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Question

Difficulty: MediumFraction Operations

A hiker plans to complete a trail that is 153415\frac{3}{4} miles long. If she hikes at a consistent average speed of 3123\frac{1}{2} miles per hour, how many hours will it take her to complete the entire trail?

Answer: 4.5 hours

Answer

4.5
The correct answer is found by converting the total distance (153415\frac{3}{4}) and the speed (3123\frac{1}{2}) into improper fractions (634\frac{63}{4} and 72\frac{7}{2}), and then dividing the distance by the speed. This translates to 634×27\frac{63}{4} \times \frac{2}{7}, which cross-simplifies perfectly to 92\frac{9}{2} or 4.54.5 hours.

Step-by-Step Solution

1
Convert the mixed numbers into improper fractions.
1534=63415\frac{3}{4} = \frac{63}{4} and 312=723\frac{1}{2} = \frac{7}{2}
Fraction arithmetic requires mixed numbers to be in improper fraction form before dividing.
2
Set up the division equation to solve for time.
Time = 634÷72\frac{63}{4} \div \frac{7}{2}
Time is calculated by dividing total distance by the average speed.
3
Change the division to multiplication by using the reciprocal of the divisor.
634×27\frac{63}{4} \times \frac{2}{7}
Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal.
4
Multiply the fractions and simplify the result.
63×24×7=12628=92=4.5\frac{63 \times 2}{4 \times 7} = \frac{126}{28} = \frac{9}{2} = 4.5
Simplifying the fraction yields the exact number of hours.

Key Concept

Dividing mixed numbers by converting to improper fractions and multiplying by the reciprocal.
Estimated Time:1m 0s
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Topics

Fraction Operations

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