The ratio between the incomes of two people is 5:3. The ratio between their expenditures is 8:5, while the ratio between their savings is 4:3. What will be the ratio between their combined Incomes and Combined Expenditures. (Answer = 32:39) Please tell me how to do it.

\(\displaystyle \frac{I_A}{I_B}=\frac{5}{3}\ \Rightarrow\ I_B=\frac{3}{5}I_A\)

\(\displaystyle \frac{E_A}{E_B}=\frac{8}{5}\ \Rightarrow\ E_B=\frac{5}{8}E_A\)

Therefore

\(\displaystyle \frac{I_A+I_B}{E_A+E_B}=\frac{I_A+\frac{3}{5}I_A}{E_A+\frac{5}{8}E_A}=\frac{\frac{8}{5}I_A}{\frac{13}{8}E_A}\)

\(\displaystyle =\frac{I_A}{E_A}\ \frac{8}{5}\ \frac{8}{13}=\frac{64}{65}\ \frac{I_A}{E_A}\)

We can obtain the relationship between \(\displaystyle I_A\ and\ E_A\) from the savings ratio

\(\displaystyle \frac{S_A}{S_B}=\frac{I_A-E_A}{I_B-E_B}=\frac{4}{3}\)

\(\displaystyle \frac{I_A-E_A}{\frac{3}{5}I_A-\frac{5}{8}E_A}=\frac{4}{3}\)

\(\displaystyle 3I_A-3E_A=4\left(\frac{3}{5}I_A\right)-4\left(\frac{5}{8}\right)E_A\)

\(\displaystyle 3I_A-3E_A=\frac{12}{5}I_A-\frac{20}{8}E_A\)

\(\displaystyle \left(\frac{15}{5}-\frac{12}{5}\right)I_A=\left(\frac{24}{8}-\frac{20}{8}\right)E_A\)

\(\displaystyle \frac{3}{5}I_A=\frac{4}{8}E_A\)

\(\displaystyle 24I_A=20E_A\)

\(\displaystyle E_A=\frac{24}{20}I_A\)

Expenditure exceeds income, hence the savings is <0

Therefore we can use the above to calculate the required ratio

\(\displaystyle \frac{64}{65}\ \frac{I_A}{E_A}=\frac{64}{65}\ \frac{I_A}{\frac{24}{20}I_A}\)

\(\displaystyle =\frac{64}{65}\ \frac{20}{24}=\frac{64}{65}\ \frac{5}{6}=\frac{32}{3}\ \frac{5}{65}=\frac{32}{3}\ \frac{1}{13}\)

\(\displaystyle =\frac{32}{39}\)