Number Theory and Operations Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB) — Sayfa 8

Soru 141Soru →

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}

Cevabı ve açıklamayı göster

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

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 142Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

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.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

Tahmini Süre:1m 15s

Soru 143Soru →

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}

Cevabı ve açıklamayı göster

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

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 144Soru →

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}

Cevabı ve açıklamayı göster

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

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Order of Operations with Fractions

Soru 145Soru →

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

Cevabı ve açıklamayı göster

Cevap: $-8.52$

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Decimal Operations with Order of Operations

Soru 146Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Fraction Division and Applied Rates

Tahmini Süre:50s

Soru 147Soru →

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.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 148Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 23.1

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Tahmini Süre:1m 0s

Soru 149Soru →

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

Cevabı ve açıklamayı göster

Cevap: 16.62

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Decimal Arithmetic and Order of Operations

Soru 150Soru →

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

Cevabı ve açıklamayı göster

Cevap: 17.5

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Order of Operations with Decimals

Tahmini Süre:1m 0s

Soru 151Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 107.75

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Tahmini Süre:1m 0s

Soru 152Soru →

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

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Order of Operations

Tahmini Süre:45s

Soru 153Soru →

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}

Cevabı ve açıklamayı göster

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

Cevap

\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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

Adding Fractions with Unlike Denominators

Soru 154Soru →

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}

Cevabı ve açıklamayı göster

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

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Tahmini Süre:45s

Soru 155Soru →

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

2.925

9.08

7.55

10.95

Cevabı ve açıklamayı göster

Cevap: $7.55$

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Decimal Operations

Tahmini Süre:45s

Soru 156Soru →

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}

Cevabı ve açıklamayı göster

Cevap: $3.4 \times 10^2$

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Multiplying Numbers in Scientific Notation and Normalization

Soru 157Soru →

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

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 158Soru →

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

Aşağıdaki boşlukları doldurun

The value of the expression

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

is .

Cevabı ve açıklamayı göster

Cevap

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

.

Adım Adım Çözüm

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).

Anahtar Kavram

Absolute Value and Order of Operations

Soru 159Soru →

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

Cevabı ve açıklamayı göster

Cevap: 9.2

Cevap

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

).

Adım Adım Çözüm

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.

Anahtar Kavram

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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