A communications platoon is upgrading its field radio systems. The unit has a single conti… — Armed Services Vocational Aptitude Battery (ASVAB)

A communications platoon is upgrading its field radio systems. The unit has a single continuous spool of copper grounding wire that measures $42 \frac{1}{2}$ meters in length. Each radio system requires exactly $2 \frac{5}{6}$ meters of this wire for a proper ground connection. Calculate the maximum number of complete radio systems the platoon can ground using this spool.

Answer:The platoon can ground a total of 【15】 complete radio systems.

Answer

To determine how many complete systems can be grounded, the total length of the wire (

42 \frac{1}{2}

meters) must be divided by the length required per system (

2 \frac{5}{6}

meters). Converting both to improper fractions gives

\frac{85}{2} \div \frac{17}{6}

. Multiplying by the reciprocal yields

\frac{85}{2} \times \frac{6}{17}

, which simplifies to

5 \times 3 = 15

Step-by-Step Solution

Convert both mixed numbers to improper fractions.

42 \frac{1}{2} = \frac{85}{2}

and

2 \frac{5}{6} = \frac{17}{6}

Multiplication and division of fractions require improper fractions rather than mixed numbers.

Set up the division equation.

\frac{85}{2} \div \frac{17}{6}

To find how many items of a specific size fit into a total quantity, you must divide the total by the size of one item.

Rewrite the division problem as multiplication by taking the reciprocal of the divisor.

\frac{85}{2} \times \frac{6}{17}

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

Simplify by cross-canceling common factors and multiply.

Divide

85

17

to get

5

. Divide

6

2

to get

3

. Then,

5 \times 3 = 15

Cross-canceling before multiplying keeps the numbers small and manageable without a calculator.

Key Concept

Dividing a total quantity by a fractional size using improper fractions and reciprocal multiplication.

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