Radhika purchased one dozen bangles. One day she slipped on the floor fell down. What cannot be the ratio of broken to unbroken bangles :

Options

A. 1:2
B. 2:3
C. 1:3
D. 5:1

Answer

To solve the problem, we need to determine what cannot be the ratio of broken to unbroken bangles after Radhika's fall.

1. **Understanding the Total Bangles**:
Radhika purchased one dozen bangles, which means she has a total of 12 bangles.

2. **Defining Broken and Unbroken Bangles**:
Let’s denote the number of broken bangles as \( B \) and the number of unbroken bangles as \( U \). Since there are a total of 12 bangles, we can express this relationship as:
\[ B + U = 12 \]

3. **Exploring Possible Ratios**:
The ratio of broken to unbroken bangles can be expressed as:
\[ \text{Ratio} = \frac{B}{U} \]
To find valid ratios, we need to consider different possible values for \( B \) and \( U \) that satisfy the equation \( B + U = 12 \).

4. **Calculating Different Scenarios**:
- If \( B = 0 \), then \( U = 12 \) → Ratio = \( 0:12 \) or \( 0:1 \)
- If \( B = 1 \), then \( U = 11 \) → Ratio = \( 1:11 \)
- If \( B = 2 \), then \( U = 10 \) → Ratio = \( 2:10 \) or \( 1:5 \)
- If \( B = 3 \), then \( U = 9 \) → Ratio = \( 3:9 \) or \( 1:3 \)
- If \( B = 4 \), then \( U = 8 \) → Ratio = \( 4:8 \) or \( 1:2 \)
- If \( B = 5 \), then \( U = 7 \) → Ratio = \( 5:7 \)
- If \( B = 6 \), then \( U = 6 \) → Ratio = \( 6:6 \) or \( 1:1 \)
- If \( B = 7 \), then \( U = 5 \) → Ratio = \( 7:5 \)
- If \( B = 8 \), then \( U = 4 \) → Ratio = \( 8:4 \) or \( 2:1 \)
- If \( B = 9 \), then \( U = 3 \) → Ratio = \( 9:3 \) or \( 3:1 \)
- If \( B = 10 \), then \( U = 2 \) → Ratio = \( 10:2 \) or \( 5:1 \)
- If \( B = 11 \), then \( U = 1 \) → Ratio = \( 11:1 \)
- If \( B = 12 \), then \( U = 0 \) → Ratio = \( 12:0 \) (undefined)

5. **Identifying Invalid Ratios**:
Now, we need to check which ratios are impossible given the total of 12 bangles. The ratio of broken to unbroken bangles must be in whole numbers.

The ratio \( 2:3 \) implies that for every 2 broken bangles, there are 3 unbroken bangles. This means a total of \( 2 + 3 = 5 \) parts. If we consider this ratio, we can calculate the total number of bangles:
\[ \text{Total parts} = 2 + 3 = 5 \implies \text{Total bangles} = 5k \text{ for some integer } k \]
For \( k = 1 \), total = 5; for \( k = 2 \), total = 10; for \( k = 3 \), total = 15. None of these give us 12 bangles.

6. **Conclusion**:
Therefore, the ratio of broken to unbroken bangles that cannot occur is \( 2:3 \).

**Final Answer**: The ratio of broken to unbroken bangles that cannot be is \( 2:3 \).