Number Theory and Operations Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 6

Question 101Question →

A naval shipyard allocates its weekly steel supply for submarine repair, surface ship repair, and new construction in a ratio of $3:4:5$ . If the surface ship repair division receives $240$ tons of steel, what is the total amount of steel allocated to all three divisions combined?

480

576

720

960

Show answer & explanation

Answer: 720

Answer

The total amount of steel allocated to all three divisions is 720 tons.

The ratio of steel for submarine repair, surface ship repair, and new construction is

3:4:5

, which means there are

12

total parts (

3+4+5

). We are given that surface ship repair, the middle term representing

4

parts of the ratio, receives

240

tons. To find the value of a single part, divide

240

by

4

, yielding

60

tons. Finally, multiply this unit rate by the

12

total parts to find the total steel allocation:

60 \times 12 = 720

tons.

Step-by-Step Solution

1

Identify the number of ratio parts assigned to surface ship repair and calculate the total number of parts.

Surface ship repair = 4 parts. Total parts = 3 + 4 + 5 = 12 parts.

To find the value of a single part, we must correctly match the given amount to its corresponding ratio value.

2

Calculate the amount of steel that corresponds to one ratio part.

240 tons / 4 parts = 60 tons per part.

Finding this unit rate allows us to scale up to the total amount.

3

Multiply the unit rate by the total number of parts.

60 tons/part * 12 parts = 720 tons.

The question asks for the total amount of steel allocated to all three divisions combined.

Key Concept

Solving multi-part ratio problems by finding a unit rate

Alternative Method

You can set up a direct proportion: 4 parts / 240 tons = 12 total parts / x tons. Cross-multiplying gives 4x = 2880, so dividing both sides by 4 yields x = 720.

Estimated Time:1m 0s

Question 102Question →

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

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

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

Show answer & explanation

Answer: 5.75

Answer

The final load factor is exactly 5.75.

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

\frac{23}{8}

and

\frac{1}{2}

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

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

).

Step-by-Step Solution

1

Convert the mixed numbers in the expression to improper fractions.

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

,

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

, and

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

.

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

2

Perform the subtraction inside the first set of parentheses:

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

.

Convert

\frac{11}{4}

to

\frac{22}{8}

. Then,

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

.

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

3

Perform the multiplication inside the second set of parentheses:

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

.

Multiply the numerators and denominators:

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

. This simplifies to

\frac{1}{2}

.

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

4

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

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

.

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

5

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

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

. As a decimal,

5 \frac{3}{4} = 5.75

.

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

Key Concept

Compound Fraction Operations and Order of Operations

Question 103Question →

Arrange the following values from least to greatest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from least to greatest is:

620

,

8,500

,

1.2 \times 10^4

,

3.0 \times 10^5

.

By converting all values to standard form, it is clear that

620

is the smallest, followed by

8,500

, then

12,000

(which is

1.2 \times 10^4

), and finally

300,000

(which is

3.0 \times 10^5

).

Step-by-Step Solution

1

Convert all values to standard numerical form to easily compare their sizes.

The values become:

12,000

(from item 1),

620

(from item 2),

300,000

(from item 3), and

8,500

(from item 4).

Translating scientific notation into standard form allows for direct visual comparison of digit lengths and magnitudes.

2

Group and compare the numbers by their total number of digits.

620

has three digits,

8,500

has four digits,

12,000

has five digits, and

300,000

has six digits.

For whole positive numbers, a value with fewer digits is always smaller than a value with more digits.

3

Order the values sequentially starting from the smallest.

The final order is

620

<

8,500

<

12,000

<

300,000

.

This establishes the ascending sequence requested by the question.

Key Concept

Comparing Magnitudes in Standard and Scientific Notation

Alternative Method

Instead of using standard form, you can convert all numbers to scientific notation:

6.2 \times 10^2

,

8.5 \times 10^3

,

1.2 \times 10^4

,

3.0 \times 10^5

. Because the exponents (

2, 3, 4, 5

) are already in ascending order, you can immediately determine the correct sequence without looking at the coefficients.

Estimated Time:45s

Question 104Question →

A cybersecurity team analyzes network traffic and categorizes detected threats into low, medium, and high severity in a ratio of $9:5:2$ . If the team logged exactly $145$ medium severity threats during a shift, what was the total number of threats logged across all three categories?

Show answer & explanation

Answer: 464

Answer

464

The correct answer is found by first determining the value of a single ratio part. Since

5

parts represent

145

medium threats, one part equals

29

threats. Multiplying this base unit by the total number of parts (

16

) gives the total of

464

threats.

Step-by-Step Solution

1

Determine the total number of parts in the ratio.

9 + 5 + 2 = 16

parts

To find the total amount of threats, we first need to know how many total parts make up the complete ratio.

2

Find the numerical value of a single part.

145 \div 5 = 29

threats per part

We know that

5

parts equal

145

medium threats. Dividing the count by the parts gives the baseline value of one part.

3

Calculate the total number of threats.

29 \times 16 = 464

threats

Multiplying the value of one part by the total number of parts yields the total count across all categories.

Key Concept

Solving Part-to-Whole Ratio Problems

Question 105Question →

A student is required to identify a calculation that yields a value greater than zero. Which of the following arithmetic expressions produces a positive integer result?

-5 + (-2)

4 - 3 \times 2

-|-8| + 5

(-3) \times (-4) - 10

Show answer & explanation

Answer: $(-3) \times (-4) - 10$

Answer

The expression

(-3) \times (-4) - 10

produces a positive integer.

The expression

(-3) \times (-4) - 10

evaluates to a positive integer because the product of

-3

and

-4

is a positive

12

, and subtracting

10

from

12

leaves

2

.

Step-by-Step Solution

1

Evaluate the first expression:

-5 + (-2)

.

-5 + (-2) = -7

.

Adding a negative integer to a negative integer results in a smaller negative integer.

2

Evaluate the second expression:

4 - 3 \times 2

.

4 - 6 = -2

.

Following the order of operations (PEMDAS), multiplication is performed before subtraction.

3

Evaluate the third expression:

-|-8| + 5

.

-8 + 5 = -3

.

The absolute value of

-8

is positive

8

. The negative sign outside makes it

-8

. Adding

5

results in a negative integer.

4

Evaluate the fourth expression:

(-3) \times (-4) - 10

.

12 - 10 = 2

.

Multiplying two negative integers yields a positive integer (

12

). Subtracting

10

from

12

leaves a positive integer.

Key Concept

Integer operations combining sign rules, absolute value, and the order of operations.

Estimated Time:50s

Question 106Question →

What is the value of the following expression?

8.4 - 2.4 \times 1.5 + 4.5 \div 0.5

13.8

27

-4.2

5.7

Show answer & explanation

Answer: 13.8

Answer

13.8

Applying the order of operations correctly requires first evaluating the multiplication (

2.4 \times 1.5 = 3.6

) and the division (

4.5 \div 0.5 = 9

). Substituting these values back simplifies the equation to

8.4 - 3.6 + 9

. Processing the remaining subtraction and addition from left to right yields

4.8 + 9 = 13.8

.

Step-by-Step Solution

1

Evaluate the multiplication part of the expression.

2.4 \times 1.5 = 3.6

According to the order of operations, multiplication must be performed before addition or subtraction.

2

Evaluate the division part of the expression.

4.5 \div 0.5 = 9

Division shares the same precedence as multiplication and must also be evaluated before addition and subtraction.

3

Substitute the results back into the original expression.

8.4 - 3.6 + 9

This simplifies the expression so that only addition and subtraction remain.

4

Evaluate the remaining addition and subtraction from left to right.

8.4 - 3.6 = 4.8

, and

4.8 + 9 = 13.8

Addition and subtraction have equal precedence and are solved strictly from left to right.

Key Concept

Order of Operations with Decimals

Question 107Question →

A specialized aerospace alloy is manufactured by combining titanium and aluminum in a ratio of $7:4$ by weight. If a batch of this alloy contains $140\text{ kilograms}$ of titanium, what is the total weight of the batch?

80\text{ kg}

220\text{ kg}

245\text{ kg}

385\text{ kg}

Show answer & explanation

Answer: $220\text{ kg}$

Answer

The total weight of the batch is

220\text{ kg}

.

The given ratio indicates the alloy consists of

7

parts titanium and

4

parts aluminum, making

11

parts in total. Because the

7

parts of titanium weigh

140\text{ kg}

, each single part weighs

20\text{ kg}

(

140 \div 7

). Multiplying the

11

total parts by

20\text{ kg}

per part results in a total weight of

220\text{ kg}

.

Step-by-Step Solution

1

Determine the weight of a single ratio part based on the known titanium content.

140\text{ kg} \div 7\text{ parts} = 20\text{ kg per part}

Since the ratio of titanium to aluminum is

7:4

, the

140\text{ kg}

corresponds exactly to the

7

parts of titanium.

2

Calculate the total number of parts in the alloy mixture.

7\text{ parts (titanium)} + 4\text{ parts (aluminum)} = 11\text{ total parts}

The total weight is composed of both the titanium and the aluminum parts combined.

3

Multiply the total number of parts by the weight of a single part.

11\text{ parts} \times 20\text{ kg per part} = 220\text{ kg}

This yields the combined weight of the entire batch.

Key Concept

Solving Part-to-Whole Ratio Problems

Estimated Time:1m 0s

Question 108Question →

What is the value of the expression $12 - 2|-4 + 1|$ ?

6

18

30

2

Show answer & explanation

Answer: 6

Answer

6

The expression is evaluated by first simplifying the terms inside the absolute value to get |-3|, which equals positive 3. Multiplying this by the coefficient 2 gives 6. Finally, subtracting 6 from 12 yields the correct answer of 6.

Step-by-Step Solution

1

Simplify the expression inside the absolute value bars.

-4 + 1 = -3

According to the order of operations, operations inside absolute value bars (which act as grouping symbols) must be performed first.

2

Evaluate the absolute value of the result.

|-3| = 3

The absolute value of a number represents its distance from zero, which is always positive.

3

Multiply the result by the coefficient outside the absolute value.

2 \times 3 = 6

Multiplication must be performed before subtraction according to the order of operations (PEMDAS).

4

Subtract the product from the initial value.

12 - 6 = 6

This completes the evaluation of the expression to find the final numerical value.

Key Concept

Absolute Value and Order of Operations

Question 109Question →

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

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order is

\frac{7}{8}

,

1 \frac{1}{4}

,

\frac{3}{2}

, and

1 \frac{3}{4}

.

The proper fraction

\frac{7}{8}

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

\frac{3}{2}

to

1 \frac{1}{2}

(or

1.5

) allows easy comparison with

1 \frac{1}{4}

(1.25) and

1 \frac{3}{4}

(1.75), confirming the correct ascending sequence.

Step-by-Step Solution

1

Identify values less than 1.

\frac{7}{8}

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

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

2

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

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

.

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

3

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

1 \frac{1}{4}

,

1 \frac{1}{2}

, and

1 \frac{3}{4}

).

The common denominator is 4.

1 \frac{1}{2}

becomes

1 \frac{2}{4}

.

Common denominators allow you to directly compare the numerators.

4

Order the mixed numbers by their fractional parts.

1 \frac{1}{4}

is smaller than

1 \frac{2}{4}

(which is

\frac{3}{2}

), and

1 \frac{2}{4}

is smaller than

1 \frac{3}{4}

.

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

Key Concept

Ordering mixed numbers and improper fractions

Question 110Question →

A botanist has $120$ tulip bulbs, $168$ daffodil bulbs, and $216$ crocus bulbs. She wants to plant them in separate rows such that each row contains only one type of flower, and every row has the exact same number of bulbs. What is the greatest number of bulbs she can plant in a single row?

Show answer & explanation

Answer: 24

Answer

24

The greatest number of bulbs per row must perfectly divide the total count of each flower type without remainders. Therefore, we calculate the Greatest Common Factor (GCF) of 120, 168, and 216. The prime factorizations reveal the common factors are

2^3

and

3^1

, which multiply to 24.

Step-by-Step Solution

1

Identify the mathematical operation required by the scenario.

The problem asks for the greatest number that divides 120, 168, and 216 evenly. This means finding the Greatest Common Factor (GCF).

Each row must have the same number of bulbs and only one flower type, meaning the row size must be a common divisor of all three total quantities.

2

Find the prime factorization of 120, 168, and 216.

120 = 2^3 \times 3 \times 5

,

168 = 2^3 \times 3 \times 7

, and

216 = 2^3 \times 3^3

.

Prime factorization is a systematic method to determine the common factors of multiple large numbers.

3

Identify the common prime factors and select their lowest powers.

The common prime factors are 2 and 3. Their lowest powers across the three numbers are

2^3

and

3^1

.

The GCF is formed by multiplying the lowest powers of all prime factors that appear in every number's factorization.

4

Calculate the GCF.

2^3 \times 3 = 8 \times 3 = 24

.

Multiplying the selected prime factors yields the greatest possible number of bulbs per row.

Key Concept

Greatest Common Factor (GCF)

Question 111Question →

Evaluate the following mathematical expressions involving decimals. What is the correct sequence of these expressions when ordered from least to greatest value?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest is:

0.15 \times 40 - 14.4 \div 1.2

, followed by

10.5 - 3.2 \times 2.5

, then

(4.2 - 1.8) \div 0.6 + 0.5

, and finally

2.4 \times 0.5 + 1.8 \div 0.3

.

Evaluating each expression using the correct order of operations yields final decimal values of

7.2

,

2.5

,

-6.0

, and

4.5

. Arranging these values numerically from least to greatest provides the correct structural sequence.

Step-by-Step Solution

1

Evaluate the first expression:

2.4 \times 0.5 + 1.8 \div 0.3

.

1.2 + 6 = 7.2

According to the order of operations, multiplication and division are performed left-to-right before addition.

2

Evaluate the second expression:

10.5 - 3.2 \times 2.5

.

10.5 - 8.0 = 2.5

Multiplication must be performed before subtraction.

3

Evaluate the third expression:

0.15 \times 40 - 14.4 \div 1.2

.

6.0 - 12 = -6.0

Perform multiplication and division first. Then subtract the larger value from the smaller value, resulting in a negative number.

4

Evaluate the fourth expression:

(4.2 - 1.8) \div 0.6 + 0.5

.

2.4 \div 0.6 + 0.5 = 4 + 0.5 = 4.5

Operations inside parentheses take highest priority, followed by division, and finally addition.

5

Order the evaluated results from least to greatest.

-6.0 < 2.5 < 4.5 < 7.2

Arranging the numerical values based on their algebraic magnitude provides the final sequence.

Key Concept

Order of Operations with Decimals

Alternative Method

Convert all terminating decimals to fractions (e.g.,

0.5 = \frac{1}{2}

,

0.3 = \frac{3}{10}

) before performing the operations to avoid decimal placement and alignment errors.

Estimated Time:1m 30s

Question 112Question →

What is the value of the following expression?

$|-4 - |-9||$

Show answer & explanation

Answer: 13

Answer

13

The correct answer is found by first evaluating the innermost absolute value,

|-9| = 9

. Substituting this value back into the expression gives

|-4 - 9|

. Simplifying the subtraction inside yields

|-13|

. Finally, the absolute value of

-13

is

13

.

Step-by-Step Solution

1

Evaluate the inner absolute value.

|-9| = 9

According to the order of operations, we must simplify the expression from the inside out. The absolute value of a negative number is its positive counterpart.

2

Substitute and simplify the remaining expression.

|-4 - 9| = |-13|

Replacing

|-9|

with

9

gives the subtraction

-4 - 9

, which simplifies to

-13

.

3

Evaluate the outer absolute value.

|-13| = 13

The absolute value represents the distance from zero on the number line, which is always non-negative.

Key Concept

Absolute Value

Question 113Question →

A commercial bakery prepares a special bread dough using flour, water, and sourdough starter in a ratio of $11:6:3$ by weight. If a single batch uses $42$ pounds of water, what is the total weight of the dough for this batch, in pounds?

Show answer & explanation

Answer: 140; 140 pounds; 140 lbs; 140 lb

Answer

140

Since the water represents

6

parts of the ratio and weighs

42

pounds, each part weighs exactly

7

pounds (

42 \div 6 = 7

). The entire batch of dough is composed of

20

total parts (

11

for flour +

6

for water +

3

for starter). Multiplying the

20

total parts by the

7

pounds per part results in a total batch weight of

140

pounds.

Step-by-Step Solution

1

Find the value of one part in the ratio.

42 \div 6 = 7

pounds per part.

Water corresponds to

6

parts of the

11:6:3

ratio, and its total weight is

42

pounds.

2

Calculate the total number of parts in the entire dough mixture.

11 + 6 + 3 = 20

total parts.

The total weight is the sum of the parts for flour, water, and starter.

3

Multiply the total number of parts by the weight of a single part.

20 \times 7 = 140

pounds.

Scaling the

20

total parts by the

7

pounds per part gives the total weight of the batch.

Key Concept

Solving proportions by finding the unit rate of a single part and multiplying it by the total number of parts.

Question 114Question →

A civil engineering project requires a specific concrete mixture consisting of cement, sand, and gravel in a ratio of $2:3:6$ by weight. If a batch of this concrete mixture weighs a total of $6,600\text{ lbs}$ , how many more pounds of gravel are used than cement?

1,800

2,400

3,300

3,600

Show answer & explanation

Answer: $2,400$

Answer

The concrete mixture requires

2,400

more pounds of gravel than cement.

The entire mixture consists of

11

equal parts (

2

cement

+ 3

sand

+ 6

gravel). Since the total mixture weighs

6,600\text{ lbs}

, one part weighs

600\text{ lbs}

(

6,600 \div 11

). The difference between the amount of gravel (

6

parts) and cement (

2

parts) is

4

parts. Multiplying this difference by the weight of one part yields

4 \times 600 = 2,400\text{ lbs}

.

Step-by-Step Solution

1

Find the total number of parts in the ratio.

2 + 3 + 6 = 11\text{ parts}

.

The total given weight of the concrete mixture must be distributed across all the ratio parts combined.

2

Calculate the weight of a single proportional part.

6,600 \div 11 = 600\text{ lbs}

.

Dividing the total mixture weight by the total number of parts determines the exact value of one unit.

3

Determine the difference in parts between the gravel and the cement.

6 - 2 = 4\text{ parts}

.

The question asks 'how many more pounds' of gravel are used than cement, which corresponds to the difference in their ratio shares.

4

Calculate the final difference in weight.

4 \times 600 = 2,400\text{ lbs}

.

Multiplying the part difference by the weight per part gives the final amount.

Key Concept

Solving part-to-whole ratio scenarios and calculating the difference between specific components.

Estimated Time:1m 0s

Question 115Question →

The prime factorizations of two integers, $X$ and $Y$ , are given below:

$X = 2^3 \times 3^2 \times 5$
$Y = 2^2 \times 3^3$

Based on this information, what is the greatest common factor (GCF) of $X$ and $Y$ ?

36

180

1080

6

Show answer & explanation

Answer: $36$

Answer

The greatest common factor is

36

.

To determine the greatest common factor (GCF) from prime factorizations, we must identify the prime bases shared by both numbers and raise each to its lowest given exponent. The shared bases for

X

and

Y

are

2

and

3

. Taking the lowest exponent for each gives

2^2

and

3^2

. Their product,

4 \times 9

, equals

36

.

Step-by-Step Solution

1

Identify the common prime bases in both factorizations.

The prime bases present in both

X

and

Y

are

2

and

3

. The prime base

5

is only in

X

.

The greatest common factor (GCF) only includes prime factors that are shared by all given numbers.

2

Determine the lowest exponent for each common prime base.

For base

2

, the exponents are

3

and

2

, so the lowest is

2

. For base

3

, the exponents are

2

and

3

, so the lowest is

2

.

To find the greatest common factor, we must use the smallest available power of each common prime to ensure it divides both numbers perfectly.

3

Multiply the resulting prime powers to calculate the GCF.

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

.

The product of these prime powers yields the greatest integer that divides both

X

and

Y

without a remainder.

Key Concept

Calculating the greatest common factor (GCF) using prime factorization by finding the lowest powers of all common prime bases.

Question 116Question →

A military motor pool maintains a fleet of tactical trucks, transport vans, and passenger cars in a ratio of $8:5:3$ . If there are exactly $112$ total vehicles in this fleet, how many transport vans does the motor pool maintain?

Show answer & explanation

Answer: 35

Answer

The motor pool maintains 35 transport vans.

The problem outlines a part-to-part-to-part ratio of

8:5:3

and a total quantity of

112

. To solve this, first add the parts of the ratio together (

8 + 5 + 3 = 16

). Next, divide the total quantity by the total ratio parts (

112 \div 16 = 7

) to determine that each 'part' represents

7

actual vehicles. Since the ratio for transport vans is the middle number (

5

), multiply

5

by

7

to correctly get

35

transport vans.

Step-by-Step Solution

1

Determine the total number of parts in the given ratio.

8 + 5 + 3 = 16

parts

Since the total number of vehicles in the fleet is given, we must find the total sum of the ratio parts to determine the size of a single part.

2

Calculate the exact value of a single ratio part.

112 \div 16 = 7

vehicles per part

Dividing the total quantity of vehicles by the total number of ratio parts gives the multiplier for the ratio.

3

Find the number of transport vans.

5 \times 7 = 35

transport vans

The ratio value specifically assigned to transport vans is

5

. Multiplying this by the value of one part gives the final count.

Key Concept

Solving part-to-whole ratio problems by establishing the total parts before finding specific component quantities.

Estimated Time:1m 0s

Question 117Question →

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

Show answer & explanation

Answer: 9.3

Answer

9.3

The initial length of the wire is

15\frac{4}{5}

meters and the used portion is

6\frac{1}{2}

meters. By converting these to decimals, you get

15.8

meters and

6.5

meters. Subtracting the used portion from the initial portion yields

15.8 - 6.5 = 9.3

meters. Alternatively, using a common denominator of 10 gives

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

, which also equals 9.3.

Step-by-Step Solution

1

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

15\frac{4}{5} = 15.8

and

6\frac{1}{2} = 6.5

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

2

Subtract the length of wire used from the initial length.

15.8 - 6.5 = 9.3

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

Key Concept

Fraction Operations

Estimated Time:45s

Question 118Question →

Arrange the following pairs of integers in ascending order based on the value of their Greatest Common Factor (GCF).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order based on the Greatest Common Factor (GCF) is:

32

and

40

(GCF = 8),

36

and

48

(GCF = 12),

42

and

56

(GCF = 14),

45

and

60

(GCF = 15).

By evaluating the Greatest Common Factor for each pair, we find that the GCF of

32

and

40

is

8

; the GCF of

36

and

48

is

12

; the GCF of

42

and

56

is

14

; and the GCF of

45

and

60

is

15

. The correct ascending order is

8

,

12

,

14

,

15

.

Step-by-Step Solution

1

Find the GCF for

45

and

60

.

The GCF is

15

.

The factors of

45

are

1, 3, 5, 9, 15, 45

. The factors of

60

include

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

. The largest shared factor is

15

.

2

Find the GCF for

32

and

40

.

The GCF is

8

.

The largest number that divides exactly into both

32

and

40

is

8

.

3

Find the GCF for

42

and

56

.

The GCF is

14

.

The largest number that divides exactly into both

42

and

56

is

14

(

42 = 14 \times 3

,

56 = 14 \times 4

).

4

Find the GCF for

36

and

48

.

The GCF is

12

.

The largest number that divides exactly into both

36

and

48

is

12

.

5

Order the pairs based on the calculated GCFs.

The sequence from smallest to largest GCF is

8

,

12

,

14

,

15

.

Arranging the values in ascending order determines the correct sequence.

Key Concept

Calculating the Greatest Common Factor (GCF) of pairs of integers.

Alternative Method

You can also use prime factorization for each number to find the GCF by multiplying the lowest powers of all common prime factors between the pairs.

Estimated Time:1m 30s

Question 119Question →

In a piece of heavy machinery, two interlocking gears are rotating together. The drive gear has $28$ teeth, and the driven gear has $42$ teeth. At a certain moment, a marked tooth on the drive gear aligns exactly with a marked tooth on the driven gear. How many teeth must pass the alignment point before these two specific marked teeth align with each other again?

84

14

1176

70

Show answer & explanation

Answer: $84$

Answer

The gears will align again after

84

teeth pass the alignment point.

To find the moment when two rotating cycles synchronize, the Least Common Multiple (LCM) of their cycle lengths (the number of teeth) must be determined. By finding the prime factorizations (

28 = 2^2 \times 7

and

42 = 2 \times 3 \times 7

) and multiplying the highest powers of all unique prime factors present (

2^2 \times 3 \times 7

), the LCM is calculated as

84

. Thus, exactly

84

teeth must pass for the marked teeth to realign.

Step-by-Step Solution

1

Identify the mathematical concept required to solve the synchronization problem.

The problem asks for the next time repeating cycles coincide, which requires finding the Least Common Multiple (LCM) of

28

and

42

.

The LCM determines the smallest positive integer that is a multiple of both numbers, representing the first point where both gear cycles overlap.

2

Find the prime factorization of the number of teeth on each gear.

For the drive gear:

28 = 2^2 \times 7

. For the driven gear:

42 = 2 \times 3 \times 7

.

Prime factorization breaks down each number into its core primes, allowing for an accurate calculation of the LCM.

3

Calculate the LCM using the identified prime factors.

LCM =

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

.

The LCM is calculated by taking the highest power of each prime factor that appears in any of the original factorizations.

Key Concept

Factors, Multiples, and Primes

Question 120Question →

What is the sum of all distinct prime factors of the integer $546$ ?

Show answer & explanation

Answer: 25

Answer

25

The integer

546

factors completely into the prime numbers

2

,

3

,

7

, and

13

. Since the question asks for the sum of all distinct prime factors, adding these together yields

2 + 3 + 7 + 13 = 25

.

Step-by-Step Solution

1

Find the prime factorization of

546

by continuously dividing by the smallest possible prime numbers.

546 = 2 \times 273 = 2 \times 3 \times 91 = 2 \times 3 \times 7 \times 13

To identify all prime factors, we must break the number down into its foundational prime components.

2

Identify the distinct prime factors from the factorization.

The distinct prime factors are

2

,

3

,

7

, and

13

.

The question specifically asks for the sum of the 'distinct' prime factors.

3

Calculate the sum of these distinct prime factors.

2 + 3 + 7 + 13 = 25

Adding the identified distinct prime factors yields the final requested value.

Key Concept

Prime Factorization and Properties of Integers

Number Theory and Operations

Answer

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