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Difficulty: MediumFraction Multiplication and Division in Context

A field communications unit has a spool containing 301430 \frac{1}{4} yards of specialized copper wire. They need to cut the wire into smaller segments that are each 1381 \frac{3}{8} yards long to connect field radios. How many complete segments of wire can the unit cut?

Answer:The unit can cut 【22】 complete segments of wire.

Answer

22
The correct answer is 22 because you must divide the total amount of wire by the length of one segment. Converting both mixed numbers to improper fractions gives 1214\frac{121}{4} and 118\frac{11}{8}. Dividing 1214\frac{121}{4} by 118\frac{11}{8} is equivalent to multiplying 1214\frac{121}{4} by 811\frac{8}{11}, which simplifies to exactly 22.

Step-by-Step Solution

1
Convert the total wire length from a mixed number to an improper fraction.
3014=30×4+14=121430 \frac{1}{4} = \frac{30 \times 4 + 1}{4} = \frac{121}{4} yards.
Improper fractions are required to perform fraction division.
2
Convert the length of each segment from a mixed number to an improper fraction.
138=1×8+38=1181 \frac{3}{8} = \frac{1 \times 8 + 3}{8} = \frac{11}{8} yards.
Improper fractions are required to perform fraction division.
3
Set up the division problem to find the total number of segments.
1214÷118\frac{121}{4} \div \frac{11}{8}
Dividing the total length by the length of a single segment gives the total number of segments.
4
Perform the division by multiplying the first fraction by the reciprocal of the second fraction.
1214×811\frac{121}{4} \times \frac{8}{11}
The rule for dividing fractions is to multiply by the reciprocal of the divisor.
5
Simplify the expression by cross-canceling and multiply.
12111×84=11×2=22\frac{121}{11} \times \frac{8}{4} = 11 \times 2 = 22 segments.
Simplifying before multiplying makes the calculation easier and yields the final whole number of segments.

Key Concept

Dividing mixed numbers in a real-world word problem context
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Topics

Fraction Multiplication and Division in Context

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