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