Manish, rahul and Bharti have some stones with each of the . Five times the number of stones with Rahul equals seven times the number of stones with Manish while five times the number of stones with Manish equals seven time the number of stones with Bharti. What is the minimum number of stones that can be there with all three of them put together?

Options

A. 113
B. 109
C. 93
D. 97

Answer

To solve the problem step by step, let's denote the number of stones with Manish, Rahul, and Bharti as M, R, and B respectively.

### Step 1: Set up the equations based on the problem statement.
From the problem, we have two relationships:
1. \( 5R = 7M \) (Five times the number of stones with Rahul equals seven times the number of stones with Manish)
2. \( 5M = 7B \) (Five times the number of stones with Manish equals seven times the number of stones with Bharti)

### Step 2: Rearrange the equations to express R and B in terms of M.
From the first equation, we can express R in terms of M:
\[ R = \frac{7M}{5} \]

From the second equation, we can express B in terms of M:
\[ B = \frac{5M}{7} \]

### Step 3: Find a common multiple to eliminate fractions.
To eliminate the fractions, we can find a common multiple for the denominators (5 and 7). The least common multiple (LCM) of 5 and 7 is 35.

### Step 4: Express M, R, and B in terms of a common variable.
Let’s express M in terms of a new variable k:
- Let \( M = 35k \)

Now substituting M into the equations for R and B:
\[ R = \frac{7(35k)}{5} = 49k \]
\[ B = \frac{5(35k)}{7} = 25k \]

### Step 5: Calculate the total number of stones.
Now we can find the total number of stones with all three:
\[ \text{Total} = M + R + B = 35k + 49k + 25k = 109k \]

### Step 6: Find the minimum number of stones.
To find the minimum number of stones, we need to set \( k = 1 \):
\[ \text{Total} = 109 \times 1 = 109 \]

Thus, the minimum number of stones that can be there with all three of them put together is **109**.

### Final Answer:
The minimum number of stones that can be there with all three of them put together is **109**.

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