To determine whether each pair of ratios forms a proportion, we need to check if the two ratios are equivalent. Two ratios \( \frac{a}{b} \) and \( \frac{c}{d} \) form a proportion if:
\[
\frac{a}{b} = \frac{c}{d}
\]
This is equivalent to checking if the cross-products are equal:
\[
a \cdot d = b \cdot c
\]
Let's solve each pair step by step.
---
Pair 1: \( \frac{5}{15} \) and \( \frac{3}{9} \)
1. Simplify each fraction:
- \( \frac{5}{15} = \frac{1}{3} \)
- \( \frac{3}{9} = \frac{1}{3} \)
2. Compare the simplified fractions:
- Both are equal: \( \frac{1}{3} = \frac{1}{3} \)
Conclusion: These ratios form a proportion.
---
Pair 2: \( \frac{2}{3} \) and \( \frac{18}{27} \)
1. Simplify the second fraction:
- \( \frac{18}{27} = \frac{2}{3} \) (dividing numerator and denominator by 9)
2. Compare the fractions:
- Both are equal: \( \frac{2}{3} = \frac{2}{3} \)
Conclusion: These ratios form a proportion.
---
Pair 3: \( \frac{68}{7} \) and \( \frac{19}{3} \)
1. Check the cross-products:
- \( 68 \cdot 3 = 204 \)
- \( 7 \cdot 19 = 133 \)
2. Compare the cross-products:
- \( 204 \neq 133 \)
Conclusion: These ratios do not form a proportion.
---
Pair 4: \( \frac{6}{12} \) and \( \frac{2}{4} \)
1. Simplify each fraction:
- \( \frac{6}{12} = \frac{1}{2} \)
- \( \frac{2}{4} = \frac{1}{2} \)
2. Compare the simplified fractions:
- Both are equal: \( \frac{1}{2} = \frac{1}{2} \)
Conclusion: These ratios form a proportion.
---
Pair 5: \( \frac{12}{8} \) and \( \frac{4}{5} \)
1. Simplify the first fraction:
- \( \frac{12}{8} = \frac{3}{2} \)
2. Check the cross-products:
- \( 3 \cdot 5 = 15 \)
- \( 2 \cdot 4 = 8 \)
3. Compare the cross-products:
- \( 15 \neq 8 \)
Conclusion: These ratios do not form a proportion.
---
Pair 6: \( \frac{3}{27} \) and \( \frac{1}{9} \)
1. Simplify the first fraction:
- \( \frac{3}{27} = \frac{1}{9} \) (dividing numerator and denominator by 3)
2. Compare the fractions:
- Both are equal: \( \frac{1}{9} = \frac{1}{9} \)
Conclusion: These ratios form a proportion.
---
Pair 7: \( \frac{18}{4} \) and \( \frac{9}{2} \)
1. Simplify the first fraction:
- \( \frac{18}{4} = \frac{9}{2} \) (dividing numerator and denominator by 2)
2. Compare the fractions:
- Both are equal: \( \frac{9}{2} = \frac{9}{2} \)
Conclusion: These ratios form a proportion.
---
Pair 8: \( \frac{4}{5} \) and \( \frac{20}{25} \)
1. Simplify the second fraction:
- \( \frac{20}{25} = \frac{4}{5} \) (dividing numerator and denominator by 5)
2. Compare the fractions:
- Both are equal: \( \frac{4}{5} = \frac{4}{5} \)
Conclusion: These ratios form a proportion.
---
Final Answers:
\[
\boxed{
\begin{array}{ll}
1. \text{Yes} & 2. \text{Yes} \\
3. \text{No} & 4. \text{Yes} \\
5. \text{No} & 6. \text{Yes} \\
7. \text{Yes} & 8. \text{Yes} \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of ratios and proportions worksheet.
