1. ## Help about ratio and proportion

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.

2. Define:
$$I_1,I_2$$ Incomes of person 1 and 2
$$E_1,E_2$$ Expenditure of persons 1 and 2.

You are given the following ratios:
$\frac{I_1}{I_2} = \frac{5}{3}$

$\frac{E_1}{E_2} = \frac{8}{5}$

$\frac{I_1 - E_1}{I_2 - E_2} = \frac{4}{3}$

This is 3 equations in 4 unknowns, so we can choose one of the values. I will set $I_1 = 1$

Now:
$\frac{1}{I_2} = \frac{5}{3}$

$\frac{E_1}{E_2} = \frac{8}{5}$

$\frac{I_1 - E_1}{I_2 - E_2} = \frac{4}{3}$

You should be able to solve that system of simultaneous equations to get
$I_1 = 1$
$I_2 = 0.6$
$E_1 = 1.2$
$E_2 = 0.75$

The ratio you want is

$\frac{I_1 + E_1}{I_2 + E_2} = \frac{1.6}{1.95} = \frac{32}{39}$

Its interesting to note that the savings amount comes out as negative for both people...but thats another story.

If you are interested in why i can set $I_1 = 1$ without breaking the answer, its because we are only interested in ratios, not the actual levels of income. I am effectively redefining the currency unit of the problem so that $I_1$ has 1 unit.

3. Originally Posted by navikhi
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.
$\frac{I_A}{I_B}=\frac{5}{3}\ \Rightarrow\ I_B=\frac{3}{5}I_A$

$\frac{E_A}{E_B}=\frac{8}{5}\ \Rightarrow\ E_B=\frac{5}{8}E_A$

Therefore

$\frac{I_A+I_B}{E_A+E_B}=\frac{I_A+\frac{3}{5}I_A}{ E_A+\frac{5}{8}E_A}=\frac{\frac{8}{5}I_A}{\frac{13 }{8}E_A}$

$=\frac{I_A}{E_A}\ \frac{8}{5}\ \frac{8}{13}=\frac{64}{65}\ \frac{I_A}{E_A}$

We can obtain the relationship between $I_A\ and\ E_A$ from the savings ratio

$\frac{S_A}{S_B}=\frac{I_A-E_A}{I_B-E_B}=\frac{4}{3}$

$\frac{I_A-E_A}{\frac{3}{5}I_A-\frac{5}{8}E_A}=\frac{4}{3}$

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

$3I_A-3E_A=\frac{12}{5}I_A-\frac{20}{8}E_A$

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

$\frac{3}{5}I_A=\frac{4}{8}E_A$

$24I_A=20E_A$

$E_A=\frac{24}{20}I_A$

Expenditure exceeds income, hence the savings is <0

Therefore we can use the above to calculate the required ratio

$\frac{64}{65}\ \frac{I_A}{E_A}=\frac{64}{65}\ \frac{I_A}{\frac{24}{20}I_A}$

$=\frac{64}{65}\ \frac{20}{24}=\frac{64}{65}\ \frac{5}{6}=\frac{32}{3}\ \frac{5}{65}=\frac{32}{3}\ \frac{1}{13}$

$=\frac{32}{39}$