Number Theory and Operations Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 5

Consider the process of evaluating the numerical expression below using the standard order of operations:

14 + 3 \times (7 - 2^2) - 8 \div 4

How should the following intermediate simplified forms be arranged to represent the exact chronological sequence of the step-by-step evaluation?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The expressions should be ordered starting with the exponent evaluation, followed by the parentheses, then multiplication and division, and finally addition.

The proper application of the order of operations (PEMDAS/BODMAS) ensures that we first simplify inside the parentheses (handling the exponent first, then subtraction), next process multiplication and division from left to right, and finally address addition and subtraction from left to right. This produces the exact sequence of intermediate expressions shown in the correct arrangement.

Step-by-Step Solution

1

Evaluate the exponent inside the grouping symbols.

2^2

simplifies to

4

, changing the expression to

14 + 3 \times (7 - 4) - 8 \div 4

.

Order of operations dictates that exponents inside parentheses take precedence over other operations.

2

Complete the operation inside the parentheses.

7 - 4

simplifies to

3

, yielding

14 + 3 \times 3 - 8 \div 4

.

All operations within grouping symbols must be fully resolved before proceeding to external operations.

3

Evaluate multiplication and division.

3 \times 3

becomes

9

and

8 \div 4

becomes

2

, resulting in

14 + 9 - 2

.

Multiplication and division share the same priority and are executed from left to right before addition or subtraction.

4

Perform the addition.

14 + 9

becomes

23

, leaving

23 - 2

.

Addition and subtraction share the lowest priority and are evaluated from left to right.

5

Perform the final subtraction.

23 - 2

simplifies to the final answer of

21

.

This is the final remaining mathematical operation.

Key Concept

Order of Operations (PEMDAS)

Question 82Question →

Determine the final inventory count and the net percentage change by filling in the blanks.

Fill in the blanks below

A military supply depot initially stored

5,000

tactical vests. During a regional exercise, the inventory was reduced by

30\%

. Afterward, a new shipment increased the remaining stock by

40\%

. The depot now holds tactical vests. Compared to the original inventory, this represents a net of \%.

Show answer & explanation

Answer

The depot now holds 4,900 vests, representing a net decrease of 2%.

The final inventory is

4,900

vests, which is

100

vests fewer than the original

5,000

. This equates to a net decrease of

2\%

. In successive percentage problems, it is crucial to recalculate the base amount for each step.

Step-by-Step Solution

1

Calculate the inventory after the initial

30\%

reduction.

5,000 - (0.30 \times 5,000) = 5,000 - 1,500 = 3,500

vests.

The first event reduces the starting stock by

30\%

, establishing a new base amount for the subsequent calculation.

2

Calculate the inventory after the

40\%

increase.

3,500 + (0.40 \times 3,500) = 3,500 + 1,400 = 4,900

vests.

The

40\%

increase applies only to the remaining stock (

3,500

), not the original

5,000

.

3

Determine the net change from the original inventory.

5,000 - 4,900 = 100

vests fewer than the original.

Comparing the final inventory (

4,900

) to the original inventory (

5,000

) reveals a net decrease.

4

Calculate the net percentage change.

(\frac{100}{5,000}) \times 100\% = 0.02 \times 100\% = 2\%

.

The percentage change is found by dividing the absolute change by the original base amount, then multiplying by

100\%

.

Key Concept

Successive Percentages and Changing Base Values

Question 83Question →

What is the smallest positive integer that is a multiple of both $8$ and $12$ ?

Show answer & explanation

Answer: 24

Answer

24

The smallest positive integer that is a multiple of both

8

and

12

is their least common multiple (LCM). The multiples of

8

are

8, 16, 24, 32, \dots

and the multiples of

12

are

12, 24, 36, \dots

. The smallest number that appears in both lists is

24

.

Step-by-Step Solution

1

Find the positive multiples of

8

.

8, 16, 24, 32, 40, \dots

We need to determine a number that is a multiple of

8

.

2

Find the positive multiples of

12

.

12, 24, 36, 48, \dots

We need the number to also be a multiple of

12

.

3

Identify the smallest common multiple from both lists.

24

The question asks for the smallest positive integer that is a multiple of both numbers.

Key Concept

Least Common Multiple (LCM)

Estimated Time:45s

Question 84Question →

In a medical supply drop, the ratio of bandages to antiseptic bottles is $7:3$ . If there are $210$ antiseptic bottles in the drop, how many bandages are included?

Show answer & explanation

Answer: 490

Answer

490

The proportion must match the quantity of bandages to antiseptics. Since the ratio is

7

parts bandages for every

3

parts antiseptics, and the actual number of antiseptics is

210

(

70

times the ratio amount), the number of bandages must also be

70

times its ratio amount. Therefore,

7 \times 70 = 490

bandages.

Step-by-Step Solution

1

Identify the known ratio and the known quantity.

The ratio of bandages to antiseptics is

7:3

, and there are

210

antiseptics.

To correctly set up a mathematically accurate proportion.

2

Set up the proportion equation.

\frac{7}{3} = \frac{x}{210}

To relate the given simplified ratio to the actual numbers.

3

Solve the proportion for

x

.

x = \frac{7 \times 210}{3} = 7 \times 70 = 490

To find the unknown number of bandages.

Key Concept

Solving basic proportions from a given part-to-part ratio

Estimated Time:45s

Question 85Question →

A carpenter has two pieces of wood. One piece is $18$ inches long and the other is $24$ inches long. She wants to cut both pieces into smaller segments of equal length with no wood left over. What is the greatest possible length, in inches, of each segment?

3

6

42

72

Show answer & explanation

Answer: 6

Answer

The greatest possible length of each segment is 6 inches.

To find the greatest possible length for the cut segments, we must find the greatest common factor (GCF) of 18 and 24. The factors of 18 are 1, 2, 3, 6, 9, 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The largest number appearing in both lists is 6, making it the correct answer.

Step-by-Step Solution

1

Identify the goal of the problem.

We need to divide both 18 and 24 into equal segments of the maximum possible length. This means finding the Greatest Common Factor (GCF) of 18 and 24.

The phrase 'cut into equal length' implies finding a common factor, and 'greatest possible length' specifies the GCF.

2

List the factors of 18.

The factors of 18 are 1, 2, 3, 6, 9, and 18.

These are all the integers that divide evenly into 18.

3

List the factors of 24.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

These are all the integers that divide evenly into 24.

4

Identify the greatest factor common to both lists.

The common factors are 1, 2, 3, and 6. The greatest among these is 6.

This is the largest segment size that will evenly divide both pieces of wood.

Key Concept

The Greatest Common Factor (GCF) is the largest positive integer that divides evenly into two or more given numbers. It is commonly used when splitting items into equal, maximum-sized groups.

Estimated Time:1m 0s

Question 86Question →

A pilot logs $120$ flight hours during the first quarter of the year. If this amount represents $40\%$ of their annual flight hour requirement, how many additional flight hours must the pilot complete by the end of the year to exactly meet the requirement?

48

72

180

300

Show answer & explanation

Answer: 180

Answer

The pilot must complete 180 additional flight hours.

The correct answer is found by first determining the total annual requirement. Since 120 hours represents 40% of the total, the total is calculated by dividing 120 by 0.40, which equals 300 hours. The question asks for the additional hours needed, so you must subtract the 120 hours already flown from the 300 total hours, leaving exactly 180 hours.

Step-by-Step Solution

1

Set up an equation to find the total annual flight hour requirement.

0.40 \times \text{Total} = 120

The problem states that 120 hours is exactly 40% of the full annual requirement.

2

Solve for the total annual requirement by dividing the completed hours by the decimal percentage.

\text{Total} = \frac{120}{0.40} = 300

hours

Dividing the part by the percentage (as a decimal) yields the whole base amount.

3

Subtract the hours already flown from the total annual requirement to find the remaining hours needed.

300 - 120 = 180

hours

The question specifically asks for the 'additional' flight hours required to finish the year.

Key Concept

Reverse Percentage and Successive Subtraction

Question 87Question →

During a field exercise, a combat engineer platoon's daily fuel consumption increased by $20\%$ in the first week, and then decreased by $25\%$ in the second week. If the final fuel consumption was $4,500$ gallons per day, what was the initial daily fuel consumption in gallons?

Show answer & explanation

Answer: 5000

Answer

5000

The correct calculation requires treating the percentage changes as sequential multipliers. The initial amount increases by

20\%

(multiplier of

1.20

) and the new amount decreases by

25\%

(multiplier of

0.75

). The combined net multiplier is

1.20 \times 0.75 = 0.90

, meaning the final value is

90\%

of the initial value. To find the initial value, we divide the final amount (

4,500

) by

0.90

, which yields

5,000

.

Step-by-Step Solution

1

Determine the multipliers for the successive percentage changes.

A

20\%

increase is represented by

1.20

, and a

25\%

decrease is represented by

0.75

.

Successive percentage changes must be multiplied consecutively, not added together.

2

Set up an equation representing the net change.

x \times 1.20 \times 0.75 = 4500

The initial amount (

x

) multiplied by each successive change equals the final amount.

3

Calculate the combined multiplier.

1.20 \times 0.75 = 0.90

Simplifying the multipliers shows that the net change is a

10\%

decrease overall.

4

Solve the equation for the initial amount.

x = \frac{4500}{0.90} = 5000

Dividing the final consumption by the combined multiplier isolates

x

to find the original amount.

Key Concept

Successive Percentage Changes and Reverse Percentages

Question 88Question →

Let $x$ be a negative even integer and $y$ be a negative odd integer. Which of the following expressions will always result in a positive odd integer?

x \times y + 1

x + y

x \times y

|x| + y

Show answer & explanation

Answer: $x \times y + 1$

Answer

x \times y + 1

When multiplying two negative integers, the result is positive. When multiplying an even integer by an odd integer, the result is always even. Therefore, the product of

x

and

y

is a positive even integer. Adding 1 to any even integer results in an odd integer. Thus, the expression always yields a positive odd integer.

Step-by-Step Solution

1

Determine the sign and parity of the product

x \times y

.

x \times y

is a positive even integer.

Multiplying two negative numbers yields a positive number. Multiplying an even number by any integer yields an even number.

2

Evaluate the result of adding 1 to the product.

x \times y + 1

is a positive odd integer.

Adding 1 to any even integer changes its parity to odd. Adding 1 to a positive number keeps it positive.

3

Check the validity of the other provided expressions.

Only

x \times y + 1

guarantees a positive odd result.

Other expressions result in negative odd values, positive even values, or values that can fluctuate in sign based on specific magnitudes.

Key Concept

Parity and Sign Properties of Integers

Question 89Question →

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 90Question →

A map uses a scale where $3\text{ cm}$ represents $20\text{ km}$ . If two towns are separated by $15\text{ cm}$ on the map, what is the actual distance between them in kilometers?

4

32

100

300

Show answer & explanation

Answer: $100$

Answer

100

The correct answer is determined by setting up a direct proportion between the map distance and the actual distance:

\frac{3}{20} = \frac{15}{x}

. Cross-multiplying yields

3x = 300

. Dividing both sides by

3

isolates

x

, resulting in

100\text{ km}

. Alternatively, you can observe that the map distance increased by a factor of

5

(

3 \times 5 = 15

), so the actual distance must also increase by a factor of

5

(

20 \times 5 = 100

).

Step-by-Step Solution

1

Identify the given scale ratio from the problem.

The ratio of map distance to actual distance is

3\text{ cm} : 20\text{ km}

.

This baseline ratio establishes the mathematical relationship needed to find any other actual distance.

2

Set up a proportion matching map measurements to actual measurements.

\frac{3}{20} = \frac{15}{x}

Keeping the units aligned (map distance in the numerator, actual distance in the denominator) ensures the proportion is solvable.

3

Solve for the unknown actual distance

x

using cross-multiplication.

3x = 15 \times 20 \rightarrow 3x = 300

Cross-multiplication transforms the proportion into a standard linear equation.

4

Isolate

x

by dividing both sides of the equation by

3

.

x = \frac{300}{3} = 100

Dividing both sides balances the equation and provides the final actual distance in kilometers.

Key Concept

Direct Proportions and Scale Factors

Question 91Question →

A logistics unit packs $3$ boxes of rations for every $4$ boxes of water. If they pack $12$ boxes of rations, how many boxes of water do they pack?

9

13

16

28

Show answer & explanation

Answer: $16$

Answer

16

The ratio of rations to water is 3 to 4. Since the actual number of rations (12) is exactly 4 times the ratio number (3), the number of water boxes must also be 4 times its corresponding ratio number. Multiplying 4 by 4 yields 16 boxes of water.

Step-by-Step Solution

1

Set up the proportion comparing rations to water.

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

This establishes the mathematical relationship between the known ratio and the actual amounts.

2

Determine the scale factor for the rations.

12 \div 3 = 4

Finding how many times the base ratio was multiplied to get the actual number of rations.

3

Apply the scale factor to the water portion of the ratio.

4 \times 4 = 16

To maintain the proportion, both parts of the ratio must be multiplied by the same scale factor.

Key Concept

Solving basic proportions by finding and applying a scale factor.

Estimated Time:45s

Question 92Question →

The sum of three consecutive negative even integers is $-102$ . What is the absolute value of the smallest of these integers?

Show answer & explanation

Answer: 36

Answer

36

By defining the consecutive even integers algebraically, the smallest integer is correctly identified as

-36

. Taking the absolute value of

-36

yields

36

.

Step-by-Step Solution

1

Define variables for the three consecutive even integers.

Let the smallest integer be

x

. The next consecutive even integers are

x + 2

and

x + 4

.

Consecutive even integers are always 2 units apart.

2

Set up the equation.

x + (x + 2) + (x + 4) = -102

The sum of the three integers is given as

-102

.

3

Solve for

x

.

3x + 6 = -102

leads to

3x = -108

, yielding

x = -36

.

Isolating

x

provides the value of the smallest integer.

4

Calculate the absolute value.

|-36| = 36

The question specifically asks for the absolute value of the smallest integer.

Key Concept

Consecutive integers and absolute value properties

Question 93Question →

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 94Question →

An ammunition crate contains tracer rounds and standard rounds in a ratio of $5:12$ . If the crate holds a total of $510$ rounds, how many tracer rounds does it contain?

Show answer & explanation

Answer: 150; 150 rounds; 150 tracer rounds

Answer

150

By adding the ratio parts (

5 + 12 = 17

), we find that tracer rounds make up

\frac{5}{17}

of the total contents. Multiplying the total number of rounds (

510

) by this fraction yields

150

tracer rounds.

Step-by-Step Solution

1

Calculate the total number of parts in the ratio.

5 + 12 = 17

total parts.

The total quantity of rounds (

510

) corresponds to the combined parts of the ratio.

2

Determine the number of rounds represented by a single part.

510 \div 17 = 30

rounds per part.

Dividing the total items by the total parts finds the scale factor.

3

Calculate the total number of tracer rounds.

5 \times 30 = 150

tracer rounds.

The ratio states there are

5

parts of tracer rounds, so we multiply the scale factor by

5

.

Key Concept

Solving part-to-whole problems using a part-to-part ratio.

Question 95Question →

A maintenance crew uses $2$ gallons of primer for every $5$ gallons of paint. If the crew uses $10$ gallons of primer, how many gallons of paint are needed?

Show answer & explanation

Answer: 25; 25 gallons; 25 gal

Answer

25

The given ratio of primer to paint is

2:5

. Since

10

gallons of primer is

5

times the original

2

gallons, you must also multiply the

5

gallons of paint by

5

to maintain the ratio. Therefore,

5 \times 5 = 25

gallons of paint are needed.

Step-by-Step Solution

1

Set up a proportion using the given ratio of primer to paint.

\frac{2}{5} = \frac{10}{x}

To relate the base ratio to the actual amounts used, where

x

represents the unknown gallons of paint.

2

Solve the proportion for

x

.

x = \frac{5 \times 10}{2} = 25

Using cross-multiplication isolates

x

to find the required amount of paint.

Key Concept

Solving direct proportions

Question 96Question →

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 97Question →

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 98Question →

Which of the following statements about integers is always true?

The difference between any integer and its absolute value is always an even integer.

The product of any non-zero integer and its absolute value is always a positive integer.

Subtracting a negative integer from its absolute value always results in zero.

The sum of any negative integer and its absolute value is always a negative integer.

Show answer & explanation

Answer: The difference between any integer and its absolute value is always an even integer.

Answer

The statement claiming that the difference between any integer and its absolute value is always an even integer.

For any non-negative integer, the difference between the integer and its absolute value is 0, which is an even integer. For any negative integer, the difference evaluates to twice the integer (2x), which is a multiple of 2 and therefore also even. Thus, the statement holds true for all integers.

Step-by-Step Solution

1

Analyze the behavior of the expression (x - |x|) for non-negative integers.

If x is 0 or positive, |x| = x. Therefore, x - |x| = x - x = 0.

To test the property for the first distinct category of integers.

2

Analyze the behavior of the expression (x - |x|) for negative integers.

If x is negative, |x| = -x. Therefore, x - |x| = x - (-x) = x + x = 2x.

To test the property for the remaining category of integers.

3

Determine the parity (odd/even status) of the results from the previous steps.

The number 0 is an even integer. The expression 2x is inherently a multiple of 2, making it an even integer regardless of the value of x.

To conclude whether the difference is universally even.

Key Concept

Parity and Absolute Value Properties

Question 99Question →

An aviation squadron requires $3$ hours of preventive maintenance for every $8$ hours of flight time. If a specific helicopter undergoes $24$ hours of preventive maintenance during a given month, what is the total number of hours logged for both flight and maintenance combined?

33

64

88

192

Show answer & explanation

Answer: $88$

Answer

The total number of hours logged is

88

.

The given ratio of maintenance to flight time is

3:8

. Because the helicopter had

24

hours of maintenance, we find the multiplier by dividing

24

by

3

, which is

8

. We then apply this multiplier to the flight time ratio (

8 \times 8 = 64

hours). To find the total combined hours, we add the maintenance hours and flight hours together:

24 + 64 = 88

.

Step-by-Step Solution

1

Find the scale factor (multiplier) by comparing the actual maintenance hours to the ratio.

24 \div 3 = 8

This establishes how many times larger the actual numbers are compared to the base ratio.

2

Calculate the actual flight hours using the scale factor.

8 \times 8 = 64

hours of flight time

Applying the scale factor to the flight time ratio gives the actual flight time.

3

Add the actual maintenance hours and actual flight hours to find the total time.

24 + 64 = 88

total hours

The question specifically asks for the combined total of both flight and maintenance time.

Key Concept

Solving part-to-part ratio word problems and calculating a total sum.

Question 100Question →

A military supply shipment contains $3.4 \times 10^4$ ready-to-eat meals. What is the total number of meals written in standard numerical form?

Show answer & explanation

Answer: 34000

Answer

34000

The expression

3.4 \times 10^4

represents

3.4

multiplied by

10,000

. Shifting the decimal point 4 places to the right results in exactly

34,000

.

Step-by-Step Solution

1

Identify the exponent on the base 10.

The exponent is 4.

The exponent determines the number of places the decimal point must be shifted.

2

Move the decimal point 4 places to the right.

3.4 becomes 34000.

Multiplying by 10^4 is equivalent to shifting the decimal to the right by 4 places to convert the scientific notation into standard form.

Key Concept

Converting from scientific notation to standard form involves shifting the decimal point based on the value and sign of the exponent.

Number Theory and Operations

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