Question

Compare the ratios 2 to 7 and 5:7 using the circles to model.

2 circles.

Using circles to model the ratios, divide both circles into
equal parts.
For the ratio 2 to 7,
parts should be shaded.
For the ratio 5:7,
parts should be shaded.
The ratio 2 to 7 is
the ratio 5:7.

Answer 1

To compare the ratios 2 to 7 and 5:7 using circles, we will model each ratio with circles divided into equal parts.

### Step 1: Model the Ratio 2 to 7

1. **Divide the Circle**: Draw a circle and divide it into 9 equal parts (because 2 + 7 = 9).
2. **Shade the Parts**: Shade 2 out of the 9 equal parts to represent the ratio 2 to 7.

### Step 2: Model the Ratio 5:7

1. **Divide the Circle**: Draw another circle and divide it into 12 equal parts (because 5 + 7 = 12).
2. **Shade the Parts**: Shade 5 out of the 12 equal parts to represent the ratio 5:7.

### Visual Representation:

- **For the ratio 2 to 7**:
- 1st Circle: Shaded 2 parts out of 9
- Visual will look like 2 shaded sections and 7 unshaded sections in a total of 9.

- **For the ratio 5:7**:
- 2nd Circle: Shaded 5 parts out of 12
- Visual will look like 5 shaded sections and 7 unshaded sections in a total of 12.

### Comparing the Ratios:

- **Ratio of shaded to total parts for 2 to 7**:
- Shaded = 2, Total = 9
- Fraction = \( \frac{2}{9} \)

- **Ratio of shaded to total parts for 5:7**:
- Shaded = 5, Total = 12
- Fraction = \( \frac{5}{12} \)

### Conclusion:

- Both ratios can be expressed as fractions, and thus we can compare \( \frac{2}{9} \) and \( \frac{5}{12} \).
- To determine which one is greater, you can convert these to have a common denominator, or you can compare by cross-multiplying.

**Cross-Multiplying to Compare**:
- \( 2 \times 12 = 24 \)
- \( 5 \times 9 = 45 \)

Since 24 < 45, it follows that:

- The ratio 2 to 7 is **less than** the ratio 5:7.

Thus, we conclude that **the ratio 2 to 7 is less than the ratio 5:7.**