The expenditure of A is four times the savings of B and the savings of A is 3/20th of the expenditure of B. If the ratio of the income of A to B is 19:24 and the difference between the income of A and B is 2500, then find the savings of A

a) Rs.1740

b) Rs.2400

c) Rs.900

d) Rs.1500

e) None of these

Let's denote the variables as follows:

- Let \( \text{Income of A} = I_A \)
- Let \( \text{Income of B} = I_B \)
- Let \( \text{Expenditure of A} = E_A \)
- Let \( \text{Expenditure of B} = E_B \)
- Let \( \text{Savings of A} = S_A \)
- Let \( \text{Savings of B} = S_B \)

Given:

1. The ratio of incomes \( I_A : I_B = 19 : 24 \).
2. The difference in their incomes is \( I_B - I_A = 2500 \).
3. The expenditure of A is four times the savings of B, i.e., \( E_A = 4S_B \).
4. The savings of A is \( \frac{3}{20} \) of the expenditure of B, i.e., \( S_A = \frac{3}{20}E_B \).

### Step 1: Find the incomes of A and B

Given the ratio \( I_A : I_B = 19 : 24 \) and the difference \( I_B - I_A = 2500 \):

Let \( I_A = 19x \) and \( I_B = 24x \).

So, 
\[
I_B - I_A = 24x - 19x = 5x = 2500
\]
\[
x = \frac{2500}{5} = 500
\]

Thus,
\[
I_A = 19x = 19 \times 500 = 9500 \quad \text{and} \quad I_B = 24x = 24 \times 500 = 12000
\]

### Step 2: Use the expenditure and savings relations

Since savings are given by income minus expenditure:
\[
S_A = I_A - E_A \quad \text{and} \quad S_B = I_B - E_B
\]

From the given relations:

1. \( E_A = 4S_B \)
2. \( S_A = \frac{3}{20}E_B \)

### Step 3: Solve for \( S_A \)

First, express \( E_A \) in terms of \( S_B \) and substitute it:
\[
E_A = 4S_B
\]
Thus,
\[
S_A = I_A - E_A = 9500 - 4S_B
\]

Also, from \( S_A = \frac{3}{20}E_B \):
\[
E_B = I_B - S_B = 12000 - S_B
\]
\[
S_A = \frac{3}{20}(12000 - S_B) = \frac{3}{20} \times (12000 - S_B) = 1800 - \frac{3}{20}S_B
\]

Now, equating the two expressions for \( S_A \):
\[
9500 - 4S_B = 1800 - \frac{3}{20}S_B
\]

### Step 4: Solve the equation

To solve for \( S_B \), first clear the fraction by multiplying the entire equation by 20:
\[
20(9500 - 4S_B) = 20(1800 - \frac{3}{20}S_B)
\]
\[
190000 - 80S_B = 36000 - 3S_B
\]
Now, bring like terms together:
\[
190000 - 36000 = 80S_B - 3S_B
\]
\[
154000 = 77S_B
\]
\[
S_B = \frac{154000}{77} = 2000
\]

### Step 5: Find \( S_A \)

Substitute \( S_B = 2000 \) into \( S_A = 9500 - 4S_B \):
\[
S_A = 9500 - 4 \times 2000 = 9500 - 8000 = 1500
\]

Therefore, the savings of A is \( \boxed{1500} \).

Would you like more details or have any questions? Here are 5 related questions for you to consider:

1. What would be the savings of B given the same scenario?
2. How would the result change if the income ratio of A to B were different?
3. Can you derive the expenditure of both A and B from the information given?
4. How does the problem change if the difference in income between A and B is altered?
5. How can these equations be set up to solve a similar problem with different ratios?

**Tip:** In problems involving ratios, always consider how changing one variable impacts the others.

Learn how to solve a complex problem involving income, expenditure, and savings relationships between individuals A and B. This problem covers ratio and proportion, linear equations, and practical applications of mathematical concepts in financial scenarios. Suitable for high school students.

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