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Answer :
To compare the fractions [tex]\(\frac{14}{15}\)[/tex] and [tex]\(\frac{19}{20}\)[/tex], let's rewrite both fractions so they have a common denominator. Then we can determine which fraction is larger by comparing their numerators.
Step 1: Find the least common denominator (LCD) of 15 and 20.
- The least common denominator is the smallest number that both denominators divide into evenly.
- The prime factorization of 15 is [tex]\(3 \times 5\)[/tex].
- The prime factorization of 20 is [tex]\(2^2 \times 5\)[/tex].
- The LCD will be [tex]\(2^2 \times 3 \times 5 = 60\)[/tex].
Step 2: Rewrite each fraction with the common denominator of 60.
- To convert [tex]\(\frac{14}{15}\)[/tex] to a fraction with a denominator of 60:
[tex]\[
\frac{14}{15} = \frac{14 \times 4}{15 \times 4} = \frac{56}{60}
\][/tex]
- To convert [tex]\(\frac{19}{20}\)[/tex] to a fraction with a denominator of 60:
[tex]\[
\frac{19}{20} = \frac{19 \times 3}{20 \times 3} = \frac{57}{60}
\][/tex]
Step 3: Compare the two fractions by looking at their numerators.
- We have [tex]\(\frac{56}{60}\)[/tex] and [tex]\(\frac{57}{60}\)[/tex].
- Since [tex]\(56 < 57\)[/tex], we can determine that:
[tex]\[
\frac{56}{60} < \frac{57}{60}
\][/tex]
Therefore, [tex]\(\frac{14}{15} < \frac{19}{20}\)[/tex].
Summarizing, we have:
[tex]\[
\frac{14}{15} = \frac{56}{60}; \quad \frac{19}{20} = \frac{57}{60}
\][/tex]
And:
[tex]\[
\frac{14}{15} < \frac{19}{20}
\][/tex]
Step 1: Find the least common denominator (LCD) of 15 and 20.
- The least common denominator is the smallest number that both denominators divide into evenly.
- The prime factorization of 15 is [tex]\(3 \times 5\)[/tex].
- The prime factorization of 20 is [tex]\(2^2 \times 5\)[/tex].
- The LCD will be [tex]\(2^2 \times 3 \times 5 = 60\)[/tex].
Step 2: Rewrite each fraction with the common denominator of 60.
- To convert [tex]\(\frac{14}{15}\)[/tex] to a fraction with a denominator of 60:
[tex]\[
\frac{14}{15} = \frac{14 \times 4}{15 \times 4} = \frac{56}{60}
\][/tex]
- To convert [tex]\(\frac{19}{20}\)[/tex] to a fraction with a denominator of 60:
[tex]\[
\frac{19}{20} = \frac{19 \times 3}{20 \times 3} = \frac{57}{60}
\][/tex]
Step 3: Compare the two fractions by looking at their numerators.
- We have [tex]\(\frac{56}{60}\)[/tex] and [tex]\(\frac{57}{60}\)[/tex].
- Since [tex]\(56 < 57\)[/tex], we can determine that:
[tex]\[
\frac{56}{60} < \frac{57}{60}
\][/tex]
Therefore, [tex]\(\frac{14}{15} < \frac{19}{20}\)[/tex].
Summarizing, we have:
[tex]\[
\frac{14}{15} = \frac{56}{60}; \quad \frac{19}{20} = \frac{57}{60}
\][/tex]
And:
[tex]\[
\frac{14}{15} < \frac{19}{20}
\][/tex]
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