When the market price per kg of rice and wheat be in the ratio 3: 2, the monthly expenses of a family towards rice and wheat are in the ratio 5: 6. If the market price of rice and wheat becomes In the ratio 4:3, what will be the ratio of expenses towards them? (Assume that amount of rice and wheat consumed remains unaltered.)

Options

A. `20:29`
B. `20:27`
C. `18:25`
D. `21:37`

Answer

To solve the problem step by step, we will follow the given ratios and calculate the expenses accordingly.

### Step 1: Understand the initial conditions
We are given:
- The initial market price ratio of rice to wheat = 3:2
- The initial expense ratio of rice to wheat = 5:6

### Step 2: Calculate the quantities consumed
Let the quantity of rice consumed be \( q_r \) and the quantity of wheat consumed be \( q_w \).

From the initial expenses, we can express the quantities in terms of the expenses and prices:
- Expense on rice = \( 5x \) (for some value \( x \))
- Expense on wheat = \( 6x \)

Using the price ratio:
- Price of rice = \( 3y \) (for some value \( y \))
- Price of wheat = \( 2y \)

Now, we can find the quantities consumed:
\[ q_r = \frac{\text{Expense on rice}}{\text{Price of rice}} = \frac{5x}{3y} \]
\[ q_w = \frac{\text{Expense on wheat}}{\text{Price of wheat}} = \frac{6x}{2y} = \frac{3x}{y} \]

### Step 3: New price ratio
Now, the new market price ratio of rice to wheat is given as 4:3.
- New price of rice = \( 4z \) (for some value \( z \))
- New price of wheat = \( 3z \)

### Step 4: Calculate new expenses
Since the quantities consumed remain unchanged, we can calculate the new expenses:
- New expense on rice = \( q_r \times \text{New price of rice} = \frac{5x}{3y} \times 4z \)
- New expense on wheat = \( q_w \times \text{New price of wheat} = \frac{3x}{y} \times 3z \)

Calculating these:
1. New expense on rice:
\[ = \frac{5x}{3y} \times 4z = \frac{20xz}{3y} \]
2. New expense on wheat:
\[ = \frac{3x}{y} \times 3z = \frac{9xz}{y} \]

### Step 5: Find the ratio of new expenses
Now, we need to find the ratio of the new expenses:
\[ \text{Ratio of new expenses} = \frac{\frac{20xz}{3y}}{\frac{9xz}{y}} = \frac{20xz}{3y} \times \frac{y}{9xz} \]
This simplifies to:
\[ = \frac{20}{27} \]

### Conclusion
Thus, the ratio of expenses towards rice and wheat when the market price becomes 4:3 is:
\[ \text{Ratio of expenses} = 20:27 \]