Finding Incomes of X and Y

Video Explanation

Question

The incomes of X and Y are in the ratio 8 : 7. Their expenditures are in the ratio 19 : 16. Each saves Rs 1250. Find their incomes.

Solution

Step 1: Let Variables

Let incomes of X and Y be \(8x\) and \(7x\) respectively

Let expenditures of X and Y be \(19y\) and \(16y\) respectively

Step 2: Use Saving Concept

Saving = Income − Expenditure

— For X:

\[ 8x – 19y = 1250 \quad (1) \]

For Y:

\[ 7x – 16y = 1250 \quad (2) \]

—

Step 3: Solve Linear Equations

Multiply (2) by 8:

\[ 56x – 128y = 10000 \quad (3) \]

Multiply (1) by 7:

\[ 56x – 133y = 8750 \quad (4) \]

Subtract (4) from (3):

\[ 5y = 1250 \]

\[ y = 250 \]

Substitute into (2):

\[ 7x – 16(250) = 1250 \]

\[ 7x – 4000 = 1250 \]

\[ 7x = 5250 \]

\[ x = 750 \]

—

Step 4: Find Incomes

Income of X: \[ 8x = 8 \times 750 = 6000 \]

Income of Y: \[ 7x = 7 \times 750 = 5250 \]

—

Conclusion

\[ \text{Income of X} = 6000,\quad \text{Income of Y} = 5250 \]

Verification

X saving: \(6000 – 4750 = 1250\) ✔

Y saving: \(5250 – 4000 = 1250\) ✔