A certain sum of money is divided between X and Y in the ratio of `2(1)/(3):1(3)/(4)` and X gets Rs 440. what is Y's share?

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the given ratio
The ratio in which the money is divided between X and Y is given as \( 2 \frac{1}{3} : 1 \frac{3}{4} \). We need to convert these mixed numbers into improper fractions.

**Hint:** To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator.

### Step 2: Convert mixed numbers to improper fractions
1. For \( 2 \frac{1}{3} \):
\[ 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \]

2. For \( 1 \frac{3}{4} \):
\[ 1 \frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4} \]

### Step 3: Write the ratio in fraction form
Now we can express the ratio of X to Y as:
\[ \frac{7}{3} : \frac{7}{4} \]

### Step 4: Simplify the ratio
To simplify the ratio \( \frac{7}{3} : \frac{7}{4} \), we can multiply both sides by the least common multiple of the denominators (3 and 4), which is 12:
\[ \frac{7}{3} \times 12 : \frac{7}{4} \times 12 = 28 : 21 \]
Now simplify \( 28 : 21 \) by dividing both terms by 7:
\[ 4 : 3 \]

### Step 5: Set up the equation for X's share
We know that X's share is Rs 440. Since the ratio of X to Y is \( 4 : 3 \), we can express X's share in terms of the total parts of the ratio:
\[ \text{Let total parts} = 4 + 3 = 7 \]
Thus, X's share can be expressed as:
\[ \text{X's share} = \frac{4}{7} \times \text{Total money} \]

### Step 6: Set up the equation to find the total money
Since X gets Rs 440, we can set up the equation:
\[ \frac{4}{7} \times \text{Total money} = 440 \]

### Step 7: Solve for total money
To find the total money, we rearrange the equation:
\[ \text{Total money} = 440 \times \frac{7}{4} \]
Calculating this gives:
\[ \text{Total money} = 440 \times 1.75 = 770 \]

### Step 8: Calculate Y's share
Now that we have the total money, we can find Y's share using the ratio:
\[ \text{Y's share} = \frac{3}{7} \times 770 \]
Calculating this gives:
\[ \text{Y's share} = 330 \]

### Final Answer
Y's share is Rs 330.

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