Production of wheat is`2(1)/(4)` times that of rice but price of rice is `1(1)/(4)` times that of wheat . If a farmer produces wheat at the place of rice then his income will be change by ?

Options

A. `1(4)/(5)`
​
B. 4/5
C. `1(3)/(5)`
D. 3/5

Answer

To solve the problem step by step, we will first convert the mixed fractions into improper fractions for easier calculations.

### Step 1: Define the production and price of rice and wheat

Let’s denote:
- The production of rice = \( R \)
- The production of wheat = \( W \)

According to the problem:
- The production of wheat is \( 2 \frac{1}{4} \) times that of rice, which can be written as:
\[ W = 2 \frac{1}{4} R = \frac{9}{4} R \]

- The price of rice is \( 1 \frac{1}{4} \) times that of wheat. Let the price of wheat be \( P_w \) and the price of rice be \( P_r \):
\[ P_r = 1 \frac{1}{4} P_w = \frac{5}{4} P_w \]

### Step 2: Calculate the income from rice and wheat

- The income from rice (when the farmer produces rice) is:
\[ \text{Income from rice} = R \times P_r = R \times \frac{5}{4} P_w \]

- The income from wheat (when the farmer produces wheat) is:
\[ \text{Income from wheat} = W \times P_w = \frac{9}{4} R \times P_w \]

### Step 3: Compare the incomes

Now, we will compare the two incomes:
1. Income from rice:
\[ \text{Income from rice} = R \times \frac{5}{4} P_w = \frac{5}{4} R P_w \]

2. Income from wheat:
\[ \text{Income from wheat} = \frac{9}{4} R \times P_w = \frac{9}{4} R P_w \]

### Step 4: Calculate the change in income

To find the change in income, we subtract the income from rice from the income from wheat:
\[ \text{Change in income} = \text{Income from wheat} - \text{Income from rice} \]
\[ = \left(\frac{9}{4} R P_w\right) - \left(\frac{5}{4} R P_w\right) \]
\[ = \left(\frac{9}{4} - \frac{5}{4}\right) R P_w = \frac{4}{4} R P_w = R P_w \]

### Step 5: Calculate the percentage change in income

To find the percentage change in income:
\[ \text{Percentage change} = \left(\frac{\text{Change in income}}{\text{Income from rice}}\right) \times 100 \]
Substituting the values:
\[ = \left(\frac{R P_w}{\frac{5}{4} R P_w}\right) \times 100 = \left(\frac{4}{5}\right) \times 100 = 80\% \]

### Final Result

The income will change by \( 80\% \).