A mechanical engineer is determining the load factor for a support beam. The calculation r… — Armed Services Vocational Aptitude Battery (ASVAB)

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.

Cevap: 5.75

Cevap

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

Adım Adım Çözüm

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.

Perform the subtraction inside the first set of parentheses:

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

Convert

\frac{11}{4}

\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.

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.

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.

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.

Anahtar Kavram

Compound Fraction Operations and Order of Operations

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