Two numbers whose sum is 84 can not be in the ratio

Options

A. `5:7`
B. `13:8`
C. `1:3`
D. `3:2`

Answer

To solve the problem of finding which ratio cannot represent two numbers whose sum is 84, we can follow these steps:

### Step 1: Understand the Problem
We need to find two numbers, A and B, such that A + B = 84. We will check different ratios to see if they can represent these two numbers.

### Step 2: Analyze the Ratios
We will analyze each given ratio to see if it can be represented by two numbers that add up to 84.

### Step 3: Check Each Ratio

1. **First Ratio: 5:7**
- The sum of the parts of the ratio is 5 + 7 = 12.
- To find the actual numbers, we can divide the total sum (84) by the sum of the ratio parts (12):
\[ \text{Value of one part} = \frac{84}{12} = 7 \]
- Therefore, the two numbers are:
\[ A = 5 \times 7 = 35 \quad \text{and} \quad B = 7 \times 7 = 49 \]
- Since 35 + 49 = 84, this ratio is valid.

2. **Second Ratio: 13:8**
- The sum of the parts of the ratio is 13 + 8 = 21.
- To find the actual numbers, divide the total sum (84) by the sum of the ratio parts (21):
\[ \text{Value of one part} = \frac{84}{21} = 4 \]
- Therefore, the two numbers are:
\[ A = 13 \times 4 = 52 \quad \text{and} \quad B = 8 \times 4 = 32 \]
- Since 52 + 32 = 84, this ratio is valid.

3. **Third Ratio: 1:3**
- The sum of the parts of the ratio is 1 + 3 = 4.
- To find the actual numbers, divide the total sum (84) by the sum of the ratio parts (4):
\[ \text{Value of one part} = \frac{84}{4} = 21 \]
- Therefore, the two numbers are:
\[ A = 1 \times 21 = 21 \quad \text{and} \quad B = 3 \times 21 = 63 \]
- Since 21 + 63 = 84, this ratio is valid.

4. **Fourth Ratio: 3:2**
- The sum of the parts of the ratio is 3 + 2 = 5.
- To find the actual numbers, divide the total sum (84) by the sum of the ratio parts (5):
\[ \text{Value of one part} = \frac{84}{5} = 16.8 \]
- Therefore, the two numbers would be:
\[ A = 3 \times 16.8 = 50.4 \quad \text{and} \quad B = 2 \times 16.8 = 33.6 \]
- Since 50.4 + 33.6 = 84, but these numbers are not integers, this ratio is not valid.

### Conclusion
The ratio that cannot represent two numbers whose sum is 84 is **3:2**.

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