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
- Divide the Circle: Draw a circle and divide it into 9 equal parts (because 2 + 7 = 9).
- 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
- Divide the Circle: Draw another circle and divide it into 12 equal parts (because 5 + 7 = 12).
- Shade the Parts: Shade 5 out of the 12 equal parts to represent the ratio 5:7.
Visual Representation:
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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.
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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:
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Ratio of shaded to total parts for 2 to 7:
- Shaded = 2, Total = 9
- Fraction = \( \frac{2}{9} \)
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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.
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