To determine the distance Ruby walked before lunch, we can follow these steps:
1. Identify the total distance of Ruby's walk:
Ruby walked a total distance of \(\frac{10}{3}\) miles.
2. Identify the fraction of the walk completed before lunch:
Ruby walked \(\frac{12}{25}\) of the total distance before stopping for lunch.
3. Calculate the distance walked before lunch:
To find the distance Ruby walked before lunch, we need to multiply the total distance by the fraction of the way she walked.
Given:
[tex]\[
\text{Total distance} = \frac{10}{3} \text{ miles}
\][/tex]
[tex]\[
\text{Fraction walked before lunch} = \frac{12}{25}
\][/tex]
Therefore, the distance walked before lunch is:
[tex]\[
\text{Distance before lunch} = \left( \frac{10}{3} \right) \times \left( \frac{12}{25} \right)
\][/tex]
4. Simplify the calculation:
Multiply the numerators together and the denominators together:
[tex]\[
\text{Distance before lunch} = \frac{10 \times 12}{3 \times 25} = \frac{120}{75}
\][/tex]
5. Simplify the fraction:
The fraction \(\frac{120}{75}\) can be simplified by finding the greatest common divisor (GCD) of 120 and 75, which is 15. Simplify by dividing both the numerator and the denominator by 15:
[tex]\[
\frac{120 \div 15}{75 \div 15} = \frac{8}{5}
\][/tex]
Thus, Ruby walked [tex]\(\frac{8}{5}\)[/tex] miles before lunch.
