Incomes of $A$ and $B$ are in the ratio $4:5$ and expenditures are in the ratio $5:6$

Question

If incomes of $A$ and $B$ are in the ratio $4:5$ and expenditures are in the ratio $5:6$, whatever be the absolute value of their incomes and expenditure $B$ will save more. How?

Answer 1

Express it in the form of equations. Say A's income is $i_A$ and A's expenditure is $e_A$.

Then by the ratios above B's income is $i_B=\frac{5}{4}i_A$ and B's expenditure is $e_B=\frac{6}{5}e_A$.

How much will B save is determined by $i_B-e_B=\frac{5}{4}i_A-\frac{6}{5}e_A>\frac{6}{5}i_A-\frac{6}{5}e_A>i_A-e_A$

and the last expression is how much A will save.

Answer 2

One way to see it is savings varies linearly with expenditure, and in both extreme cases, B saves more than A, so B saves more than A under all cases between the extreme cases.

Answer 3

Given information:

• A makes $x$ and spends $y$
• B makes $\frac54x$ and spends $\frac65y$

Therefore:

• A saves $x-y$
• B saves $\frac54x-\frac65y$

Therefore:

• B saves $\frac{25}{20}x-\frac{24}{20}y$

Therefore:

• B saves $\frac{25}{20}x-\frac{25}{20}y+\frac{1}{20}y$

Therefore:

• B saves $\frac{25}{20}(x-y)+\frac{1}{20}y$

Therefore:

• B saves $\frac{25}{20}a+\frac{1}{20}y$, where $a$ is A's savings

Keep in mind that $y\geq0$, therefore B saves more than A.