The number of oranges in three baskets are in the ratio of 3:4:5. In which ratio the no. of oranges in first two baskets must be increased so that the new ratio becomes 5:4:3?

Options

A. `1:3`
B. `2:1`
C. `3:4`
D. `2:3`

Answer

To solve the problem, we need to determine the ratio in which the number of oranges in the first two baskets must be increased so that the new ratio becomes 5:4:3.

### Step-by-Step Solution:

1. **Define the Initial Ratios**:
Let the number of oranges in the three baskets be represented as:
- Basket 1: 3x
- Basket 2: 4x
- Basket 3: 5x
Here, x is a common multiplier.

2. **Define the New Ratios**:
We want the new ratio of the number of oranges to be 5:4:3. Let’s denote the increase in the number of oranges in the first basket as 'a' and in the second basket as 'b'.

Therefore, the new quantities will be:
- New Basket 1: 3x + a
- New Basket 2: 4x + b
- New Basket 3: 5x (remains unchanged)

3. **Set Up the Equation**:
According to the new ratio, we can set up the equation:
\[ \frac{3x + a}{4x + b} = \frac{5}{4} \]
\[ \frac{4x + b}{5x} = \frac{4}{3} \]

4. **Cross Multiply to Solve for a and b**:
From the first equation:
\[ 4(3x + a) = 5(4x + b) \]
Expanding gives:
\[ 12x + 4a = 20x + 5b \]
Rearranging gives:
\[ 4a - 5b = 8x \quad \text{(Equation 1)} \]

From the second equation:
\[ 3(4x + b) = 4(5x) \]
Expanding gives:
\[ 12x + 3b = 20x \]
Rearranging gives:
\[ 3b = 8x \quad \text{(Equation 2)} \]

5. **Substitute Equation 2 into Equation 1**:
From Equation 2, we have:
\[ b = \frac{8x}{3} \]
Substitute this into Equation 1:
\[ 4a - 5\left(\frac{8x}{3}\right) = 8x \]
This simplifies to:
\[ 4a - \frac{40x}{3} = 8x \]
Multiplying through by 3 to eliminate the fraction:
\[ 12a - 40x = 24x \]
Rearranging gives:
\[ 12a = 64x \]
Therefore:
\[ a = \frac{64x}{12} = \frac{16x}{3} \]

6. **Find the Ratio of Increases**:
Now we have:
- Increase in Basket 1: \( a = \frac{16x}{3} \)
- Increase in Basket 2: \( b = \frac{8x}{3} \)

The ratio of the increases \( a:b \) is:
\[ \frac{a}{b} = \frac{\frac{16x}{3}}{\frac{8x}{3}} = \frac{16}{8} = 2:1 \]

### Final Answer:
The ratio in which the number of oranges in the first two baskets must be increased is **2:1**.