A combat engineering team is tasked with excavating a defensive trench that will be 15 \fr… — Armed Services Vocational Aptitude Battery (ASVAB)

A combat engineering team is tasked with excavating a defensive trench that will be $15 \frac{3}{4}$ miles long. If the team can excavate $2 \frac{1}{3}$ miles of trench per week, how many weeks will it take them to complete the entire project?

$6 \frac{3}{4}$ Cevap
B
$36 \frac{3}{4}$
C
$9 \frac{3}{4}$
D
$13 \frac{5}{12}$

Cevap

It will take

6 \frac{3}{4}

weeks to complete the trench.

To determine the total time needed to complete a project when given the total amount of work and the rate of work per unit of time, divide the total work by the rate. Converting

15 \frac{3}{4}

and

2 \frac{1}{3}

to improper fractions gives

\frac{63}{4}

and

\frac{7}{3}

. Dividing these fractions (by multiplying by the reciprocal) yields

\frac{63}{4} \times \frac{3}{7} = \frac{27}{4}

, which simplifies to

6 \frac{3}{4}

Adım Adım Çözüm

Identify the total length of the trench and the rate of excavation.

Total length =

15 \frac{3}{4}

miles. Rate =

2 \frac{1}{3}

miles per week.

Understanding the given values is the first step in setting up the correct mathematical operation.

Determine the necessary operation to find the total time.

Time = Total Distance

\div

Rate.

When a total amount is being completed at a specific rate per time unit, division is used to find the total time required.

Convert the mixed numbers into improper fractions.

15 \frac{3}{4} = \frac{15 \times 4 + 3}{4} = \frac{63}{4}

. And

2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}

Fraction division requires improper fractions rather than mixed numbers.

Set up and perform the division by multiplying by the reciprocal.

\frac{63}{4} \div \frac{7}{3} = \frac{63}{4} \times \frac{3}{7}

Dividing by a fraction is equivalent to multiplying by its reciprocal (inverting the second fraction).

Simplify the expression and multiply.

Cross-canceling

63

and

7

leaves

9

and

1

. The equation becomes

\frac{9}{4} \times \frac{3}{1} = \frac{27}{4}

Simplifying before multiplying reduces the chance of calculation errors and results in smaller numbers.

Convert the improper fraction back to a mixed number.

\frac{27}{4} = 6 \frac{3}{4}

weeks.

The original context and answer choices are presented in mixed numbers.

Anahtar Kavram

Dividing mixed numbers in rate problems by converting to improper fractions and multiplying by the reciprocal.

Soru

Cevap

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