### Part (a): Comparing the ratios [tex]\( 5:9 \)[/tex] and [tex]\( y:12 \)[/tex]
1. Given Ratios: - Ratio [tex]\(1\)[/tex]: [tex]\( 5:9 \)[/tex] - Ratio [tex]\(2\)[/tex]: [tex]\( y:12 \)[/tex]
2. Objective: - We need to find the value of [tex]\( y \)[/tex] such that the two ratios are equivalent.
3. Method to Find [tex]\( y \)[/tex]: - To find the value of [tex]\( y \)[/tex] for the ratios to be equivalent, we solve the proportion: [tex]\[
\frac{5}{9} = \frac{y}{12}
\][/tex]
4. Solving for [tex]\( y \)[/tex]: - Cross multiply to get: [tex]\[
5 \times 12 = 9 \times y
\][/tex] - Simplify: [tex]\[
60 = 9y
\][/tex] - Divide both sides by 9: [tex]\[
y = \frac{60}{9} = 6.6667
\][/tex]
5. Conclusion: - The two ratios [tex]\( 5:9 \)[/tex] and [tex]\( y:12 \)[/tex] are equivalent when [tex]\( y = 6.6667 \)[/tex].
### Part (b): Comparing the ratios [tex]\( 15:32 \)[/tex] and [tex]\( 19:40 \)[/tex]
1. Given Ratios: - Ratio [tex]\(1\)[/tex]: [tex]\( 15:32 \)[/tex] - Ratio [tex]\(2\)[/tex]: [tex]\( 19:40 \)[/tex]
2. Objective: - We need to compare these two ratios to see how they relate to each other.
3. Simplify Each Ratio: - Simplify the ratios to their decimal forms to facilitate the comparison. - For [tex]\( 15:32 \)[/tex]: [tex]\[
\frac{15}{32} = 0.46875
\][/tex] - For [tex]\( 19:40 \)[/tex]: [tex]\[
\frac{19}{40} = 0.475
\][/tex]
4. Comparison of the Ratios: - Ratio [tex]\(1\)[/tex] as a decimal is [tex]\( 0.46875 \)[/tex]. - Ratio [tex]\(2\)[/tex] as a decimal is [tex]\( 0.475 \)[/tex].
5. Conclusion: - [tex]\( 15:32 \)[/tex] is slightly smaller than [tex]\( 19:40 \)[/tex]. - Therefore, the ratios are not exactly the same, but they are close in value.
