Number Theory and Operations Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 8

Question 141Question →

An electrician has a spool containing $25\frac{1}{6}$ yards of copper wire. They use $7\frac{1}{2}$ yards of the wire for a kitchen circuit and $8\frac{3}{4}$ yards for a living room circuit. How many yards of wire are left on the spool?

9\frac{1}{12}

9\frac{1}{2}

8\frac{11}{12}

26\frac{5}{12}

Show answer & explanation

Answer: $8\frac{11}{12}$

Answer

The correct answer is

8\frac{11}{12}

yards.

First, the total wire used must be calculated by adding

7\frac{1}{2}

and

8\frac{3}{4}

. Using a common denominator of 4,

7\frac{2}{4} + 8\frac{3}{4} = 15\frac{5}{4}

, which simplifies to

16\frac{1}{4}

. Next, subtract this total from the initial amount on the spool:

25\frac{1}{6} - 16\frac{1}{4}

. The common denominator for 6 and 4 is 12, resulting in

25\frac{2}{12} - 16\frac{3}{12}

. Since 3 cannot be subtracted from 2, you must borrow 1 from 25, turning it into

24\frac{14}{12}

. Finally,

24\frac{14}{12} - 16\frac{3}{12} = 8\frac{11}{12}

.

Step-by-Step Solution

1

Determine the total amount of wire used by adding

7\frac{1}{2}

and

8\frac{3}{4}

.

16\frac{1}{4}

yards

The total wire consumed is the sum of the pieces used for the kitchen and the living room.

2

Find a common denominator to subtract the total used wire from the initial amount:

25\frac{1}{6} - 16\frac{1}{4}

.

25\frac{2}{12} - 16\frac{3}{12}

A common denominator of 12 is needed to perform the subtraction between the fractional parts.

3

Borrow 1 from the whole number 25 to perform the subtraction.

24\frac{14}{12} - 16\frac{3}{12} = 8\frac{11}{12}

yards

Since

\frac{3}{12}

is larger than

\frac{2}{12}

, you must borrow a whole (

\frac{12}{12}

) to complete the operation.

Key Concept

Adding and subtracting mixed numbers with unlike denominators, including the borrowing method.

Question 142Question →

A pottery studio has a storage bin containing $15\frac{1}{4}$ pounds of dry clay. An instructor removes $4\frac{5}{8}$ pounds of the clay for a beginner class and another $3\frac{1}{2}$ pounds for an advanced class. Later that day, a new shipment arrives, and $5\frac{7}{8}$ pounds of clay are added to the bin. How many pounds of clay are in the bin now?

Show answer & explanation

Answer: 13

Answer

13

The correct answer is found by establishing a common denominator for all fractions, correctly borrowing a whole number to subtract the first amount of clay, subtracting the second amount, and then adding the final shipment to yield exactly 13 pounds.

Step-by-Step Solution

1

Find a common denominator for all fractions.

15\frac{1}{4} = 15\frac{2}{8}

and

3\frac{1}{2} = 3\frac{4}{8}

.

Fractions must have the same denominator before they can be added or subtracted.

2

Subtract the amount of clay used for the beginner class from the total.

15\frac{2}{8} - 4\frac{5}{8} = 14\frac{10}{8} - 4\frac{5}{8} = 10\frac{5}{8}

pounds.

Since

\frac{2}{8}

is smaller than

\frac{5}{8}

, we borrow 1 from 15 to properly execute the subtraction.

3

Subtract the amount of clay used for the advanced class from the new total.

10\frac{5}{8} - 3\frac{4}{8} = 7\frac{1}{8}

pounds.

This accounts for the second removal of clay.

4

Add the newly shipped clay to the remaining amount.

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

pounds.

The new shipment increases the total inventory.

5

Simplify the final mixed number.

13

pounds.

\frac{8}{8}

is equal to 1 whole, so

12 + 1 = 13

.

Key Concept

Addition and subtraction of mixed numbers with unlike denominators, requiring whole-number borrowing.

Estimated Time:1m 15s

Question 143Question →

A tailor has a bolt of fabric that is $20\frac{1}{4}$ yards long. He cuts $5\frac{3}{4}$ yards of fabric to make a suit jacket and another $3\frac{5}{8}$ yards to make a pair of trousers. How many yards of fabric are remaining on the bolt?

10\frac{7}{8}

11\frac{7}{12}

18\frac{1}{8}

11\frac{1}{8}

Show answer & explanation

Answer: $10\frac{7}{8}$

Answer

10\frac{7}{8}

The correct answer is found by first calculating the total fabric used (

5\frac{3}{4} + 3\frac{5}{8} = 9\frac{3}{8}

yards). Subtracting this from the original

20\frac{1}{4}

yards requires converting to a common denominator (

20\frac{2}{8} - 9\frac{3}{8}

) and then borrowing from the whole number (

19\frac{10}{8} - 9\frac{3}{8}

) to get exactly

10\frac{7}{8}

yards.

Step-by-Step Solution

1

Find the total amount of fabric used by adding

5\frac{3}{4}

and

3\frac{5}{8}

.

5\frac{3}{4} + 3\frac{5}{8} = 5\frac{6}{8} + 3\frac{5}{8} = 8\frac{11}{8} = 9\frac{3}{8}

yards.

Before finding the remaining amount, the total amount removed from the bolt must be calculated by finding a common denominator.

2

Subtract the total used fabric from the original length of the bolt.

20\frac{1}{4} - 9\frac{3}{8} = 20\frac{2}{8} - 9\frac{3}{8}

.

Subtracting the used amount from the initial amount determines how much fabric is left.

3

Borrow from the whole number to complete the subtraction.

19\frac{10}{8} - 9\frac{3}{8} = 10\frac{7}{8}

yards.

Since

2

is less than

3

in the numerators,

1

whole (

8/8

) must be borrowed from

20

to make the subtraction possible.

Key Concept

Adding and subtracting mixed numbers with unlike denominators, including borrowing.

Question 144Question →

Evaluate the following mathematical expression:

\frac{7}{8} - \frac{2}{3} \times 1\frac{1}{4}

Which of the following is the correct result?

\frac{25}{96}

1

\frac{1}{24}

-\frac{1}{24}

Show answer & explanation

Answer: $\frac{1}{24}$

Answer

The correct result is

\frac{1}{24}

.

The correct answer is found by first performing the multiplication (resulting in

\frac{5}{6}

) and then subtracting this from

\frac{7}{8}

using a common denominator of 24.

Step-by-Step Solution

1

Convert the mixed number to an improper fraction.

1\frac{1}{4} = \frac{5}{4}

Fractions must be in proper or improper form before multiplying.

2

Multiply the fractions according to the order of operations (PEMDAS).

\frac{2}{3} \times \frac{5}{4} = \frac{10}{12}

Multiplication must be performed before subtraction in mathematical expressions.

3

Simplify the resulting fraction.

\frac{10}{12} = \frac{5}{6}

Simplifying makes finding a common denominator easier in the next step.

4

Find a common denominator and subtract the fractions.

\frac{7}{8} - \frac{5}{6} = \frac{21}{24} - \frac{20}{24} = \frac{1}{24}

Fractions require a common denominator before they can be added or subtracted.

Key Concept

Order of Operations with Fractions

Question 145Question →

What is the exact value of the expression $-4.2 + 3.6 \times (1.5 - 2.7)$ ?

-8.52

0.72

0.12

-4.74

Show answer & explanation

Answer: $-8.52$

Answer

The correct value is

-8.52

.

The correct answer follows the standard order of operations and proper arithmetic rules for decimals and integers. First, the subtraction inside the parentheses is calculated as

1.5 - 2.7 = -1.2

. Next, the multiplication is performed before the addition:

3.6 \times (-1.2) = -4.32

. Finally,

-4.32

is added to

-4.2

, making sure to properly align the decimal points, which yields

-8.52

.

Step-by-Step Solution

1

Evaluate the expression inside the parentheses.

1.5 - 2.7 = -1.2

According to the order of operations (PEMDAS), operations inside parentheses must be completed first.

2

Multiply the result from step 1 by

3.6

.

3.6 \times (-1.2) = -4.32

Multiplication takes precedence over addition.

3

Add

-4.2

to the product from step 2.

-4.2 + (-4.32) = -8.52

This is the final remaining operation. Decimal points must be aligned vertically (

-4.20 + (-4.32)

) to ensure correct addition.

Key Concept

Decimal Operations with Order of Operations

Question 146Question →

A machinist needs to drill a hole completely through a steel block that is $3\frac{3}{4}$ inches thick. The drill bit advances through the steel at a constant rate of $\frac{3}{16}$ of an inch per second. How many seconds will it take to drill completely through the block?

Show answer & explanation

Answer: 20

Answer

It will take 20 seconds to drill completely through the block.

To find the time required, the total thickness of the block must be divided by the distance the drill advances per second. Converting

3\frac{3}{4}

to

\frac{15}{4}

and dividing by

\frac{3}{16}

is calculated by multiplying by the reciprocal:

\frac{15}{4} \times \frac{16}{3} = 20

.

Step-by-Step Solution

1

Convert the mixed number to an improper fraction.

3\frac{3}{4} = \frac{15}{4}

Fraction multiplication and division are performed using improper fractions rather than mixed numbers.

2

Set up the division equation to solve for time.

Time = Distance

\div

Rate =

\frac{15}{4} \div \frac{3}{16}

The time required is equal to the total distance (thickness) divided by the rate of progress.

3

Rewrite the division problem as multiplication by the reciprocal.

\frac{15}{4} \times \frac{16}{3}

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

4

Simplify the fractions by cross-canceling, then multiply.

5 \times 4 = 20

Cross-canceling common factors (15 and 3, 16 and 4) before multiplying makes the final calculation simpler.

Key Concept

Fraction Division and Applied Rates

Estimated Time:50s

Question 147Question →

A logistics specialist is calculating the weight of four different cargo pallets in kilograms. Based on the provided mathematical expressions for each pallet, arrange them in order from lightest to heaviest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from lightest to heaviest is Pallet S (

47.95

kg), Pallet Q (

48.1

kg), Pallet P (

48.2

kg), and Pallet R (

48.4

kg).

The correct sequence is determined by first applying the proper order of operations (PEMDAS) to compute the exact weight of each pallet. Multiplying and dividing before adding and subtracting yields the values: Pallet S is

47.95

kg, Pallet Q is

48.1

kg, Pallet P is

48.2

kg, and Pallet R is

48.4

kg. Comparing these final decimal figures starting from the greatest place value provides the correct ascending order.

Step-by-Step Solution

1

Calculate the exact weight of Pallet S.

9.5 \times 5.4 - 3.35 = 51.3 - 3.35 = 47.95

Applying the order of operations, multiplication is performed before subtraction.

2

Calculate the exact weight of Pallet Q.

12.5 + 7.12 \times 5 = 12.5 + 35.6 = 48.1

Multiplication precedes addition. Aligning the decimals properly ensures correct addition.

3

Calculate the exact weight of Pallet P.

60.2 - 4.8 \times 2.5 = 60.2 - 12 = 48.2

Multiply first to get exactly

12

, then subtract from

60.2

.

4

Calculate the exact weight of Pallet R.

58.08 \div 1.2 = 580.8 \div 12 = 48.4

Shift the decimal point one place to the right in both the dividend and divisor to compute the division.

5

Compare the final decimal values to determine the ascending order.

47.95 < 48.1 < 48.2 < 48.4

Comparing digits from left to right (tens, ones, tenths, hundredths) confirms the sequence.

Key Concept

Applying the order of operations (PEMDAS) to evaluate decimal expressions and comparing relative decimal values.

Question 148Question →

A military field generator consumes diesel fuel at a rate of $3.25$ gallons per hour. The generator's fuel tank initially contains $50.4$ gallons of fuel. If the generator runs continuously for $8.4$ hours, how many gallons of fuel remain in the tank?

Show answer & explanation

Answer: 23.1

Answer

23.1

To find the remaining fuel, you first need to determine the total amount used. Multiplying the hourly rate of

3.25

gallons by

8.4

hours gives

27.3

gallons consumed. Subtracting this from the initial

50.4

gallons (

50.4 - 27.3

) leaves exactly

23.1

gallons.

Step-by-Step Solution

1

Calculate the total amount of fuel consumed by the generator during the run time.

3.25 \times 8.4 = 27.3

gallons

The total fuel consumed is found by multiplying the hourly consumption rate by the total number of hours the generator operates.

2

Calculate the remaining fuel by subtracting the consumed fuel from the initial amount.

50.4 - 27.3 = 23.1

gallons

The fuel left in the tank is the difference between the starting volume and the volume that was burned.

Key Concept

Solving multi-step practical arithmetic problems involving decimal multiplication and decimal subtraction.

Estimated Time:1m 0s

Question 149Question →

A student is asked to simplify the expression below.

8.4 - 2.1 \times (-3.2) + 1.5

What is the correct simplified value?

-18.66

3.18

7.71

16.62

Show answer & explanation

Answer: 16.62

Answer

The correct simplified value is 16.62.

The exact evaluation requires multiplying 2.1 by -3.2 first to obtain -6.72. The expression then becomes an addition problem (

8.4 + 6.72 + 1.5

). Properly aligning the decimal points and summing these terms yields the true simplified value of 16.62.

Step-by-Step Solution

1

Identify and perform the multiplication operation first.

2.1 \times (-3.2) = -6.72

Following the order of operations (PEMDAS), multiplication takes precedence over addition and subtraction.

2

Substitute the product back into the expression and resolve the subtraction of a negative value.

8.4 - (-6.72) + 1.5 = 8.4 + 6.72 + 1.5

Subtracting a negative number is mathematically identical to adding its positive counterpart.

3

Perform the addition operations from left to right, making sure to properly align the decimal points.

8.40 + 6.72 + 1.50 = 16.62

Addition operations are performed sequentially from left to right once all higher-priority operations are resolved, requiring proper place-value alignment.

Key Concept

Decimal Arithmetic and Order of Operations

Question 150Question →

What is the value of the following mathematical expression?

\frac{0.84}{0.04} + 2.5 \times (-1.4)

-32.9

17.5

24.5

18.5

Show answer & explanation

Answer: 17.5

Answer

17.5

To evaluate the expression correctly, the order of operations must be followed. First, resolve the division:

0.84 \div 0.04 = 21

. Next, resolve the multiplication:

2.5 \times (-1.4) = -3.5

. Finally, add these two results together:

21 + (-3.5) = 17.5

. Proper decimal alignment (

21.0 - 3.5

) ensures the correct final value.

Step-by-Step Solution

1

Perform the division operation.

\frac{0.84}{0.04} = \frac{84}{4} = 21

According to the order of operations (PEMDAS), division and multiplication must be evaluated before addition.

2

Perform the multiplication operation.

2.5 \times (-1.4) = -3.5

Complete the remaining multiplication before moving on to addition. Note that a positive multiplied by a negative results in a negative product.

3

Add the results of the division and multiplication together.

21 + (-3.5) = 21.0 - 3.5 = 17.5

Substitute the evaluated terms back into the expression and align the decimal points to subtract correctly.

Key Concept

Order of Operations with Decimals

Estimated Time:1m 0s

Question 151Question →

An industrial pump dispenses $3.45$ liters of a chemical per cycle. A tank initially contains $150.8$ liters of the chemical. If the pump runs for $14$ cycles to remove the chemical from the tank, and then a technician manually adds $5.25$ liters back into the tank, how many liters of chemical remain in the tank?

Show answer & explanation

Answer: 107.75

Answer

107.75

The final volume is accurately found by multiplying the removal rate by the number of cycles to determine the total removed amount, subtracting that from the initial volume, and finally adding the manually returned amount while maintaining proper decimal alignment.

Step-by-Step Solution

1

Calculate the total volume of chemical removed by the pump.

14 \times 3.45 = 48.3

The pump runs for 14 cycles, removing 3.45 liters each time, so multiplication is required to find the total removal.

2

Subtract the removed volume from the initial tank volume.

150.8 - 48.3 = 102.5

Removing the chemical decreases the total volume available in the tank.

3

Add the volume of the chemical manually poured back into the tank.

102.5 + 5.25 = 107.75

Adding chemical increases the total volume to its final amount.

Key Concept

Solving multi-step practical problems involving the multiplication, subtraction, and addition of decimals.

Estimated Time:1m 0s

Question 152Question →

Identify the correct step-by-step sequence to fully simplify the given mathematical expression. Order the operations from first to last:

15 - 3 \times (2 + 1)^2

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The sequence is to evaluate the parentheses first, apply the exponent to that result, perform the multiplication, and finally execute the subtraction.

Following the standard order of operations (PEMDAS), you must first resolve the grouping symbols (parentheses), then evaluate the exponent, perform the multiplication, and finally execute the subtraction.

Step-by-Step Solution

1

Evaluate the terms inside the parentheses:

(2 + 1)

3

Parentheses (grouping symbols) hold the highest priority in the order of operations.

2

Apply the exponent to the result:

3^2

9

Exponents must be evaluated before multiplication and subtraction.

3

Perform the multiplication:

3 \times 9

27

Multiplication is prioritized before subtraction.

4

Perform the subtraction:

15 - 27

-12

Subtraction is the final remaining operation in the expression.

Key Concept

Order of Operations

Estimated Time:45s

Question 153Question →

A runner completes $\frac{2}{5}$ of a mile as a warm-up and then runs another $\frac{1}{3}$ of a mile. What is the total distance, in miles, the runner covered?

\frac{11}{15}

\frac{3}{8}

\frac{2}{15}

\frac{1}{5}

Show answer & explanation

Answer: $\frac{11}{15}$

Answer

\frac{11}{15}

To find the total distance, the fractions must be added. Since they have unlike denominators (5 and 3), the lowest common denominator (15) must be found. Converting the fractions yields

\frac{6}{15}

and

\frac{5}{15}

. Adding the numerators provides the final total of

\frac{11}{15}

.

Step-by-Step Solution

1

Identify the operation needed.

Addition:

\frac{2}{5} + \frac{1}{3}

The question asks for the total distance covered, requiring the sum of the two distinct parts of the run.

2

Find a common denominator for 5 and 3.

The lowest common multiple of 5 and 3 is 15.

Fractions must have a common denominator before they can be added together.

3

Convert both fractions to have a denominator of 15.

\frac{2}{5} = \frac{6}{15}

and

\frac{1}{3} = \frac{5}{15}

To maintain the same value, multiply the numerator and denominator of the first fraction by 3, and the second fraction by 5.

4

Add the numerators and keep the common denominator.

\frac{6}{15} + \frac{5}{15} = \frac{11}{15}

Once denominators are the same, only the numerators are added to find the total.

Key Concept

Adding Fractions with Unlike Denominators

Question 154Question →

A chef is preparing a special sauce and needs $\frac{7}{8}$ of a cup of broth. They currently have $\frac{1}{4}$ of a cup of broth in a measuring jug. How much more broth must be added to reach the required amount?

\frac{2}{3}

\frac{3}{4}

\frac{5}{8}

1\frac{1}{8}

Show answer & explanation

Answer: $\frac{5}{8}$

Answer

The chef needs

\frac{5}{8}

of a cup of additional broth.

The correct answer is found by taking the total amount needed and subtracting the amount already available. First, convert

\frac{1}{4}

to an equivalent fraction with a denominator of 8, which is

\frac{2}{8}

. Then, subtract

\frac{2}{8}

from

\frac{7}{8}

to get

\frac{5}{8}

.

Step-by-Step Solution

1

Set up a subtraction problem to find the difference between the total amount of broth needed and the amount currently on hand.

\frac{7}{8} - \frac{1}{4}

Subtracting the available amount from the required total yields the missing remainder.

2

Find a common denominator so the fractions can be subtracted. Multiply the numerator and denominator of the second fraction by 2.

\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}

Fractions must have the same denominator (in this case, 8) before they can be added or subtracted.

3

Subtract the numerators while keeping the common denominator the same.

\frac{7}{8} - \frac{2}{8} = \frac{5}{8}

This calculation provides the final exact fraction of additional broth required.

Key Concept

Subtracting fractions with unlike denominators by first converting to a common denominator.

Estimated Time:45s

Question 155Question →

What is the final result of the expression $9.25 - 3.4 \times 0.5$ ?

2.925

9.08

7.55

10.95

Show answer & explanation

Answer: $7.55$

Answer

The correct answer is

7.55

.

The expression requires following the proper order of operations, meaning the multiplication must be calculated before the subtraction. First, multiplying

3.4

by

0.5

yields

1.7

. Then, aligning the decimals correctly and subtracting

1.7

from

9.25

provides the final answer of

7.55

.

Step-by-Step Solution

1

Identify the correct order of operations (PEMDAS/BODMAS) for the expression.

Multiplication must be performed before subtraction.

Standard mathematical rules require evaluating multiplication before addition or subtraction.

2

Multiply

3.4

by

0.5

.

3.4 \times 0.5 = 1.7

Resolving the multiplication part of the expression.

3

Subtract the product from

9.25

, ensuring the decimal points are vertically aligned.

9.25 - 1.70 = 7.55

Completing the final subtraction operation to find the total value.

Key Concept

Decimal Operations

Estimated Time:45s

Question 156Question →

A city water purification system processes $8.5 \times 10^5$ liters of water per day. A specialized chemical treatment requires $4.0 \times 10^{-4}$ grams of a purifying agent per liter of water. What is the total mass, in grams, of the purifying agent needed each day? Express the answer in proper scientific notation.

3.4 \times 10^0

3.4 \times 10^{-19}

3.4 \times 10^2

3.4 \times 10^{10}

Show answer & explanation

Answer: $3.4 \times 10^2$

Answer

The correct answer is

3.4 \times 10^2

.

The total mass is calculated by multiplying the daily volume by the required agent per liter:

(8.5 \times 10^5) \times (4.0 \times 10^{-4})

. Multiplying the coefficients gives

34

, and adding the exponents yields

10^1

. The intermediate result,

34 \times 10^1

, normalizes to

3.4 \times 10^2

by shifting the decimal one place to the left and adding

1

to the exponent.

Step-by-Step Solution

1

Set up the multiplication expression to find the total mass.

(8.5 \times 10^5) \times (4.0 \times 10^{-4})

The total mass is the product of the total daily volume of water and the mass of the purifying agent required per liter.

2

Group the coefficients and the powers of ten.

(8.5 \times 4.0) \times (10^5 \times 10^{-4})

The commutative and associative properties of multiplication allow grouping similar numerical components.

3

Multiply the coefficients and add the exponents.

34 \times 10^1

8.5 \times 4.0 = 34

, and following the product rule of exponents,

10^5 \times 10^{-4} = 10^{5 + (-4)} = 10^1

.

4

Normalize the result into proper scientific notation.

3.4 \times 10^2

The coefficient

34

must be converted to

3.4

(a number between 1 and 10). Because the decimal point is moved one place to the left, the exponent is increased by 1.

Key Concept

Multiplying Numbers in Scientific Notation and Normalization

Question 157Question →

Arrange the following fractions and mixed numbers in ascending order (from least to greatest).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order is

\frac{14}{9}

,

1\frac{5}{8}

,

1\frac{2}{3}

, and

\frac{7}{4}

.

By converting all values to improper fractions and finding a common denominator of 72, the values become

\frac{112}{72}

,

\frac{117}{72}

,

\frac{120}{72}

, and

\frac{126}{72}

. Comparing the numerators allows us to determine the correct ascending order of the original values.

Step-by-Step Solution

1

Convert all mixed numbers to improper fractions.

1\frac{5}{8} = \frac{13}{8}

and

1\frac{2}{3} = \frac{5}{3}

.

Improper fractions are easier to evaluate when finding a common denominator.

2

Determine the least common multiple (LCM) of the denominators 9, 8, 3, and 4.

The LCM is 72.

A common denominator is required to accurately compare the sizes of fractions.

3

Convert each fraction into an equivalent fraction with a denominator of 72.

\frac{14}{9} = \frac{112}{72}

,

\frac{13}{8} = \frac{117}{72}

,

\frac{5}{3} = \frac{120}{72}

,

\frac{7}{4} = \frac{126}{72}

.

Having a common denominator allows for direct comparison of the numerators.

4

Compare the numerators to order the fractions from least to greatest.

112 < 117 < 120 < 126

, which means

\frac{112}{72} < \frac{117}{72} < \frac{120}{72} < \frac{126}{72}

.

The question specifies arranging the values in ascending order.

5

Map the ordered fractions back to their original forms.

\frac{14}{9}

,

1\frac{5}{8}

,

1\frac{2}{3}

,

\frac{7}{4}

.

To provide the final answer using the exact values given in the prompt.

Key Concept

Comparing and ordering mixed numbers and improper fractions with unlike denominators by finding a common denominator.

Question 158Question →

Evaluate the following mathematical expression and determine the final integer result.

Fill in the blanks below

The value of the expression

\frac{|-15 + 3|}{2} - |-4 \times 2|

is .

Show answer & explanation

Answer

The final evaluated result is -2.

The correct result is found by first evaluating the expressions inside the absolute value symbols, which act as grouping symbols. The expression

|-15 + 3|

becomes

|-12|

, which evaluates to

12

. The second part,

|-4 \times 2|

, becomes

|-8|

, which evaluates to

8

. Substituting these simplified values back into the original expression yields

\frac{12}{2} - 8

. Following the order of operations, division is performed first, giving

6

. Finally,

6 - 8

results in

-2

.

Step-by-Step Solution

1

Evaluate the operations inside the first absolute value bars.

|-15 + 3| = |-12|

Order of operations requires treating absolute value bars as grouping symbols, meaning the arithmetic inside must be completed first.

2

Apply the absolute value to the first result.

|-12| = 12

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

3

Evaluate the operations inside the second absolute value bars.

|-4 \times 2| = |-8| = 8

Perform the multiplication inside the grouping symbol first, then take the absolute value of the negative product.

4

Substitute the evaluated absolute values back into the expression and solve.

\frac{12}{2} - 8 = 6 - 8 = -2

Perform division before subtraction according to the standard order of operations (PEMDAS).

Key Concept

Absolute Value and Order of Operations

Question 159Question →

An engineer uses the formula $V = V_0 - r \times t + C$ to calculate the final volume of fluid in a tank in liters. If the initial volume $V_0 = 12.4$ , the rate $r = 2.5$ , the time $t = 3.2$ , and a correction factor $C = 4.8$ , what is the final volume $V$ ?

16.4

9.2

-0.4

36.48

Show answer & explanation

Answer: 9.2

Answer

9.2

To find the final volume, substitute the values into the formula to get

12.4 - 2.5 \times 3.2 + 4.8

. According to the order of operations, the multiplication must be evaluated first:

2.5 \times 3.2 = 8.0

. Substituting this back gives

12.4 - 8.0 + 4.8

. Performing the remaining operations from left to right, first subtract (

12.4 - 8.0 = 4.4

), and then add (

4.4 + 4.8 = 9.2

).

Step-by-Step Solution

1

Substitute the given values into the formula.

V = 12.4 - 2.5 \times 3.2 + 4.8

This sets up the arithmetic expression that needs to be evaluated.

2

Apply the order of operations (PEMDAS) by performing the multiplication first.

2.5 \times 3.2 = 8.0

Multiplication takes precedence over addition and subtraction.

3

Rewrite the expression with the product and perform the subtraction (working left to right).

12.4 - 8.0 = 4.4

Addition and subtraction are evaluated from left to right in the order they appear.

4

Perform the final addition step.

4.4 + 4.8 = 9.2

This yields the final evaluated volume.

Key Concept

Applying the order of operations (PEMDAS) to expressions containing decimals, ensuring proper place-value alignment during addition and subtraction.

Number Theory and Operations

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