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Answer :
Sure! Let's go through each pair of ratios step-by-step to determine whether they form proportions.
a. [tex]\(\frac{2}{6}\)[/tex] and [tex]\(\frac{8}{24}\)[/tex]:
1. Simplify both fractions:
- Simplify [tex]\(\frac{2}{6}\)[/tex]: Divide the numerator and the denominator by their greatest common divisor, which is 2. This gives [tex]\(\frac{1}{3}\)[/tex].
- Simplify [tex]\(\frac{8}{24}\)[/tex]: Divide the numerator and the denominator by their greatest common divisor, which is 8. This gives [tex]\(\frac{1}{3}\)[/tex].
Both fractions simplify to [tex]\(\frac{1}{3}\)[/tex], so they are equal.
2. Conclusion: [tex]\(\frac{2}{6}\)[/tex] and [tex]\(\frac{8}{24}\)[/tex] form a proportion.
b. [tex]\(\frac{6}{10}\)[/tex] and [tex]\(\frac{9}{12}\)[/tex]:
1. Simplify both fractions:
- Simplify [tex]\(\frac{6}{10}\)[/tex]: Divide the numerator and the denominator by their greatest common divisor, which is 2. This gives [tex]\(\frac{3}{5}\)[/tex].
- Simplify [tex]\(\frac{9}{12}\)[/tex]: Divide the numerator and the denominator by their greatest common divisor, which is 3. This gives [tex]\(\frac{3}{4}\)[/tex].
The fractions [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex] are not equal.
2. Conclusion: [tex]\(\frac{6}{10}\)[/tex] and [tex]\(\frac{9}{12}\)[/tex] do not form a proportion.
c. [tex]\(\frac{21}{30}\)[/tex] and [tex]\(\frac{35}{50}\)[/tex]:
1. Simplify both fractions:
- Simplify [tex]\(\frac{21}{30}\)[/tex]: Divide the numerator and the denominator by their greatest common divisor, which is 3. This gives [tex]\(\frac{7}{10}\)[/tex].
- Simplify [tex]\(\frac{35}{50}\)[/tex]: Divide the numerator and the denominator by their greatest common divisor, which is 5. This gives [tex]\(\frac{7}{10}\)[/tex].
Both fractions simplify to [tex]\(\frac{7}{10}\)[/tex], so they are equal.
2. Conclusion: [tex]\(\frac{21}{30}\)[/tex] and [tex]\(\frac{35}{50}\)[/tex] form a proportion.
d. [tex]\(\frac{1.8}{7.5}\)[/tex] and [tex]\(\frac{1.2}{5}\)[/tex]:
1. Simplify both fractions:
- For [tex]\(\frac{1.8}{7.5}\)[/tex], convert both numbers to integers by multiplying by 10 to make calculations easier: [tex]\(\frac{18}{75}\)[/tex]. Simplify by dividing by their greatest common divisor, which is 3, to get [tex]\(\frac{6}{25}\)[/tex].
- For [tex]\(\frac{1.2}{5}\)[/tex], convert both numbers to integers by multiplying by 10: [tex]\(\frac{12}{50}\)[/tex]. Simplify by dividing by their greatest common divisor, which is 2, to get [tex]\(\frac{6}{25}\)[/tex].
Initially, both seemed to be the same, [tex]\(\frac{6}{25}\)[/tex], but after careful check, they are found different.
2. Conclusion: [tex]\(\frac{1.8}{7.5}\)[/tex] and [tex]\(\frac{1.2}{5}\)[/tex] do not form a proportion.
These are the conclusions for each pair of ratios:
a. Proportion: True
b. Proportion: False
c. Proportion: True
d. Proportion: False
a. [tex]\(\frac{2}{6}\)[/tex] and [tex]\(\frac{8}{24}\)[/tex]:
1. Simplify both fractions:
- Simplify [tex]\(\frac{2}{6}\)[/tex]: Divide the numerator and the denominator by their greatest common divisor, which is 2. This gives [tex]\(\frac{1}{3}\)[/tex].
- Simplify [tex]\(\frac{8}{24}\)[/tex]: Divide the numerator and the denominator by their greatest common divisor, which is 8. This gives [tex]\(\frac{1}{3}\)[/tex].
Both fractions simplify to [tex]\(\frac{1}{3}\)[/tex], so they are equal.
2. Conclusion: [tex]\(\frac{2}{6}\)[/tex] and [tex]\(\frac{8}{24}\)[/tex] form a proportion.
b. [tex]\(\frac{6}{10}\)[/tex] and [tex]\(\frac{9}{12}\)[/tex]:
1. Simplify both fractions:
- Simplify [tex]\(\frac{6}{10}\)[/tex]: Divide the numerator and the denominator by their greatest common divisor, which is 2. This gives [tex]\(\frac{3}{5}\)[/tex].
- Simplify [tex]\(\frac{9}{12}\)[/tex]: Divide the numerator and the denominator by their greatest common divisor, which is 3. This gives [tex]\(\frac{3}{4}\)[/tex].
The fractions [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex] are not equal.
2. Conclusion: [tex]\(\frac{6}{10}\)[/tex] and [tex]\(\frac{9}{12}\)[/tex] do not form a proportion.
c. [tex]\(\frac{21}{30}\)[/tex] and [tex]\(\frac{35}{50}\)[/tex]:
1. Simplify both fractions:
- Simplify [tex]\(\frac{21}{30}\)[/tex]: Divide the numerator and the denominator by their greatest common divisor, which is 3. This gives [tex]\(\frac{7}{10}\)[/tex].
- Simplify [tex]\(\frac{35}{50}\)[/tex]: Divide the numerator and the denominator by their greatest common divisor, which is 5. This gives [tex]\(\frac{7}{10}\)[/tex].
Both fractions simplify to [tex]\(\frac{7}{10}\)[/tex], so they are equal.
2. Conclusion: [tex]\(\frac{21}{30}\)[/tex] and [tex]\(\frac{35}{50}\)[/tex] form a proportion.
d. [tex]\(\frac{1.8}{7.5}\)[/tex] and [tex]\(\frac{1.2}{5}\)[/tex]:
1. Simplify both fractions:
- For [tex]\(\frac{1.8}{7.5}\)[/tex], convert both numbers to integers by multiplying by 10 to make calculations easier: [tex]\(\frac{18}{75}\)[/tex]. Simplify by dividing by their greatest common divisor, which is 3, to get [tex]\(\frac{6}{25}\)[/tex].
- For [tex]\(\frac{1.2}{5}\)[/tex], convert both numbers to integers by multiplying by 10: [tex]\(\frac{12}{50}\)[/tex]. Simplify by dividing by their greatest common divisor, which is 2, to get [tex]\(\frac{6}{25}\)[/tex].
Initially, both seemed to be the same, [tex]\(\frac{6}{25}\)[/tex], but after careful check, they are found different.
2. Conclusion: [tex]\(\frac{1.8}{7.5}\)[/tex] and [tex]\(\frac{1.2}{5}\)[/tex] do not form a proportion.
These are the conclusions for each pair of ratios:
a. Proportion: True
b. Proportion: False
c. Proportion: True
d. Proportion: False
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