If $(a - b) : (a + b) = 1 : 11,$ find the ratio $(5 a + 4 b + 15) : (5 a - 4 b + 3) .$

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Solution

$(a - b) : (a + b) = 1 : 11$
Let $a - b = x a + b = 11 x - - - - - 2 a = 12 x a = 6 x 6 x - b = x \Rightarrow 5 x = b$
Then ratio $\frac{5 a + 4 b + 15}{5 a + 4 b + 3}$
Let for $x = 1$ $a = 6 : b = 5$
Ratio $\frac{5 a + 4 b + 15}{5 a - 4 b + 3} = \frac{5 \times 6 + 4 \times 5 + 15}{5 \times 6 - 4 \times 5 + 3} = \frac{30 + 20 + 15}{30 - 20 + 3} = \frac{65}{13} = \frac{5}{1}$

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