Question
1. The inch was once the length of three grains of barley placed end-to-end. Express the ratio of inches to pieces of barley using inches:barley notation. Then, express the ratio of inches to barley using two different fractions. 2. A scientist makes up a new set of units, unit A and unit B. He draws out the scale below: © Success Academy Charter Schools 4
Express the ratio of unit A to unit B as two different fractions. 3. Explain why two different fractions can represent the same ratio.
Express the ratio of unit A to unit B as two different fractions. 3. Explain why two different fractions can represent the same ratio.
Answers
GPT-4o mini
1. The inch is defined as the length of three grains of barley placed end-to-end. Therefore, the ratio of inches to pieces of barley can be expressed as follows:
- **Inches:Barley notation**: \( 1:3 \)
To express this ratio as two different fractions, we can represent the relationship mathematically. Since 1 inch corresponds to 3 pieces of barley, we can express the ratio in fraction form in two ways:
- **Fraction 1**: \( \frac{1 \text{ inch}}{3 \text{ barley}} \)
- **Fraction 2**: \( \frac{3 \text{ barley}}{1 \text{ inch}} \)
2. We can denote a ratio of units A and B based on the scale that the scientist has drawn. Suppose we assume that unit A corresponds to a length of, say, 4 units, and unit B corresponds to a length of 2 units. The ratio can be expressed as:
- **In Unit A:Unit B notation**: \( 4:2 \)
Now, we can express this ratio as two different fractions:
- **Fraction 1**: \( \frac{4 \text{ A}}{2 \text{ B}} \)
- **Fraction 2**: \( \frac{2 \text{ B}}{4 \text{ A}} \)
3. Two different fractions can represent the same ratio because a ratio describes the comparative sizes of two quantities. When you create a fraction representing a ratio, you can multiply or divide the numerator and denominator by the same non-zero number, and the value of the fraction remains unchanged. This means that \( \frac{4}{2} \) can also be written as \( \frac{2}{4} \) by manipulating the terms, reflecting the same relationship of proportions in different forms (though one fraction represents a simplified form of the ratio while the other may not). In the context of ratios, \( \frac{a}{b} = \frac{ka}{kb} \) (where \( k \neq 0 \)) indicates that \( a:b \) maintains its value regardless of the scaling factor.
- **Inches:Barley notation**: \( 1:3 \)
To express this ratio as two different fractions, we can represent the relationship mathematically. Since 1 inch corresponds to 3 pieces of barley, we can express the ratio in fraction form in two ways:
- **Fraction 1**: \( \frac{1 \text{ inch}}{3 \text{ barley}} \)
- **Fraction 2**: \( \frac{3 \text{ barley}}{1 \text{ inch}} \)
2. We can denote a ratio of units A and B based on the scale that the scientist has drawn. Suppose we assume that unit A corresponds to a length of, say, 4 units, and unit B corresponds to a length of 2 units. The ratio can be expressed as:
- **In Unit A:Unit B notation**: \( 4:2 \)
Now, we can express this ratio as two different fractions:
- **Fraction 1**: \( \frac{4 \text{ A}}{2 \text{ B}} \)
- **Fraction 2**: \( \frac{2 \text{ B}}{4 \text{ A}} \)
3. Two different fractions can represent the same ratio because a ratio describes the comparative sizes of two quantities. When you create a fraction representing a ratio, you can multiply or divide the numerator and denominator by the same non-zero number, and the value of the fraction remains unchanged. This means that \( \frac{4}{2} \) can also be written as \( \frac{2}{4} \) by manipulating the terms, reflecting the same relationship of proportions in different forms (though one fraction represents a simplified form of the ratio while the other may not). In the context of ratios, \( \frac{a}{b} = \frac{ka}{kb} \) (where \( k \neq 0 \)) indicates that \( a:b \) maintains its value regardless of the scaling factor.