Let the present ages of Mutua and Wambua be \( 15x \) and \( 8x \) respectively, where \( x \) is a common factor.
In 10 years, Mutua's age will be \( 15x + 10 \) and Wambua's age will be \( 8x + 10 \).
According to the problem, the ratio of their ages in 10 years will be \( 5:23 \). Therefore, we can set up the equation:
\[ \frac{15x + 10}{8x + 10} = \frac{5}{23} \]
Cross multiplying gives us:
\[ 23(15x + 10) = 5(8x + 10) \]
Expanding both sides:
\[ 345x + 230 = 40x + 50 \]
Now, move all terms involving \( x \) to one side and constant terms to the other side:
\[ 345x - 40x = 50 - 230 \]
\[ 305x = -180 \]
Now, solving for \( x \):
\[ x = -\frac{180}{305} = -\frac{36}{61} \]
This value does not make sense in the context of finding ages, suggesting that we need to double-check the setup.
Upon re-evaluating the condition that "Mutua will be 70 years old in 10 years", we can directly calculate any variables needed.
Since Mutua will be 70 in 10 years, we have:
\[ 15x + 10 = 70 \]
Subtracting 10 from both sides:
\[ 15x = 60 \]
Dividing by 15 gives:
\[ x = 4 \]
Now, we can find the present ages:
For Mutua:
\[ 15x = 15(4) = 60 , \text{years} \]
For Wambua:
\[ 8x = 8(4) = 32 , \text{years} \]
Thus, the present ages are:
- Mutua: 60 years
- Wambua: 32 years