Instead of dividing Rs 3,910 among P , Q and R in the ratio `1/4 : 1/5 : 1/8` by mistake it was divided in the ratio 4:5:8. By how much did R gain in this transaction?

Options

A. Rs 990
B. Rs 940
C. Rs 890
D. Rs 900

Answer

To solve the problem step by step, we need to find out how much R gained when the amount of Rs 3,910 was mistakenly divided in the ratio 4:5:8 instead of the intended ratio of 1/4:1/5:1/8.

### Step 1: Find the correct ratio of P, Q, and R
The intended ratio is given as:
\[ \frac{1}{4} : \frac{1}{5} : \frac{1}{8} \]
To simplify this, we can find a common denominator. The least common multiple (LCM) of 4, 5, and 8 is 40.

Now, we convert each fraction:
- For \(\frac{1}{4}\): \( \frac{1 \times 10}{4 \times 10} = \frac{10}{40} \)
- For \(\frac{1}{5}\): \( \frac{1 \times 8}{5 \times 8} = \frac{8}{40} \)
- For \(\frac{1}{8}\): \( \frac{1 \times 5}{8 \times 5} = \frac{5}{40} \)

Thus, the intended ratio becomes:
\[ 10 : 8 : 5 \]

### Step 2: Calculate the total parts in the correct ratio
Now, we add the parts of the ratio:
\[ 10 + 8 + 5 = 23 \]

### Step 3: Calculate the share of R in the correct ratio
R's share in the correct ratio is 5 parts. Therefore, R's share would be:
\[ \text{R's share} = \frac{5}{23} \times 3910 \]

Calculating R's share:
\[ \text{R's share} = \frac{5 \times 3910}{23} = \frac{19550}{23} = 850 \]

### Step 4: Calculate the share of P, Q, and R in the mistaken ratio
The mistaken ratio is:
\[ 4 : 5 : 8 \]
The total parts in this ratio are:
\[ 4 + 5 + 8 = 17 \]

Now, we calculate R's share in the mistaken ratio:
\[ \text{R's share} = \frac{8}{17} \times 3910 \]

Calculating R's share:
\[ \text{R's share} = \frac{8 \times 3910}{17} = \frac{31280}{17} = 1840 \]

### Step 5: Calculate the gain of R
Now, we find out how much R gained:
\[ \text{Gain} = \text{R's share in mistaken ratio} - \text{R's share in correct ratio} \]
\[ \text{Gain} = 1840 - 850 = 990 \]

### Final Answer
Thus, R gained Rs 990 in this transaction.