3}{12}

by properly borrowing from the whole number to get

11\frac{15}{12} - 8\frac{5}{12} = 3\frac{10}{12}

, which simplifies to

3\frac{5}{6}

.

Step-by-Step Solution

1

Find the combined length of the first two pipe sections.

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

To determine how much of the gap has already been covered.

2

Convert the fractions to a common denominator and add.

4\frac{8}{12} + 3\frac{9}{12} = 7\frac{17}{12}

The least common multiple of 3 and 4 is 12.

3

Simplify the improper fraction.

7\frac{17}{12} = 7 + 1\frac{5}{12} = 8\frac{5}{12}

feet

Converting to a proper mixed number makes the final subtraction step accurate.

4

Subtract the combined length from the total gap width.

12\frac{1}{4} - 8\frac{5}{12} = 12\frac{3}{12} - 8\frac{5}{12}

The remaining distance requires subtracting the covered distance from the total.

5

Borrow from the whole number to perform the subtraction.

11\frac{15}{12} - 8\frac{5}{12} = 3\frac{10}{12} = 3\frac{5}{6}

feet

Since

\frac{3}{12}

is smaller than

\frac{5}{12}

, you must borrow 1 whole (or

\frac{12}{12}

) from 12.

Key Concept

Fraction Operations with Mixed Numbers

Number Theory and Operations

Answer

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Common Mistakes

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Common Mistakes

Alternative Method

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Alternative Method

Answer

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