Number Theory and Operations Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 4

Question 61Question →

A custom mathematical operation is defined for all integers such that $a \star b = |a - b| - (a \times b)$ . If $a$ represents the largest negative integer and $b$ represents the smallest prime number, what is the value of the expression $a \star b$ ?

5

1

-1

8

Show answer & explanation

Answer: 5

Answer

5

The correct value is 5. First, identify the specific integers based on their definitions: the largest negative integer is -1 (so a = -1), and the smallest prime number is 2 (so b = 2). Substituting these values into the given operation yields |-1 - 2| - (-1 \times 2). Evaluate the grouping inside the absolute value first: |-1 - 2| = |-3| = 3. Next, evaluate the multiplication: -1 \times 2 = -2. Finally, subtract the product from the absolute value result: 3 - (-2), which simplifies to 3 + 2 = 5.

Step-by-Step Solution

1

Identify the numerical values of the integers a and b based on their mathematical properties.

a = -1 and b = 2

The largest negative integer is -1 (as it is closest to zero on the negative side), and the smallest prime number is 2 (the only even prime).

2

Substitute the values of a and b into the custom operation expression.

|-1 - 2| - (-1 \times 2)

Direct substitution is required to evaluate a \star b.

3

Simplify the arithmetic inside the absolute value brackets.

|-3| - (-1 \times 2)

According to the order of operations, operations inside grouping symbols must be completed first.

4

Evaluate the absolute value and perform the multiplication.

3 - (-2)

The absolute value of -3 is 3 (distance from zero is always positive), and multiplying -1 by 2 yields -2.

5

Perform the final subtraction operation.

5

Subtracting a negative number is mathematically equivalent to adding a positive number: 3 + 2 = 5.

Key Concept

Applying fundamental integer definitions (prime numbers, negative bounds) alongside absolute value rules and the order of operations.

Question 62Question →

A construction battalion is preparing a specialized concrete mix. The ratio of cement to sand by weight is $2:5$ , and the ratio of sand to gravel is $3:4$ . If the battalion needs to prepare exactly $3,280$ pounds of this concrete mix, how many pounds of gravel are required?

Show answer & explanation

Answer: 1600; 1,600; 1600 pounds; 1,600 pounds; 1600 lbs; 1,600 lbs

Answer

1600 pounds

By converting the separate part-to-part ratios (

2:5

and

3:4

) into a single continuous ratio (

6:15:20

), we determine that gravel accounts for

20

out of the

41

total parts. Multiplying the total weight (

3,280

) by the fraction

\frac{20}{41}

yields exactly

1,600

pounds.

Step-by-Step Solution

1

Determine the combined continuous ratio of cement, sand, and gravel.

Cement:Sand:Gravel =

6:15:20

Since sand is the common element, find the least common multiple of its ratio shares (

5

and

3

), which is

15

. Multiply the first ratio by

3

and the second by

5

to align them.

2

Calculate the total number of parts in the combined ratio.

6 + 15 + 20 = 41

parts

To find the fraction of the total mix each part represents, all ratio parts must be summed.

3

Determine the weight of a single part.

3,280 \div 41 = 80

pounds per part

Divide the total weight by the total number of parts.

4

Calculate the total weight of the gravel.

20 \times 80 = 1,600

pounds

Multiply the weight of one part by the number of parts of gravel in the combined ratio.

Key Concept

Combined Ratios and Part-to-Whole Relationships

Alternative Method

You can also use algebra by setting the multiplier for the first ratio as

x

. Cement =

2x

and sand =

5x

. Since the ratio of sand to gravel is

3:4

, gravel must be

\frac{4}{3}

of sand, which is

\frac{20}{3}x

. The total is

2x + 5x + \frac{20}{3}x = \frac{41}{3}x = 3,280

. Solving for

x

gives

240

. Gravel is

\frac{20}{3}(240) = 1,600

.

Estimated Time:1m 30s

Question 63Question →

A tactical backpack originally costs $\$80$ . If it is on sale for a $25\%$ discount, what is the final sale price of the backpack?

\$60

\$20

\$55

\$100

Show answer & explanation

Answer: $\$60$

Answer

The final sale price of the backpack is

\$60

.

The correct answer is found by first calculating

25\%

of

\$80

, which is

\$20

. Since it is a discount, this

\$20

is subtracted from the original

\$80

price, resulting in a final sale price of

\$60

. Alternatively, a

25\%

discount means the buyer pays

75\%

of the original price, and

0.75 \times \$80 = \$60

.

Step-by-Step Solution

1

Calculate the amount of the discount.

0.25 \times \$80 = \$20

To find the discount amount, multiply the original price by the decimal form of the percentage (

25\% = 0.25

).

2

Subtract the discount from the original price.

\$80 - \$20 = \$60

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

Key Concept

Calculating a percentage of a number and applying it as a discount.

Question 64Question →

A recruit training battalion consists of $500$ recruits. If $15\%$ of the recruits are scheduled for a dental examination today, how many recruits are scheduled for the examination?

15

75

425

750

Show answer & explanation

Answer: 75

Answer

75

To find

15\%

of

500

, the percentage must be converted to a decimal (

0.15

) and multiplied by the total base amount. Alternatively, using mental math,

10\%

of

500

is

50

, and

5\%

is

25

. Adding those together yields

50 + 25 = 75

.

Step-by-Step Solution

1

Convert the percentage to a decimal or fraction.

15\% = 0.15

Percentages must be converted into a workable decimal format before multiplying by the base number.

2

Multiply the total number of recruits by the decimal value.

500 \times 0.15 = 75

Multiplying the base amount by the percentage rate gives the final part.

Key Concept

Calculating a basic percentage of a whole number.

Question 65Question →

Compute the final value of the following fraction by correctly applying the order of operations:

\frac{-3(4 - 7)^2 + 15}{2^3 - 4 \div 2}

Which of the following is the correct result?

-6

7

-2

16

Show answer & explanation

Answer: -2

Answer

-2

The correct answer is found by evaluating the numerator and denominator independently using the order of operations. In the numerator, the subtraction inside the parentheses is performed first (4 minus 7 is -3), followed by the exponent (-3 squared is 9). Multiplying by -3 gives -27, and adding 15 results in -12. In the denominator, the exponent is evaluated first (2 cubed is 8), followed by the division (4 divided by 2 is 2). Subtracting 2 from 8 gives 6. Finally, dividing the numerator by the denominator (-12 divided by 6) yields -2.

Step-by-Step Solution

1

Simplify the expression inside the parentheses in the numerator.

4 - 7 = -3

According to the order of operations, operations inside grouping symbols must be completed first.

2

Evaluate the exponents in both the numerator and the denominator.

Numerator: (-3)^2 = 9. Denominator: 2^3 = 8.

Exponents are the next highest priority after grouping symbols.

3

Perform multiplication and division operations from left to right.

Numerator: -3 \times 9 = -27. Denominator: 4 \div 2 = 2.

Multiplication and division take precedence over addition and subtraction.

4

Perform the remaining addition and subtraction operations.

Numerator: -27 + 15 = -12. Denominator: 8 - 2 = 6.

Addition and subtraction are the final steps before resolving the main fraction bar.

5

Divide the simplified numerator by the simplified denominator.

-12 \div 6 = -2

The fraction bar acts as a final division operation grouping the entire numerator and denominator.

Key Concept

Order of operations with rational expressions and signed numbers

Question 66Question →

A maintenance squad has $250$ spark plugs in their inventory. After a week of vehicle repairs, the inventory is reduced by $40\%$ . How many spark plugs are left in the inventory?

Show answer & explanation

Answer: 150

Answer

150

Since

40\%

of the inventory was used,

60\%

remains. Multiplying the original

250

spark plugs by

0.60

results in

150

spark plugs left.

Step-by-Step Solution

1

Calculate the percentage of the inventory that remains.

100\% - 40\% = 60\%

The total starting inventory represents

100\%

. Subtracting the

40\%

reduction gives the percentage that is left.

2

Calculate

60\%

of the initial

250

spark plugs.

250 \times 0.60 = 150

Multiplying the initial amount by the decimal form of the remaining percentage gives the final quantity.

Key Concept

Calculating the remaining quantity after a percentage decrease.

Question 67Question →

Evaluate the following mathematical expression:

|-12| - (-4) \times 2

Which of the following is the correct calculated value?

32

4

-4

20

Show answer & explanation

Answer: $20$

Answer

The correct result of the expression is

20

.

Following the order of operations, first evaluate the absolute value:

|-12| = 12

. Next, perform the multiplication:

-4 \times 2 = -8

. Finally, perform the subtraction:

12 - (-8)

. Since subtracting a negative number is the same as adding a positive number, the expression becomes

12 + 8 = 20

.

Step-by-Step Solution

1

Evaluate the absolute value

|-12| = 12

Absolute value represents the distance from zero on a number line, so it is always non-negative.

2

Perform the multiplication

-4 \times 2 = -8

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

3

Subtract the product from the absolute value result

12 - (-8) = 12 + 8 = 20

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

Key Concept

Applying the order of operations (PEMDAS), evaluating absolute values, and following integer sign rules.

Question 68Question →

What is the absolute difference between the smallest positive integer and the largest negative integer?

Show answer & explanation

Answer: 2

Answer

2

The smallest positive integer is 1 and the largest negative integer is -1. The absolute difference between them is found by subtracting the two values and taking the absolute value:

|1 - (-1)| = |1 + 1| = 2

.

Step-by-Step Solution

1

Identify the smallest positive integer.

1

The integer 0 is neither positive nor negative. The positive integers start at 1, making it the smallest positive integer.

2

Identify the largest negative integer.

-1

Negative integers increase in value as they get closer to zero. Therefore, -1 is the largest negative integer.

3

Calculate the absolute difference.

2

The absolute difference represents the total distance between the two numbers on the number line. Calculate this as

|1 - (-1)| = |1 + 1| = 2

.

Key Concept

Properties of Integers (Number Line and Absolute Value)

Question 69Question →

In a mechanical winch system, three interlocking gears are used to lift cargo. Gear A completes $4$ revolutions for every $3$ revolutions Gear B completes. Gear B completes $2$ revolutions for every $5$ revolutions Gear C completes. If Gear C spins at a rate of $120$ revolutions per minute, how many revolutions will Gear A complete in one minute?

48

64

90

160

Show answer & explanation

Answer: $64$

Answer

Gear A will complete

64

revolutions in one minute.

The correct answer successfully steps through the proportion chain. First, finding Gear B's revolutions (

120 \times \frac{2}{5} = 48

), and then using Gear B's value to find Gear A's revolutions (

48 \times \frac{4}{3} = 64

).

Step-by-Step Solution

1

Find the number of revolutions for Gear B using its ratio with Gear C.

Gear B completes

48

revolutions.

The ratio of Gear B to Gear C is

2:5

. Setting up the proportion

\frac{B}{120} = \frac{2}{5}

gives

B = \frac{120 \times 2}{5} = 48

.

2

Find the number of revolutions for Gear A using its ratio with Gear B.

Gear A completes

64

revolutions.

The ratio of Gear A to Gear B is

4:3

. Setting up the proportion

\frac{A}{48} = \frac{4}{3}

gives

A = \frac{48 \times 4}{3} = 64

.

Key Concept

Multi-step ratios and proportions

Estimated Time:1m 0s

Question 70Question →

An important part of algebraic evaluation is applying operations in the correct sequence. Consider the mathematical expression provided below:

-3^2 + 4 \times [15 \div (2 - 5)] - (-6)

What is the final calculated value of this expression?

Show answer & explanation

Answer: -23

Answer

-23

Following the strict order of operations, we first resolve the grouping symbols from the inside out to get

-5

for the bracketed term. Next, we evaluate the isolated exponent

-3^2

as

-9

. Then, we multiply the

4

by our bracketed result of

-5

to get

-20

. Finally, we perform addition and subtraction from left to right:

-9 + (-20) + 6

, which results in

-23

.

Step-by-Step Solution

1

Evaluate innermost parentheses

(2 - 5) = -3

Order of operations dictates solving expressions inside the innermost grouping symbols first.

2

Evaluate brackets

[15 \div (-3)] = -5

Continue evaluating operations inside the next level of grouping symbols.

3

Evaluate exponent

-3^2 = -9

Exponents are evaluated before multiplication and division. The negative sign is outside the base, so only the 3 is squared.

4

Perform multiplication

4 \times (-5) = -20

Multiplication and division are performed from left to right before any addition or subtraction.

5

Perform addition and subtraction

-9 + (-20) - (-6) = -23

Add and subtract from left to right. Subtracting a negative number is mathematically equivalent to adding a positive number.

Key Concept

Applying the order of operations (PEMDAS) to expressions with nested grouping symbols, exponents, and signed numbers.

Question 71Question →

During a physical fitness assessment, $68$ recruits met the advanced standard. If these recruits represent $17\%$ of the total training company, what is the total number of recruits in the company?

Show answer & explanation

Answer: 400

Answer

400

The correct total is found by recognizing the relationship:

\text{Part} = \text{Rate} \times \text{Base}

. We are given the Part (

68

) and the Rate (

17\%

or

0.17

). To find the Base, we divide the Part by the Rate:

68 \div 0.17 = 400

.

Step-by-Step Solution

1

Set up an algebraic equation relating the given part to the unknown whole.

0.17 \times x = 68

The problem states that

17\%

of the total class (

x

) is equal to

68

recruits.

2

Isolate the variable

x

to solve for the total.

x = \frac{68}{0.17}

Dividing both sides of the equation by

0.17

will leave

x

by itself.

3

Perform the division without a calculator.

x = 400

Multiply the numerator and denominator by

100

to remove the decimal, resulting in

\frac{6800}{17}

. Since

17 \times 4 = 68

,

\frac{6800}{17}

equals

400

.

Key Concept

Finding the original base when a percentage rate and its corresponding amount are known.

Estimated Time:1m 0s

Question 72Question →

A military mess hall prepares meals using a ratio of $4$ pounds of potatoes for every $15$ soldiers. How many pounds of potatoes are needed to feed $60$ soldiers?

Show answer & explanation

Answer: 16; 16 pounds; 16 lbs; 16 lb

Answer

16

Since the ratio of potatoes to soldiers is constant, and the number of soldiers increases by a scale factor of

4

(from

15

to

60

), the pounds of potatoes must also be multiplied by

4

(

4 \times 4 = 16

).

Step-by-Step Solution

1

Set up a proportion matching pounds of potatoes to the number of soldiers.

\frac{4}{15} = \frac{x}{60}

This proportion relates the known ratio (

4

pounds for

15

soldiers) to the unknown quantity

x

for

60

soldiers.

2

Solve for the unknown quantity

x

.

x = \frac{4 \times 60}{15} = 4 \times 4 = 16

Solving the equation gives the total pounds of potatoes needed to maintain the same ratio.

Key Concept

Direct Proportions

Estimated Time:45s

Question 73Question →

What is the evaluated result of the expression shown below?

24 \div 2^3 \times 3 - (5 - 8)^2 + |-10|

10

2

28

-10

Show answer & explanation

Answer: 10

Answer

The correct result is 10, achieved by evaluating grouping symbols and exponents first, followed by left-to-right division and multiplication, and finally left-to-right subtraction and addition.

The correct answer is 10. By following the standard order of operations, grouping symbols and exponents are resolved first to yield

24 \div 8 \times 3 - 9 + 10

. Next, executing multiplication and division strictly from left to right gives

9 - 9 + 10

. Finally, executing addition and subtraction strictly from left to right yields

0 + 10 = 10

.

Step-by-Step Solution

1

Simplify the expressions inside the parentheses and absolute value bars.

24 \div 2^3 \times 3 - (-3)^2 + 10

Grouping symbols must be resolved first according to the standard order of operations.

2

Evaluate the exponential terms.

24 \div 8 \times 3 - 9 + 10

Exponents are the second priority step and must be resolved before multiplication or division.

3

Perform multiplication and division sequentially from left to right.

First,

24 \div 8 = 3

. Then,

3 \times 3 = 9

. The expression becomes

9 - 9 + 10

.

Multiplication and division share the exact same priority level and must be executed strictly from left to right.

4

Perform addition and subtraction sequentially from left to right.

First,

9 - 9 = 0

. Then,

0 + 10 = 10

.

Addition and subtraction share the same priority level and must also be executed strictly from left to right.

Key Concept

Order of Operations (PEMDAS/BODMAS) and Left-to-Right Priority

Estimated Time:1m 0s

Question 74Question →

A military base assigns personnel to guard duty and patrol duty in a ratio of $5:2$ . If there are $40$ personnel assigned to guard duty, what is the total number of personnel assigned to both duties combined?

16

56

100

140

Show answer & explanation

Answer: 56

Answer

The total number of personnel assigned to both duties is 56.

The correct answer accurately determines that 5 parts equal 40 personnel, meaning 1 part is equal to 8 personnel. The total number of personnel represents 7 parts in total (5 + 2). Multiplying 7 total parts by 8 personnel per part results in 56 total personnel.

Step-by-Step Solution

1

Determine the value of one part of the ratio.

1 part = 8 personnel

The ratio of guard to patrol duty is 5:2, meaning 5 parts correspond to the 40 personnel on guard duty. Dividing 40 by 5 gives the value of a single part.

2

Calculate the total number of parts in the given ratio.

Total parts = 7

The total personnel is the sum of guard duty parts (5) and patrol duty parts (2).

3

Multiply the total parts by the value of one part.

7 × 8 = 56 personnel

Multiplying the total number of parts by the personnel value of each individual part gives the combined total.

Key Concept

Solving part-to-whole ratio problems by determining the value of a single unit part.

Question 75Question →

A logistics facility implements an energy-saving program that reduces its daily electricity consumption by $20\%$ . Six months later, expanded operations require the facility to increase its new, reduced consumption by $30\%$ . What is the net percentage change in the facility's daily electricity consumption compared to the original amount?

4\%

increase

10\%

increase

24\%

increase

16\%

decrease

Show answer & explanation

Answer: $4\%$ increase

Answer

The net change is a

4\%

increase compared to the original electricity consumption.

To find the net percentage change, each change must be applied sequentially. A

20\%

decrease leaves

80\%

of the original amount. A subsequent

30\%

increase is calculated on this

80\%

(

0.30 \times 80\% = 24\%

). Adding this to the reduced amount gives

80\% + 24\% = 104\%

. The final amount is

104\%

of the original, representing a

4\%

net increase.

Step-by-Step Solution

1

Establish an initial baseline value to simplify calculations.

Assume the original daily electricity consumption is

100

units.

Using

100

makes percentage calculations intuitive and straightforward.

2

Calculate the consumption after the initial

20\%

reduction.

100 - (0.20 \times 100) = 80

units.

The facility reduces its original consumption by

20\%

.

3

Calculate the final consumption after the subsequent

30\%

increase.

80 + (0.30 \times 80) = 80 + 24 = 104

units.

The

30\%

increase must be applied to the new, reduced base of

80

units, not the original

100

units.

4

Determine the net percentage change from the original baseline.

104 - 100 = +4

, which represents a

4\%

increase.

Comparing the final amount (

104

) to the original amount (

100

) yields the exact net percentage change.

Key Concept

Successive Percentage Changes

Question 76Question →

When the expression $\frac{(8.0 \times 10^4) \times (6.0 \times 10^{-7})}{1.5 \times 10^3}$ is evaluated and the result is written in standard scientific notation as $a \times 10^n$ , where $1 \le a < 10$ , what is the value of the integer $n$ ?

Show answer & explanation

Answer: -5

Answer

The exponent is -5.

Evaluating the numerator yields

48 \times 10^{-3}

. Dividing this by the denominator

1.5 \times 10^3

results in

32 \times 10^{-6}

. To write this in standard scientific notation, the coefficient must be strictly greater than or equal to 1 and strictly less than 10, so 32 is rewritten as

3.2 \times 10^1

. Applying this shift makes the final expression

3.2 \times 10^{-5}

. Thus, the final exponent

n

is -5.

Step-by-Step Solution

1

Multiply the expressions in the numerator

48 \times 10^{-3}

Following the order of operations, simplify the numerator first by multiplying the coefficients (

8.0 \times 6.0 = 48

) and adding the exponents of the base 10 (

4 + (-7) = -3

).

2

Divide by the denominator

32 \times 10^{-6}

Divide the resulting numerator's coefficient by the denominator's coefficient (

48 \div 1.5 = 32

) and subtract the denominator's exponent from the numerator's exponent (

-3 - 3 = -6

).

3

Normalize the coefficient to standard scientific notation

3.2 \times 10^{-5}

Standard scientific notation requires the coefficient (

a

) to be between 1 and 10. We rewrite

32

as

3.2 \times 10^1

. Multiplying this by the existing

10^{-6}

changes the overall exponent to

1 + (-6) = -5

.

4

Identify the final value of

n

-5

The question asks for the integer

n

from the format

a \times 10^n

. Comparing our final result

3.2 \times 10^{-5}

to this format,

n = -5

.

Key Concept

Operations with scientific notation and coefficient normalization

Question 77Question →

A flashing red light blinks every $12$ seconds, and a flashing blue light blinks every $15$ seconds. If both lights blink together at exactly 8:00 PM, how many seconds will pass before they blink together again?

3

27

60

180

Show answer & explanation

Answer: 60

Answer

60

To find when two periodic events will occur together again, the least common multiple (LCM) of their individual intervals must be determined. The LCM of

12

and

15

is

60

, meaning the lights will blink together every

60

seconds.

Step-by-Step Solution

1

Identify the goal of the problem.

Find the Least Common Multiple (LCM) of the two blinking intervals to determine when the lights will sync up again.

Because the lights blink periodically, their next simultaneous blink occurs at a common multiple of both intervals.

2

Find the prime factorization of 12 and 15.

12 = 2^2 \times 3

, and

15 = 3 \times 5

.

Prime factorization is a reliable method for finding the LCM.

3

Calculate the LCM using the prime factors.

LCM = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60

.

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

Key Concept

Least Common Multiple (LCM) applied to periodic events

Question 78Question →

Three automated beacons are programmed to flash at specific intervals. Beacon A flashes every $24$ seconds, Beacon B flashes every $36$ seconds, and Beacon C flashes every $45$ seconds. If all three beacons flash simultaneously at exactly 8:00:00 PM, how many more times will they flash together within exactly one hour?

6

10

330

1,200

Show answer & explanation

Answer: 10

Answer

10

To determine when multiple periodic events occur at the exact same time, we must find their Least Common Multiple (LCM). Using prime factorization, we find the LCM of

24

,

36

, and

45

by taking the highest power of each distinct prime factor present:

2^3 \times 3^2 \times 5 = 360

seconds. This means the beacons align and flash together once every

360

seconds. Because the total timeframe is exactly one hour, we convert this duration to seconds (

60 \text{ minutes} \times 60 \text{ seconds/minute} = 3,600 \text{ seconds}

). Finally, dividing the total time by the simultaneous interval (

3,600 \div 360

) results in exactly

10

additional synchronized flashes.

Step-by-Step Solution

1

Determine the prime factorization of each beacon's flashing interval.

24 = 2^3 \times 3

,

36 = 2^2 \times 3^2

, and

45 = 3^2 \times 5

Finding the prime factors is necessary to calculate the Least Common Multiple (LCM), which dictates when the independent events align.

2

Calculate the Least Common Multiple (LCM) of the three intervals.

\text{LCM} = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360

seconds

The LCM represents the exact time interval in seconds at which all three beacons will flash together again.

3

Convert the total given time window (one hour) into the same units (seconds).

1 \text{ hour} = 60 \text{ minutes} \times 60 \text{ seconds/minute} = 3,600 \text{ seconds}

Both the flashing interval and the total time frame must be in identical units before dividing.

4

Divide the total time window by the LCM interval.

3,600 \div 360 = 10

This division yields the total number of times the beacons will synchronize during the designated one-hour period.

Key Concept

Applying the Least Common Multiple (LCM) and unit conversions to solve periodic real-world alignment problems.

Question 79Question →

Determine the total number of positive factors for each of the following integers. Then, place the integers in order from the one with the fewest factors to the one with the most factors.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence is: The integer

7

(2 factors), The integer

9

(3 factors), The integer

8

(4 factors), and The integer

12

(6 factors).

By counting the positive divisors of each number, we find that

7

has two factors,

9

has three factors,

8

has four factors, and

12

has six factors. Placing them in ascending order of their factor counts yields

7, 9, 8

, and

12

.

Step-by-Step Solution

1

List all positive factors for the integer

12

.

The factors of

12

are

1, 2, 3, 4, 6

, and

12

. There are

6

factors in total.

Finding the total count of factors is required to compare and order the integers.

2

List all positive factors for the integer

7

.

The factors of

7

are

1

and

7

. There are

2

factors in total.

Determining the factor count for each number allows for proper sorting.

3

List all positive factors for the integer

8

.

The factors of

8

are

1, 2, 4

, and

8

. There are

4

factors in total.

We must evaluate every integer provided in the list.

4

List all positive factors for the integer

9

.

The factors of

9

are

1, 3

, and

9

. There are

3

factors in total.

This completes the factor counting for all items.

5

Sort the integers based on their factor counts from fewest to most.

The sequence is

2

factors (The integer

7

), then

3

factors (The integer

9

), then

4

factors (The integer

8

), and finally

6

factors (The integer

12

).

The question asks to place the integers in order from fewest factors to most factors.

Key Concept

Factors and Prime Numbers

Estimated Time:45s

Question 80Question →

Determine the solutions for $x$ in the given absolute value equation.

Fill in the blanks below

Given the equation

2|5 - x| - 4 = 10

, the smaller value of

x

is and the larger value of

x

is .

Show answer & explanation

Answer

The smaller value is -2 and the larger value is 12.

To solve

2|5 - x| - 4 = 10

, first isolate the absolute value term. Adding 4 to both sides yields

2|5 - x| = 14

. Dividing both sides by 2 gives

|5 - x| = 7

. The absolute value of an expression equals a positive number when the expression itself equals that positive number or its negative counterpart. Therefore, we set up two cases:

5 - x = 7

and

5 - x = -7

. Solving the first case gives

-x = 2

, so

x = -2

. Solving the second case gives

-x = -12

, so

x = 12

. Comparing the two solutions, the smaller value is

-2

and the larger value is

12

.

Step-by-Step Solution

1

Isolate the absolute value expression.

Add 4 to both sides to get

2|5 - x| = 14

, then divide by 2 to get

|5 - x| = 7

.

The absolute value must be isolated before separating the equation into cases.

2

Set up the positive case equation.

5 - x = 7

The first case assumes the expression inside the absolute value is positive or zero.

3

Solve the positive case.

Subtract 5 to get

-x = 2

, which means

x = -2

.

This finds the first potential value for

x

.

4

Set up the negative case equation.

5 - x = -7

The second case accounts for the expression inside the absolute value being negative.

5

Solve the negative case.

Subtract 5 to get

-x = -12

, which means

x = 12

.

This finds the second potential value for

x

.

Key Concept

Solving Linear Absolute Value Equations

Estimated Time:1m 30s

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