The sum of two natural numbers is 64. Which of the following can not be the ratio of these two numbers.

Options

A. `3:5`
B. `1:3`
C. `7:9`
D. `3:4`

Answer

To solve the problem, we need to determine which of the given ratios cannot represent two natural numbers that sum to 64. Let's analyze each ratio step by step.

### Step 1: Understand the Problem
We know the sum of two natural numbers is 64. If we denote these two numbers as \( a \) and \( b \), we have:
\[ a + b = 64 \]

### Step 2: Express the Ratio
If the ratio of the two numbers is given as \( m:n \), we can express the two numbers in terms of a variable \( x \):
\[ a = mx \]
\[ b = nx \]
Thus, we can rewrite the sum:
\[ mx + nx = 64 \]
This simplifies to:
\[ (m+n)x = 64 \]

### Step 3: Solve for \( x \)
From the equation \( (m+n)x = 64 \), we can solve for \( x \):
\[ x = \frac{64}{m+n} \]

### Step 4: Check Each Ratio
Now we need to check each ratio to see if \( x \) results in a natural number (whole number).

1. **Ratio 3:5**
- \( m+n = 3+5 = 8 \)
- \( x = \frac{64}{8} = 8 \) (Natural number)

2. **Ratio 1:3**
- \( m+n = 1+3 = 4 \)
- \( x = \frac{64}{4} = 16 \) (Natural number)

3. **Ratio 7:9**
- \( m+n = 7+9 = 16 \)
- \( x = \frac{64}{16} = 4 \) (Natural number)

4. **Ratio 3:4**
- \( m+n = 3+4 = 7 \)
- \( x = \frac{64}{7} \approx 9.14 \) (Not a natural number)

### Conclusion
The ratio that cannot represent two natural numbers whose sum is 64 is **3:4**.