Which ratios are in the simplest from ?
a . `1:5`
b. `9:15`
c. `5:13`
d. `16:4`

Answer

To determine which ratios are in simplest form, we need to check each ratio to see if the greatest common factor (GCF) of the two numbers in the ratio is 1. If the GCF is 1, the ratio is in simplest form. Let’s evaluate each option step by step.

### Step 1: Evaluate the ratio `1:5`
- **Factors of 1**: The only factor is 1.
- **Factors of 5**: The factors are 1 and 5.
- **Common factors**: The only common factor is 1.
- **Conclusion**: Since the common factor is 1, the ratio `1:5` is in simplest form.

### Step 2: Evaluate the ratio `9:15`
- **Factors of 9**: The factors are 1, 3, and 9.
- **Factors of 15**: The factors are 1, 3, 5, and 15.
- **Common factors**: The common factors are 1 and 3.
- **Conclusion**: Since the common factor is not 1 (it is 3), the ratio `9:15` is not in simplest form.

### Step 3: Evaluate the ratio `5:13`
- **Factors of 5**: The only factors are 1 and 5.
- **Factors of 13**: The only factors are 1 and 13.
- **Common factors**: The only common factor is 1.
- **Conclusion**: Since the common factor is 1, the ratio `5:13` is in simplest form.

### Step 4: Evaluate the ratio `16:4`
- **Factors of 16**: The factors are 1, 2, 4, 8, and 16.
- **Factors of 4**: The factors are 1, 2, and 4.
- **Common factors**: The common factors are 1, 2, and 4.
- **Conclusion**: Since the common factor is not 1 (it is 2 and 4), the ratio `16:4` is not in simplest form.

### Summary of Results:
- `1:5` is in simplest form.
- `9:15` is not in simplest form.
- `5:13` is in simplest form.
- `16:4` is not in simplest form.

### Final Answer:
The ratios that are in simplest form are:
- a. `1:5`
- c. `5:13`