Sure, let's go through a step-by-step solution to determine how much older Mia is than Fahi given that Fahi's age is one-sixth of Mia's age and the sum of their ages is 91.
1. Define Variables:
Let \(\( M \)\) represent Mia's age.
Let \(\( F \)\) represent Fahi's age.
2. Set Up Equations:
Given that Fahi's age is one-sixth of Mia's age, we can write:
\(\[ F = \frac{1}{6} M \]\)
Also given that the sum of their ages is 91, we can write:
\(\[ F + M = 91 \]\)
3. Substitute Fahi's Age (F) into the Sum Equation:
Substitute \(\( F \)\) with \(\( \frac{1}{6} M \)\) in the second equation:
\(\[ \frac{1}{6} M + M = 91 \]\)
4. Combine Like Terms:
Combine the terms involving \(\( M \)\):
\(\[ \frac{1}{6} M + \frac{6}{6} M = 91 \]\)
\(\[ \frac{7}{6} M = 91 \]\)
5. Solve for Mia's Age (M):
To solve for \(\( M \)\), multiply both sides by 6 to clear the fraction:
\(\[ 7M = 546 \]\)
Now, divide both sides by 7:
\(\[ M = 78 \]\)
So, Mia is 78 years old.
6. Determine Fahi's Age (F):
Recall that \(\( F \)\) is one-sixth of \(\( M \)\):
\(\[ F = \frac{1}{6} M \]\)
Substitute \(\( M = 78 \)\):
\(\[ F = \frac{1}{6} \times 78 \]\)
\(\[ F = 13 \]\)
So, Fahi is 13 years old.
7. Calculate the Age Difference:
To find how much older Mia is than Fahi, subtract Fahi's age from Mia's age:
\(\[ \text{Age Difference} = M - F \]\)
\(\[ \text{Age Difference} = 78 - 13 \]\)
\(\[ \text{Age Difference} = 65 \]\)
Therefore, Mia is 65 years older than Fahi.
