In a house, there are dogs, cats and parrot in the ratio `3 : 7 : 5`. If the number of cats was more than the number of dogs by a multiple of both 9 and 7, what is the minimum of pets in the house?

Options

A. 945
B. 630
C. 252
D. 238

Answer

To solve the problem step by step, we will follow the given information and derive the minimum number of pets in the house.

### Step 1: Understand the Ratio
The ratio of dogs, cats, and parrots is given as \(3 : 7 : 5\). This means:
- Let the number of dogs be \(3k\)
- Let the number of cats be \(7k\)
- Let the number of parrots be \(5k\)

### Step 2: Express the Difference
We know that the number of cats is more than the number of dogs. We can express this difference mathematically:
\[ \text{Difference} = 7k - 3k = 4k \]
This difference \(4k\) must be a multiple of both 9 and 7.

### Step 3: Find the Least Common Multiple (LCM)
To find a common multiple of 9 and 7, we calculate the least common multiple (LCM):
\[ \text{LCM}(9, 7) = 63 \]
Thus, \(4k\) must be a multiple of 63.

### Step 4: Set Up the Equation
Since \(4k\) is a multiple of 63, we can write:
\[ 4k = 63m \]
where \(m\) is a positive integer.

### Step 5: Solve for \(k\)
Now, we can solve for \(k\):
\[ k = \frac{63m}{4} \]
For \(k\) to be a natural number, \(63m\) must be divisible by 4. Since 63 is not divisible by 4, we need to find the smallest \(m\) such that \(63m\) is divisible by 4.

### Step 6: Find the Minimum Value of \(m\)
To find the smallest \(m\) that makes \(63m\) divisible by 4, we can check the multiples of 63:
- \(m = 4\) gives \(63 \times 4 = 252\), which is divisible by 4.

Thus, the smallest \(m\) is 4.

### Step 7: Calculate \(k\)
Now substituting \(m = 4\) back into the equation for \(k\):
\[ k = \frac{63 \times 4}{4} = 63 \]

### Step 8: Calculate the Total Number of Pets
Now we calculate the total number of pets:
\[ \text{Total Pets} = 3k + 7k + 5k = 15k = 15 \times 63 = 945 \]

### Conclusion
The minimum number of pets in the house is **945**.

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