Answer :
To find which pair of fractions is not equivalent to [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{1}{10}\)[/tex], let's check each option to see if the fractions are the same when simplified to their lowest terms.
Step-by-step solution:
1. Equivalent Fractions:
- Two fractions are equivalent if they simplify to the same lowest terms.
2. Understand the Given Fractions:
- [tex]\(\frac{2}{5}\)[/tex] is already in its simplest form.
- [tex]\(\frac{1}{10}\)[/tex] is also already in its simplest form.
3. Checking Each Option:
(A) [tex]\(\frac{12}{30}\)[/tex] and [tex]\(\frac{3}{30}\)[/tex]:
- Simplify [tex]\(\frac{12}{30}\)[/tex]:
Divide both numerator and denominator by their greatest common divisor, which is 6:
[tex]\(\frac{12 \div 6}{30 \div 6} = \frac{2}{5}\)[/tex].
- Simplify [tex]\(\frac{3}{30}\)[/tex]:
Divide both numerator and denominator by 3:
[tex]\(\frac{3 \div 3}{30 \div 3} = \frac{1}{10}\)[/tex].
- This matches the original fractions.
(B) [tex]\(\frac{6}{15}\)[/tex] and [tex]\(\frac{5}{15}\)[/tex]:
- Simplify [tex]\(\frac{6}{15}\)[/tex]:
Divide both numerator and denominator by 3:
[tex]\(\frac{6 \div 3}{15 \div 3} = \frac{2}{5}\)[/tex].
- Simplify [tex]\(\frac{5}{15}\)[/tex]:
Divide both numerator and denominator by 5:
[tex]\(\frac{5 \div 5}{15 \div 5} = \frac{1}{3}\)[/tex], not [tex]\(\frac{1}{10}\)[/tex].
- One of these fractions does not match.
(C) [tex]\(\frac{20}{50}\)[/tex] and [tex]\(\frac{10}{100}\)[/tex]:
- Simplify [tex]\(\frac{20}{50}\)[/tex]:
Divide both numerator and denominator by 10:
[tex]\(\frac{20 \div 10}{50 \div 10} = \frac{2}{5}\)[/tex].
- Simplify [tex]\(\frac{10}{100}\)[/tex]:
Divide both numerator and denominator by 10:
[tex]\(\frac{10 \div 10}{100 \div 10} = \frac{1}{10}\)[/tex].
- This matches the original fractions.
(D) [tex]\(\frac{8}{20}\)[/tex] and [tex]\(\frac{2}{20}\)[/tex]:
- Simplify [tex]\(\frac{8}{20}\)[/tex]:
Divide both numerator and denominator by 4:
[tex]\(\frac{8 \div 4}{20 \div 4} = \frac{2}{5}\)[/tex].
- Simplify [tex]\(\frac{2}{20}\)[/tex]:
Divide both numerator and denominator by 2:
[tex]\(\frac{2 \div 2}{20 \div 2} = \frac{1}{10}\)[/tex].
- This matches the original fractions.
4. Conclusion:
The pair of fractions in option (B) are not equivalent to the given fractions [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{1}{10}\)[/tex]. The first fraction matches, but the second fraction simplifies to [tex]\(\frac{1}{3}\)[/tex], which does not match [tex]\(\frac{1}{10}\)[/tex]. Hence, the correct answer is B.
Step-by-step solution:
1. Equivalent Fractions:
- Two fractions are equivalent if they simplify to the same lowest terms.
2. Understand the Given Fractions:
- [tex]\(\frac{2}{5}\)[/tex] is already in its simplest form.
- [tex]\(\frac{1}{10}\)[/tex] is also already in its simplest form.
3. Checking Each Option:
(A) [tex]\(\frac{12}{30}\)[/tex] and [tex]\(\frac{3}{30}\)[/tex]:
- Simplify [tex]\(\frac{12}{30}\)[/tex]:
Divide both numerator and denominator by their greatest common divisor, which is 6:
[tex]\(\frac{12 \div 6}{30 \div 6} = \frac{2}{5}\)[/tex].
- Simplify [tex]\(\frac{3}{30}\)[/tex]:
Divide both numerator and denominator by 3:
[tex]\(\frac{3 \div 3}{30 \div 3} = \frac{1}{10}\)[/tex].
- This matches the original fractions.
(B) [tex]\(\frac{6}{15}\)[/tex] and [tex]\(\frac{5}{15}\)[/tex]:
- Simplify [tex]\(\frac{6}{15}\)[/tex]:
Divide both numerator and denominator by 3:
[tex]\(\frac{6 \div 3}{15 \div 3} = \frac{2}{5}\)[/tex].
- Simplify [tex]\(\frac{5}{15}\)[/tex]:
Divide both numerator and denominator by 5:
[tex]\(\frac{5 \div 5}{15 \div 5} = \frac{1}{3}\)[/tex], not [tex]\(\frac{1}{10}\)[/tex].
- One of these fractions does not match.
(C) [tex]\(\frac{20}{50}\)[/tex] and [tex]\(\frac{10}{100}\)[/tex]:
- Simplify [tex]\(\frac{20}{50}\)[/tex]:
Divide both numerator and denominator by 10:
[tex]\(\frac{20 \div 10}{50 \div 10} = \frac{2}{5}\)[/tex].
- Simplify [tex]\(\frac{10}{100}\)[/tex]:
Divide both numerator and denominator by 10:
[tex]\(\frac{10 \div 10}{100 \div 10} = \frac{1}{10}\)[/tex].
- This matches the original fractions.
(D) [tex]\(\frac{8}{20}\)[/tex] and [tex]\(\frac{2}{20}\)[/tex]:
- Simplify [tex]\(\frac{8}{20}\)[/tex]:
Divide both numerator and denominator by 4:
[tex]\(\frac{8 \div 4}{20 \div 4} = \frac{2}{5}\)[/tex].
- Simplify [tex]\(\frac{2}{20}\)[/tex]:
Divide both numerator and denominator by 2:
[tex]\(\frac{2 \div 2}{20 \div 2} = \frac{1}{10}\)[/tex].
- This matches the original fractions.
4. Conclusion:
The pair of fractions in option (B) are not equivalent to the given fractions [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{1}{10}\)[/tex]. The first fraction matches, but the second fraction simplifies to [tex]\(\frac{1}{3}\)[/tex], which does not match [tex]\(\frac{1}{10}\)[/tex]. Hence, the correct answer is B.
