Sure! Let's go through each part step-by-step.
### 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.
