Number Theory and Operations Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 3

Question 41Question →

A logistics convoy is transporting supplies to a remote base. On the first day of the trip, they complete $30\%$ of the total distance. On the second day, they complete $40\%$ of the remaining distance. If they still have $210$ miles left to travel, what is the total distance of the trip in miles?

350

500

700

750

Show answer & explanation

Answer: 500

Answer

The total distance of the trip is 500 miles.

The correct answer accurately accounts for the successive percentages by applying the second day's percentage to the remaining distance. After traveling 30% of the trip, 70% remains. Leaving 40% of this remaining 70% completed means 60% of the 70% is left uncompleted, which equals 42% of the total distance. Setting 42% equal to the 210 remaining miles gives a total distance of 500 miles.

Step-by-Step Solution

1

Determine the percentage of the trip remaining after the first day.

100\% - 30\% = 70\%

The second day's travel is based on the remaining distance, not the total distance.

2

Calculate the percentage of the remaining distance that is left after the second day.

100\% - 40\% = 60\%

Since they completed 40% of the remaining distance, 60% of that remaining distance is still uncompleted.

3

Calculate the percentage of the total distance that the final remaining 210 miles represents.

0.70 \times 0.60 = 0.42

, which is

42\%

of the total distance.

Successive percentages are applied by multiplying the decimal forms of the remaining portions.

4

Set up an equation to solve for the total distance, D.

0.42D = 210 \implies D = 210 \div 0.42 = 500

Dividing the remaining miles by the remaining percentage of the total trip yields the original total distance.

Key Concept

Calculating successive percentage changes by adjusting the base for each subsequent calculation.

Question 42Question →

Evaluate each of the following arithmetic expressions. Then, arrange the expressions in order from the smallest resulting value to the largest resulting value.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from smallest to largest is:

10 - 2 \times (4 - 1)^2

,

10 - (2 \times 4) - 1^2

,

10 - 2 \times 4 - (-1)^3

, and

(10 - 2) \times 4 - 1^2

.

Evaluating each expression using the correct order of operations yields -8, 31, 1, and 3 respectively. Arranging these integer values from smallest to largest dictates the correct order.

Step-by-Step Solution

1

Evaluate the first expression:

10 - 2 \times (4 - 1)^2

-8

Applying PEMDAS: solve inside parentheses

(3)

, apply the exponent

(3^2 = 9)

, multiply

(2 \times 9 = 18)

, and subtract

(10 - 18 = -8)

.

2

Evaluate the second expression:

(10 - 2) \times 4 - 1^2

31

Applying PEMDAS: solve inside parentheses

(8)

, apply the exponent

(1^2 = 1)

, multiply

(8 \times 4 = 32)

, and subtract

(32 - 1 = 31)

.

3

Evaluate the third expression:

10 - (2 \times 4) - 1^2

1

Applying PEMDAS: multiply inside parentheses

(8)

, apply the exponent

(1^2 = 1)

, and subtract from left to right

(10 - 8 = 2; 2 - 1 = 1)

.

4

Evaluate the fourth expression:

10 - 2 \times 4 - (-1)^3

3

Applying PEMDAS: apply the exponent

((-1)^3 = -1)

, multiply

(2 \times 4 = 8)

, and subtract from left to right

(10 - 8 = 2; 2 - (-1) = 3)

.

5

Arrange the evaluated numbers from smallest to largest

-8 < 1 < 3 < 31

Comparing the integer values yields the final logical order of the respective expressions.

Key Concept

Evaluating arithmetic expressions sequentially by applying the correct order of operations (PEMDAS) and then sorting the numerical values.

Question 43Question →

A naval fleet has a certain number of vessels. Currently, $20\%$ of the fleet is undergoing long-term maintenance. Of the operational vessels remaining, $25\%$ are currently deployed on active missions. If there are exactly $33$ operational vessels remaining at the home port, what is the total number of vessels in the entire fleet?

44

60

55

165

Show answer & explanation

Answer: $55$

Answer

The total number of vessels in the entire fleet is

55

.

The correct answer is found by tracking the remaining percentages step-by-step. Since 20% of the fleet is in maintenance, 80% remains operational. The 25% deployment rate applies to this 80%, meaning 20% of the total fleet is deployed (0.25 * 0.80 = 0.20). Subtracting the deployed vessels leaves 60% of the original fleet at the port. Finally, dividing the 33 vessels by 0.60 yields the original total fleet size of 55.

Step-by-Step Solution

1

Determine the percentage of the fleet that is not in maintenance.

100\% - 20\% = 80\%

The

25\%

deployment rate applies only to the remaining operational vessels.

2

Calculate the percentage of the total fleet that is deployed.

25\%

of

80\% = 0.25 \times 80\% = 20\%

To find the deployed amount relative to the whole fleet, multiply the successive percentages.

3

Determine the percentage of the total fleet that remains at the home port.

80\% - 20\% = 60\%

Subtracting the deployed vessels from the operational vessels gives the portion remaining at port.

4

Set up an equation and solve for the total number of vessels in the fleet (

x

).

0.60x = 33 \Rightarrow x = \frac{33}{0.60} = 55

The

60\%

of the total fleet is equal to the

33

vessels explicitly counted at the port.

Key Concept

Successive Percentage Calculations and Reversing Percentages

Question 44Question →

In a military motor pool, the ratio of trucks to jeeps is $2:3$ , and the ratio of jeeps to motorcycles is $4:5$ . If there are exactly $140$ vehicles among these three types, how many more motorcycles are there than trucks?

28

30

42

60

Show answer & explanation

Answer: 28

Answer

There are 28 more motorcycles than trucks.

To solve this, the two separate ratios must be unified. By finding the least common multiple for the overlapping term (jeeps, which are represented by 3 in the first ratio and 4 in the second), we scale the ratios to 8:12 and 12:15. This creates a combined ratio of 8:12:15 for trucks to jeeps to motorcycles. Adding these parts gives 35 total proportional units. Dividing the actual total (140 vehicles) by 35 units reveals that each unit equals 4 vehicles. The difference between motorcycles (15 units) and trucks (8 units) is 7 units. Multiplying 7 units by 4 vehicles per unit yields 28.

Step-by-Step Solution

1

Identify the common entity between the two given ratios to create a single, unified ratio.

Jeeps are the common entity. Trucks : Jeeps =

2:3

, and Jeeps : Motorcycles =

4:5

.

You cannot compare trucks and motorcycles directly or find the total parts until all three vehicles share the same proportional scale.

2

Find the least common multiple (LCM) for the 'Jeeps' portion of both ratios and scale them.

The LCM of

3

and

4

is

12

. Multiply the first ratio by

4

to get

8:12

. Multiply the second ratio by

3

to get

12:15

. The combined ratio of Trucks : Jeeps : Motorcycles is

8:12:15

.

Scaling ensures the common term is equal in both ratios, allowing them to be combined safely.

3

Calculate the value of a single ratio part based on the total number of vehicles.

Total parts =

8 + 12 + 15 = 35

. Divide the total vehicles by total parts:

140 \div 35 = 4

vehicles per part.

This establishes the multiplier needed to convert ratio units into actual vehicle counts.

4

Find the difference in parts between motorcycles and trucks, then multiply by the part value.

Motorcycles (

15

parts) minus trucks (

8

parts) equals

7

parts.

7 \times 4 = 28

vehicles.

Subtracting the parts before multiplying by the scale factor is more efficient than finding the total of each vehicle first, though both methods yield 28.

Key Concept

Combining multiple distinct ratios using a least common multiple to form a unified proportional relationship.

Question 45Question →

Simplify the given mathematical expression. Which of the following represents its final value?

-5 - 3 \times (-2) + 4

-11

-7

5

20

Show answer & explanation

Answer: 5

Answer

5

According to the order of operations, multiplication must be evaluated before addition and subtraction. First, multiply 3 by -2 to get -6. Subtracting this negative product from -5 gives -5 - (-6), which correctly simplifies to -5 + 6 = 1. Finally, add 4 to reach the correct final answer of 5.

Step-by-Step Solution

1

Identify the operations in the expression and determine their correct order of execution.

The expression contains subtraction, multiplication, and addition. Multiplication must be performed first.

According to the standard order of operations (PEMDAS), multiplication has a higher priority than addition and subtraction.

2

Perform the multiplication step.

3 \times (-2) = -6

. The expression becomes

-5 - (-6) + 4

.

Calculate the product before processing the outer subtraction.

3

Simplify the subtraction of a negative number.

-5 - (-6) + 4

becomes

-5 + 6 + 4

.

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

4

Evaluate the remaining addition and subtraction from left to right.

-5 + 6 = 1

, and

1 + 4 = 5

.

Addition and subtraction hold the same operational priority and are resolved sequentially from left to right.

Key Concept

Order of Operations with Signed Integers

Alternative Method

Instead of subtracting the product of 3 and -2, you can treat the subtraction symbol as a negative sign attached to the 3. This changes the multiplication step to (-3) * (-2) = 6. The entire expression then immediately becomes -5 + 6 + 4, which easily sums to 5.

Estimated Time:45s

Question 46Question →

A naval shipyard increases its dry dock maintenance workforce by $12.5\%$ to meet new project demands. If the current workforce size after the increase is $810$ personnel, what was the original size of the workforce before the increase?

Show answer & explanation

Answer: 720

Answer

720

The scenario requires a reverse percentage calculation. The final count of

810

personnel represents the original workforce plus a

12.5\%

increase, meaning

810

is exactly

112.5\%

(or

\frac{9}{8}

) of the original amount. To find the original amount, you set up the equation

\frac{9}{8}x = 810

and solve for

x

by multiplying

810

by the reciprocal

\frac{8}{9}

, which equals

720

.

Step-by-Step Solution

1

Identify the relationship between the final amount and the initial amount.

A

12.5\%

increase means the final amount is

100\% + 12.5\% = 112.5\%

of the original amount.

To correctly set up an algebraic representation of the percentage growth based on the unknown starting value.

2

Convert the new percentage multiplier into a fraction.

Knowing that

12.5\% = \frac{1}{8}

, the multiplier for a

12.5\%

increase is

1 + \frac{1}{8} = \frac{9}{8}

.

Using fractions makes mental arithmetic much simpler and faster without a calculator.

3

Set up the equation to solve for the original amount.

Letting

x

represent the original workforce size, the equation is

\frac{9}{8}x = 810

.

This translates the word problem into a solvable mathematical equation.

4

Solve the equation for

x

.

x = 810 \times \frac{8}{9} = 90 \times 8 = 720

.

Isolating

x

provides the initial workforce size before the increase.

Key Concept

Reverse percentage calculations and percentage-to-fraction conversions.

Estimated Time:1m 0s

Question 47Question →

If $x = -3$ and $y = -5$ , what is the value of the following expression?

x^2 + 2x \times |y| - (x - y)

-41

-23

13

37

Show answer & explanation

Answer: -23

Answer

The correct answer is -23.

By accurately applying the rules for negative integers, (-3)^2 evaluates to a positive 9. The absolute value |-5| evaluates to a positive 5. Following the order of operations, the multiplication 2(-3) * 5 is performed next, yielding -30. Finally, subtracting the evaluation of the parenthetical group (2) from the sum of 9 and -30 results in exactly -23.

Step-by-Step Solution

1

Substitute the values of x and y into the expression.

(-3)^2 + 2(-3) \times |-5| - (-3 - (-5))

Direct substitution is the first step in evaluating algebraic expressions.

2

Evaluate the terms inside parentheses and absolute value brackets.

|-5| = 5

and

(-3 - (-5)) = (-3 + 5) = 2

. The expression becomes:

(-3)^2 + 2(-3) \times 5 - 2

According to the order of operations, grouping symbols like absolute value and parentheses must be resolved first.

3

Evaluate the exponent.

(-3)^2 = 9

. The expression becomes:

9 + 2(-3) \times 5 - 2

Exponents must be calculated before multiplication, addition, and subtraction. A negative number squared yields a positive result.

4

Perform the multiplication.

2(-3) \times 5 = -6 \times 5 = -30

. The expression becomes:

9 + (-30) - 2

Multiplication is performed before addition and subtraction.

5

Perform the remaining addition and subtraction from left to right.

9 - 30 - 2 = -21 - 2 = -23

Addition and subtraction are the final steps in the order of operations.

Key Concept

Evaluating expressions using properties of negative integers and absolute value

Estimated Time:1m 0s

Question 48Question →

A fuel tank at a military base is initially filled to $80\%$ of its total capacity. After a training exercise, the fuel level decreases by $35\%$ of its current volume. If $260$ gallons of fuel remain in the tank after the exercise, what is the total capacity of the fuel tank in gallons?

Show answer & explanation

Answer: 500

Answer

500

To find the total capacity, we must track the percentage of fuel at each stage relative to the total capacity. Initially, the tank holds

80\%

of its capacity, or

0.80

. A decrease of

35\%

means

65\%

of that fuel remains. Multiplying the initial proportion (

0.80

) by the remaining proportion (

0.65

) yields

0.52

. This indicates that the final

260

gallons represent exactly

52\%

of the tank's total capacity. Setting up the equation

0.52C = 260

and dividing

260

by

0.52

gives a total capacity of

500

gallons.

Step-by-Step Solution

1

Define a variable for the total capacity and express the initial fuel volume.

Initial volume =

0.80C

The tank is starting at

80\%

of its total capacity (

C

).

2

Determine the percentage of fuel remaining after the exercise.

Remaining percentage =

100\% - 35\% = 65\%

A

35\%

decrease means that

65\%

of the fuel that was in the tank before the exercise is still there.

3

Express the final remaining volume in terms of the total capacity (

C

).

0.65 \times 0.80C = 0.52C

Applying the

65\%

remaining proportion to the initial

0.80C

volume gives the final percentage of the total capacity.

4

Set up an equation using the known final volume and solve for

C

.

0.52C = 260 \Rightarrow C = \frac{260}{0.52} = 500

Equating the algebraic expression to the

260

gallons given in the problem isolates the unknown base capacity.

Key Concept

Calculating successive percentages and identifying an unknown base value.

Question 49Question →

An aviation squadron reduces its weekly flight hours by $15\%$ to conserve fuel. If the new weekly flight schedule is $340$ hours, what was the original number of weekly flight hours before the reduction?

400

391

289

40

Show answer & explanation

Answer: 400

Answer

The original number of weekly flight hours was 400.

The problem describes a reverse percentage scenario. Because the hours were reduced by 15%, the current 340 hours represent 85% (100% - 15%) of the original schedule. To find the original amount, you divide the current amount by the decimal form of the remaining percentage: 340 / 0.85 = 400.

Step-by-Step Solution

1

Determine what percentage the new hours represent compared to the original.

The new hours represent

100\% - 15\% = 85\%

of the original hours.

A reduction of 15% means 85% of the original amount remains.

2

Set up an equation where the original hours are represented by the variable

x

.

0.85x = 340

This translates the relationship into an algebraic equation to solve for the unknown original base.

3

Solve for

x

by dividing both sides by

0.85

.

x = \frac{340}{0.85} = 400

Isolating

x

provides the original number of flight hours.

Key Concept

Reverse Percentage Calculations

Alternative Method

You can set this up as a proportion:

\frac{85}{100} = \frac{340}{x}

. Cross-multiply to get

85x = 34000

, and divide by 85 to find

x = 400

.

Estimated Time:1m 0s

Question 50Question →

If $x = -3$ , $y = -5$ , and $z = 2$ , what is the value of the expression $-2|x^2 - |y - 3z|| + 4x$ ?

-26

-16

-52

-8

Show answer & explanation

Answer: -16

Answer

-16

The correct answer is found by strictly adhering to the order of operations and properties of absolute values. First, the inner absolute value |-5 - 3(2)| evaluates to |-11|, which is 11. The expression then simplifies to -2|(-3)² - 11| + 4(-3). Squaring -3 gives 9, so the expression inside the outer absolute value becomes 9 - 11, which is -2. The absolute value of -2 is 2. Multiplying this by the leading coefficient of -2 gives -4, and adding 4(-3), which is -12, results in -16.

Step-by-Step Solution

1

Substitute the given variable values into the expression.

-2|(-3)² - |-5 - 3(2)|| + 4(-3)

To prepare the expression for evaluation by replacing variables with their known numeric values.

2

Evaluate the innermost operations following the order of operations, starting with the inner absolute value.

-5 - 3(2) = -5 - 6 = -11, so |-11| = 11

The order of operations (PEMDAS) requires evaluating multiplication before subtraction, and operations inside the innermost grouping symbols first.

3

Evaluate the exponent and substitute the result of the inner absolute value back into the expression.

-2|9 - 11| - 12

Exponents must be evaluated before subtraction within the remaining absolute value.

4

Evaluate the subtraction inside the remaining absolute value and apply the absolute value function.

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

The absolute value of any non-zero integer is its positive counterpart.

5

Multiply by the leading coefficient and add the final term.

-2(2) - 12 = -4 - 12 = -16

Multiplication must be performed before subtraction to find the final value.

Key Concept

Evaluating multi-step algebraic expressions with nested absolute values

Question 51Question →

Determine the exact value of the expression below by applying the correct sequence of mathematical operations:

20 - 2 \times (3 - 5)^3 + \frac{-16}{4}

Which of the following represents the correct numerical result?

0

-148

32

80

Show answer & explanation

Answer: 32

Answer

32

Following the proper order of operations, we first simplify the parentheses to get -2. Next, we apply the exponent:

(-2)^3

equals -8. Then we perform multiplication and division:

-2 \times -8

gives a positive 16, and dividing -16 by 4 gives -4. Finally, substituting these values back into the expression leaves

20 + 16 - 4

, which evaluates to 32.

Step-by-Step Solution

1

Evaluate the expression inside the parentheses:

(3 - 5)

-2

According to the order of operations, operations inside parentheses must be resolved first.

2

Apply the exponent to the result of the parentheses:

(-2)^3

-8

Exponents must be evaluated before multiplication or division.

3

Perform the multiplication:

-2 \times (-8)

16

Multiplication is performed left-to-right before any addition or subtraction. Note that multiplying two negative numbers yields a positive result.

4

Perform the division:

\frac{-16}{4}

-4

Division is also resolved before addition and subtraction. A negative divided by a positive yields a negative result.

5

Complete the final addition and subtraction left-to-right:

20 + 16 - 4

32

Once all higher-order operations are complete, add and subtract from left to right.

Key Concept

Order of Operations with Signed Numbers and Exponents

Question 52Question →

A tool kit originally costs $\$40$ . Its price is increased to $\$50$ . What is the percentage increase in the price of the tool kit?

20\%

125\%

25\%

250\%

Show answer & explanation

Answer: $25\%$

Answer

25\%

The correct percentage increase is found by determining the difference between the new price and the old price (

\$10

), and then dividing that difference by the original price (

\$40

). The fraction

10/40

reduces to

1/4

, which equals

0.25

or

25\%

.

Step-by-Step Solution

1

Calculate the absolute increase in price.

\$50 - \$40 = \$10

The percentage increase is always based on the amount of change in value.

2

Divide the price increase by the original price to find the decimal rate.

10 \div 40 = 0.25

Percentage change must be calculated relative to the starting amount, not the final amount.

3

Convert the decimal rate to a percentage.

0.25 \times 100 = 25\%

Multiplying a decimal by 100 formats it as a standard percentage.

Key Concept

Percentage Increase

Estimated Time:45s

Question 53Question →

An aviation mechanic is preparing $60$ liters of a de-icing fluid mixture. The current mixture has a ratio of pure chemical to water of $3:2$ . To meet severe weather regulations, the mechanic must alter the mixture so that the ratio of pure chemical to water is $4:1$ . Assuming no water is removed, how many liters of pure chemical must be added to the current mixture?

12

60

96

204

Show answer & explanation

Answer: 60

Answer

60

The correct approach requires recognizing that the amount of water remains constant at 24 liters. To achieve a new ratio of 4 parts chemical to 1 part water, the final mixture must contain 4 times as much chemical as water (

24 \times 4 = 96

liters). Since there are already 36 liters of chemical in the mixture, you must add

96 - 36 = 60

liters of pure chemical.

Step-by-Step Solution

1

Determine the total number of parts in the initial ratio.

3 + 2 = 5

total parts.

To find the volume of a single part, you must first know how many total parts make up the 60-liter mixture.

2

Calculate the initial volume of pure chemical and water.

60 \div 5 = 12

liters per part. Pure chemical =

3 \times 12 = 36

liters. Water =

2 \times 12 = 24

liters.

Finding the absolute volumes allows you to establish the constant value (water) that will be used for the new proportion.

3

Set up a proportion for the new mixture using the constant water volume.

Let

x

be the added chemical. The new ratio is

\frac{36 + x}{24} = \frac{4}{1}

.

Since no water is removed, the water volume remains 24 liters. The new chemical amount must maintain a 4:1 relationship with this 24 liters.

4

Solve the proportion for

x

.

36 + x = 24 \times 4 \Rightarrow 36 + x = 96 \Rightarrow x = 60

liters.

Isolating

x

provides the exact volume of pure chemical that needs to be added to reach the desired target ratio.

Key Concept

Multi-step proportion scaling with a constant part and a changing total.

Question 54Question →

Arrange the mathematical steps in the correct sequence to evaluate the following expression according to the standard order of operations (PEMDAS):

12 - 4 \times \left| -5 + 2^3 \right|

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence starts with evaluating the exponent inside the absolute value, followed by addition, resolving the absolute value, multiplication, and finally subtraction.

Following the standard order of operations (PEMDAS), operations inside grouping symbols are prioritized. The absolute value bars act as a grouping symbol. Inside, the exponent is evaluated first, then the addition. The absolute value is then applied to the result. Finally, outside the grouping symbol, multiplication takes precedence over subtraction.

Step-by-Step Solution

1

Evaluate the exponent inside the absolute value.

12 - 4 \times \left| -5 + 8 \right|

Exponents within grouping symbols (absolute value) must be evaluated first.

2

Perform the addition inside the absolute value.

12 - 4 \times \left| 3 \right|

Complete the operations inside the grouping symbols.

3

Resolve the absolute value.

12 - 4 \times 3

The absolute value of

3

is

3

.

4

Perform the multiplication.

12 - 12

Multiplication precedes subtraction.

5

Perform the subtraction.

0

Subtraction is the final remaining operation.

Key Concept

Order of Operations (PEMDAS)

Question 55Question →

What is the value of the expression $-3 \times |-2| - (-5)$ ?

-1

11

-21

-11

Show answer & explanation

Answer: -1

Answer

The correct answer is -1.

The expression must be solved by first finding the absolute value of

-2

, which is

2

. Multiplying

-3

by

2

gives

-6

. Finally, subtracting

-5

from

-6

is the exact same as adding

5

to

-6

, which results in

-1

.

Step-by-Step Solution

1

Evaluate the absolute value expression.

|-2| = 2

Absolute value represents the distance of a number from zero, which is always non-negative.

2

Perform the multiplication.

-3 \times 2 = -6

According to the order of operations (PEMDAS), multiplication must be completed before addition and subtraction. Multiplying a negative integer by a positive integer yields a negative integer.

3

Perform the subtraction.

-6 - (-5) = -6 + 5 = -1

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

Key Concept

Integer Sign Rules and Order of Operations

Question 56Question →

Read the scenario and fill in the blank with the correct numeric value.

Fill in the blanks below

A recruit is on a

12

-mile training march. If the recruit has completed

75\%

of the total distance, they have marched miles.

Show answer & explanation

Answer

9

To find

75\%

of a

12

-mile march, convert the percentage into a decimal (

0.75

) or a simplified fraction (

\frac{3}{4}

). Multiplying the total distance of

12

miles by

\frac{3}{4}

yields

9

miles. Therefore, the recruit has completed

9

miles of the march.

Step-by-Step Solution

1

Identify the total distance and the percentage of the distance completed.

The total distance is

12

miles, and the percentage completed is

75\%

.

To calculate the distance completed, the whole amount and the percentage rate must be identified.

2

Convert the percentage to a fraction or a decimal.

75\% = \frac{75}{100} = \frac{3}{4}

, or

0.75

.

Percentages must be converted into standard numerical forms to perform arithmetic operations.

3

Multiply the total distance by the fraction or decimal.

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

, or

12 \times 0.75 = 9

.

Finding a percentage of a number requires multiplying the total by the percentage rate.

Key Concept

Calculating a percentage of a whole number.

Question 57Question →

The product of two negative integers is always a positive integer.

True

False

Show answer & explanation

Answer: True

Answer

True

The statement is correct because the mathematical rule for multiplying two integers with identical negative signs dictates that their product is positive.

Step-by-Step Solution

1

Recall the rules for multiplying integers with the same sign.

A negative number multiplied by another negative number equals a positive number.

This is a fundamental property of integer multiplication.

2

Apply the rule to a concrete example to verify.

For instance,

-3 \times -4 = 12

.

Using concrete integers confirms that the product is indeed a positive integer.

Key Concept

Sign Rules for Integer Operations

Estimated Time:15s

Question 58Question →

Complete the statements below with the correct numerical values.

Fill in the blanks below

An upgraded water purification system at a naval facility processes

5,200

gallons of water per hour. This new processing rate represents a

30\%

increase over the system's previous capacity.

The previous capacity of the system was gallons per hour. The upgrade increased the processing capacity by gallons per hour.

Show answer & explanation

Answer

The previous capacity was 4,000 gallons per hour, and the upgrade increased the capacity by 1,200 gallons per hour.

A 30% increase means the new capacity is 130% of the original capacity. Dividing the new capacity (5,200) by 1.30 gives the original capacity of 4,000 gallons per hour. The numerical increase is the difference between the new and old capacities: 5,200 - 4,000 = 1,200 gallons per hour.

Step-by-Step Solution

1

Set up an equation representing the percentage increase.

Let x be the previous capacity. Since there is a 30% increase, the new capacity is 100% + 30% = 130% of the previous capacity. Therefore, 1.3x = 5200.

To find the original base amount before a percentage increase, we must express the final amount as a percentage of the original.

2

Solve the equation for the previous capacity (x).

x = 5200 / 1.3 = 4000. This is the previous capacity.

Dividing the final amount by the percentage multiplier isolates the original base.

3

Calculate the amount of the capacity increase.

5200 - 4000 = 1200. This is the amount of the increase.

Subtracting the original capacity from the final capacity yields the exact volume added by the upgrade.

Key Concept

Reverse Percentage and Percentage Increase

Question 59Question →

Arrange the following mathematical values related to factors, multiples, and primes in ascending order (from least to greatest).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending sequence is: the total number of positive factors of 120 (16), followed by the sum of the distinct prime factors of 330 (21), followed by the GCF of 84 and 140 (28), and finally the LCM of 2, 3, 4, and 5 (60).

Evaluating each statement yields the numerical values

16

,

21

,

28

, and

60

. Arranging these from least to greatest gives the correct order: Factor count of

120

(16), Prime sum of

330

(21), GCF of

84

and

140

(28), and LCM of

2, 3, 4,

and

5

(60).

Step-by-Step Solution

1

Calculate the total number of positive factors of

120

.

16

By finding the prime factorization (

120 = 2^3 \times 3^1 \times 5^1

) and adding

1

to each exponent before multiplying them, we get

(3+1) \times (1+1) \times (1+1) = 4 \times 2 \times 2 = 16

.

2

Calculate the sum of the distinct prime factors of

330

.

21

The prime factorization is

330 = 33 \times 10 = 3 \times 11 \times 2 \times 5

. The distinct primes are

2, 3, 5,

and

11

. Their sum is

2 + 3 + 5 + 11 = 21

.

3

Determine the greatest common factor (GCF) of

84

and

140

.

28

Using prime factorizations (

84 = 2^2 \times 3 \times 7

and

140 = 2^2 \times 5 \times 7

), the common prime factors are

2^2

and

7

. Multiplying these yields

4 \times 7 = 28

.

4

Determine the least common multiple (LCM) of

2, 3, 4,

and

5

.

60

The LCM must include the highest power of each prime factor present in the numbers:

2^2

(from

4

),

3^1

, and

5^1

.

4 \times 3 \times 5 = 60

.

5

Order the evaluated results from least to greatest.

16 < 21 < 28 < 60

Sorting the calculated values provides the final correct arrangement of the mathematical statements.

Key Concept

Applying Prime Factorization to Calculate Factor Counts, GCF, and LCM

Alternative Method

Instead of full prime factorization for the GCF, the Euclidean algorithm can be used:

140 - 84 = 56

. Next,

84 - 56 = 28

. Next,

56 - 28 = 28

. The GCF is

28

.

Estimated Time:2m 30s

Question 60Question →

If $15\%$ of $y$ is equal to $25\%$ of $180$ , what is the value of $y$ ?

27

30

300

108

Show answer & explanation

Answer: 300

Answer

The correct value of

y

is

300

.

The wording translates to the algebraic equation

0.15y = 0.25 \times 180

. First, evaluate the right side:

25\%

of

180

is

45

. The equation simplifies to

0.15y = 45

. To find the value of

y

, divide

45

by

0.15

, which yields

300

.

Step-by-Step Solution

1

Translate the problem statement into an algebraic equation.

0.15y = 0.25 \times 180

'

15\%

of

y

' translates to

0.15y

, and '

25\%

of

180

' translates to

0.25 \times 180

.

2

Calculate

25\%

of

180

.

0.25 \times 180 = 45

Simplifying the right side of the equation makes it easier to isolate the variable.

3

Solve for

y

by dividing both sides by

0.15

.

y = \frac{45}{0.15} = 300

Dividing by the coefficient of

y

isolates the variable and provides the final answer.

Key Concept

Finding an unknown base using algebraic translation of percentage equations.

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