Understanding the Income Ratios

This problem involves ratios of incomes for two individuals, A and B, over two different years: last year and the present year. We are given three key ratios:

• The ratio of A's income to B's income in the last year.
• The ratio of A's income in the last year to A's income in the present year.
• The ratio of B's income in the last year to B's income in the present year.

We are also given the total income of A and B in the present year and asked to find the sum of A's income in the last year and B's income in the present year.

Setting Up the Problem with Variables

Let's denote the incomes as follows:

• \(A_L\): Income of A in the last year
• \(B_L\): Income of B in the last year
• \(A_P\): Income of A in the present year
• \(B_P\): Income of B in the present year

From the problem statement, we have the following relationships based on the given ratios:

1. Ratio of incomes of A and B in the last year: \(A_L : B_L = 4 : 3\)
2. Ratio of A's income in the last year and present year: \(A_L : A_P = 3 : 4\)
3. Ratio of B's income in the last year and present year: \(B_L : B_P = 5 : 6\)

We are also given that the total income in the present year is Rs. 24.12 lakhs, which means \(A_P + B_P = 24.12\).

Expressing Incomes in Terms of a Common Variable

Let's use the first ratio to express \(A_L\) and \(B_L\) in terms of a common variable, say \(k\).

Since \(A_L : B_L = 4 : 3\), we can write \(A_L = 4k\) and \(B_L = 3k\).

Now, let's use the other two ratios to express \(A_P\) and \(B_P\) in terms of the same variable \(k\).

• For A: \(A_L : A_P = 3 : 4\) \[ \frac{A_L}{A_P} = \frac{3}{4} \] Substitute \(A_L = 4k\): \[ \frac{4k}{A_P} = \frac{3}{4} \] \[ 3 \times A_P = 4 \times 4k \] \[ 3 A_P = 16k \] \[ A_P = \frac{16k}{3} \]
• For B: \(B_L : B_P = 5 : 6\) \[ \frac{B_L}{B_P} = \frac{5}{6} \] Substitute \(B_L = 3k\): \[ \frac{3k}{B_P} = \frac{5}{6} \] \[ 5 \times B_P = 3k \times 6 \] \[ 5 B_P = 18k \] \[ B_P = \frac{18k}{5} \]

So, we have \(A_L = 4k\), \(B_L = 3k\), \(A_P = \frac{16k}{3}\), and \(B_P = \frac{18k}{5}\).

Using the Total Present Year Income to Find k

We know that the total income in the present year is Rs. 24.12 lakhs. So, \(A_P + B_P = 24.12\).

Substitute the expressions for \(A_P\) and \(B_P\) in terms of \(k\):

To solve for \(k\), find a common denominator for the fractions, which is 15.

Let's perform the division:

The value of the variable \(k\) is 2.7.

Calculating the Required Incomes

We need to find the sum of A's income in the last year (\(A_L\)) and B's income in the present year (\(B_P\)).

• A's income in the last year (\(A_L\)) = \(4k\)
• B's income in the present year (\(B_P\)) = \(\frac{18k}{5}\)

Substitute the value of \(k = 2.7\):

• \(A_L = 4 \times 2.7 = 10.8\) lakhs
• \(B_P = \frac{18 \times 2.7}{5} = \frac{48.6}{5} = 9.72\) lakhs

Finding the Sum

The sum of A's income in the last year and B's income in the present year is \(A_L + B_P\).

The sum of the income of A in the last year and that of B in the present year is Rs. 20.52 lakhs.

Summary of Incomes (in lakhs)

The sum required is \(A_L + B_P = 10.8 + 9.72 = 20.52\) lakhs.

Revision Table - Income Ratio Problem

Additional Information - Ratios and Proportions

Ratios and proportions are fundamental concepts in mathematics used to compare quantities and describe relationships. A ratio \(a:b\) can be written as a fraction \(\frac{a}{b}\). A proportion is an equality between two ratios, e.g., \(\frac{a}{b} = \frac{c}{d}\).

• Direct Proportion: If two quantities increase or decrease together at a constant rate, they are directly proportional. For example, if you buy more items of the same price, the total cost increases proportionally.
• Inverse Proportion: If an increase in one quantity causes a decrease in another quantity, they are inversely proportional. For example, if you increase speed, the time taken to cover a fixed distance decreases.

In this problem, we used ratios to set up proportional relationships between incomes across individuals and across years. By introducing a common variable (\(k\)), we were able to link all the income figures together and solve for the specific values using the given total income.