Calculating Income, Expenditure, and Savings Ratios

This problem involves understanding ratios related to income, expenditure, and savings for two individuals, A and B. We are given the ratio of their incomes, the ratio of their expenditures, and a specific condition relating A's income to B's expenditure. Our goal is to find the ratio of their savings.

Setting up Variables for Income and Expenditure

Let's represent the monthly incomes and expenditures using variables based on the given ratios:

• Ratio of incomes of A and B is $4:5$. Let their monthly incomes be $4x$ and $5x$, where $x$ is a common multiplier.
• Ratio of expenditures of A and B is $3:8$. Let their monthly expenditures be $3y$ and $8y$, where $y$ is another common multiplier.

So:

• Income of A = $4x$
• Income of B = $5x$
• Expenditure of A = $3y$
• Expenditure of B = $8y$

Using the Given Condition

The problem states that the income of A is equal to the expenditure of B.

Mathematically, this condition is:

Income of A = Expenditure of B

$4x = 8y$

We can use this equation to find a relationship between $x$ and $y$. Dividing both sides by 4, we get:

$x = 2y$

This equation tells us that the multiplier for income ($x$) is twice the multiplier for expenditure ($y$).

Calculating Savings for A and B

Savings is defined as Income minus Expenditure. We will calculate the savings for both A and B using our expressions for income and expenditure:

• Savings of A = Income of A - Expenditure of A = $4x - 3y$
• Savings of B = Income of B - Expenditure of B = $5x - 8y$

Substituting the Relationship between x and y

Now, we substitute the relationship $x = 2y$ into the savings expressions. This will allow us to express the savings of both A and B in terms of a single variable ($y$).

• Savings of A = $4(2y) - 3y = 8y - 3y = 5y$
• Savings of B = $5(2y) - 8y = 10y - 8y = 2y$

So, Savings of A is $5y$ and Savings of B is $2y$.

Finding the Ratio of Savings

Finally, we need to find the ratio of savings of A and B.

Ratio of Savings (A : B) = Savings of A : Savings of B

Ratio of Savings (A : B) = $5y : 2y$

Since $y$ is a common factor (and assuming income and expenditure are positive, $y > 0$), we can cancel $y$ from both sides of the ratio.

Ratio of Savings (A : B) = $5 : 2$

The ratio of savings of A and B is $5:2$.

Let's summarize the values in a table for clarity:

The ratio of savings of A to B is indeed $5y : 2y$, which simplifies to $5:2$.

Revision Table: Income Expenditure Savings Ratio

Additional Information on Ratios and Proportions

Understanding ratios and proportions is fundamental in solving problems like this. A ratio expresses how many times one number contains another. It's a way of comparing sizes. For example, an income ratio of $4:5$ means that for every $4 units of income A has, B has $5 units.

When working with ratios involving different categories (like income and expenditure here), it's important to use different variables (like $x$ and $y$) for the common multipliers unless a specific relationship between the categories is given, as was the case with the condition "Income of A = Expenditure of B". This condition allowed us to link the two sets of variables and solve the problem.

Problems involving income, expenditure, and savings often require setting up equations based on given ratios and conditions, then solving for unknown values or ratios. Always remember the core relationship: Savings = Income - Expenditure.