To find the old record, we can set up the relationship based on the information given. Let \( x \) be the previous best distance in the jump competition. According to the problem, Bill's distance was \( 1 \frac{1}{8} \) times longer than the previous record, which can be expressed as:
\[ \text{Bill's distance} = \frac{9}{8} x \]
We know that Bill's distance is \( 5 \frac{2}{5} \) m. We can convert this mixed number into an improper fraction:
\[ 5 \frac{2}{5} = \frac{5 \times 5 + 2}{5} = \frac{25 + 2}{5} = \frac{27}{5} \]
Now we can set up the equation:
\[ \frac{27}{5} = \frac{9}{8} x \]
To solve for \( x \), we can multiply both sides by \( \frac{8}{9} \):
\[ x = \frac{27}{5} \times \frac{8}{9} \]
Next, we can simplify the right side:
\[ x = \frac{27 \times 8}{5 \times 9} = \frac{216}{45} \]
We can simplify \( \frac{216}{45} \) by finding the greatest common divisor (GCD) of 216 and 45:
- The GCD of 216 and 45 is 9.
- Now, divide the numerator and denominator by their GCD:
\[ x = \frac{216 \div 9}{45 \div 9} = \frac{24}{5} \]
We can convert \( \frac{24}{5} \) back into a mixed number:
\[ \frac{24}{5} = 4 \frac{4}{5} \]
Thus, the old record was:
\[ \boxed{4 \frac{4}{5} \text{ m}} \]
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