Manny runs 15 3/4 miles each week. Zander runs 5/6 as many miles each week as Manny.

Question

How many miles does Zander run each week?

Enter your answer as a mixed number in simplest form by filling in the boxes.

Answer 1

To find out how many miles Zander runs each week, we need to calculate 5/6 of the miles Manny runs.

Manny runs \(15 \frac{3}{4}\) miles each week. First, let's convert this mixed number into an improper fraction.

1. Convert \(15 \frac{3}{4}\) into an improper fraction: \[ 15 \frac{3}{4} = 15 + \frac{3}{4} = \frac{15 \times 4}{4} + \frac{3}{4} = \frac{60}{4} + \frac{3}{4} = \frac{63}{4} \]

2. Next, calculate \( \frac{5}{6} \) of \( \frac{63}{4} \): \[ Zander's\ miles = \frac{5}{6} \times \frac{63}{4} = \frac{5 \times 63}{6 \times 4} = \frac{315}{24} \]

3. Simplify \( \frac{315}{24} \):

   • First, find the greatest common divisor (GCD) of 315 and 24.
   • The prime factorization of 315 is \( 3 \times 3 \times 5 \times 7 \) (or \( 3^2 \times 5 \times 7 \)).
   • The prime factorization of 24 is \( 2^3 \times 3 \).
   • The GCD is \( 3 \) (the only common factor).

   Now, divide both the numerator and the denominator by the GCD: \[ \frac{315 \div 3}{24 \div 3} = \frac{105}{8} \]

4. Convert \( \frac{105}{8} \) into a mixed number:

   • Divide 105 by 8. The quotient is 13 and the remainder is 1.
   • Therefore, \( \frac{105}{8} = 13 \frac{1}{8} \).

In simplest form, Zander runs \( 13 \frac{1}{8} \) miles each week.

Final answer: \[ \text{Zander runs } 13 \frac{1}{8} \text{ miles each week}. \]