Question

1. write >, <, or =.

a) \\(\frac{11}{7}\\) — \\(\frac{10}{9}\\)
b) \\(\frac{21}{8}\\) — \\(\frac{31}{12}\\)
c) \\(\frac{17}{7}\\) — \\(2\frac{3}{4}\\)
d) \\(1\frac{1}{2}\\) — \\(\frac{24}{16}\\)
e) \\(\frac{24}{5}\\) — \\(\frac{48}{10}\\)
f) \\(3\frac{4}{5}\\) — \\(\frac{78}{25}\\)

Explanation:

Part a)

Step1: Find a common denominator

The denominators are 7 and 9. The least common denominator (LCD) of 7 and 9 is \( 7\times9 = 63 \).

Step2: Convert fractions

For \( \frac{11}{7} \), multiply numerator and denominator by 9: \( \frac{11\times9}{7\times9}=\frac{99}{63} \).
For \( \frac{10}{9} \), multiply numerator and denominator by 7: \( \frac{10\times7}{9\times7}=\frac{70}{63} \).

Step3: Compare numerators

Since \( 99>70 \), we have \( \frac{11}{7}>\frac{10}{9} \).

Part b)

Step1: Find a common denominator

The denominators are 8 and 12. The LCD of 8 and 12 is 24.

Step2: Convert fractions

For \( \frac{21}{8} \), multiply numerator and denominator by 3: \( \frac{21\times3}{8\times3}=\frac{63}{24} \).
For \( \frac{31}{12} \), multiply numerator and denominator by 2: \( \frac{31\times2}{12\times2}=\frac{62}{24} \).

Step3: Compare numerators

Since \( 63 > 62 \), we have \( \frac{21}{8}>\frac{31}{12} \).

Part c)

Step1: Convert mixed number to improper fraction

\( 2\frac{3}{4}=\frac{2\times4 + 3}{4}=\frac{11}{4} \).

Step2: Find a common denominator

The denominators are 7 and 4. The LCD of 7 and 4 is 28.

Step3: Convert fractions

For \( \frac{17}{7} \), multiply numerator and denominator by 4: \( \frac{17\times4}{7\times4}=\frac{68}{28} \).
For \( \frac{11}{4} \), multiply numerator and denominator by 7: \( \frac{11\times7}{4\times7}=\frac{77}{28} \).

Step4: Compare numerators

Since \( 68<77 \), we have \( \frac{17}{7}<2\frac{3}{4} \).

Part d)

Step1: Convert mixed number to improper fraction

\( 1\frac{1}{2}=\frac{1\times2+1}{2}=\frac{3}{2} \).

Step2: Simplify \( \frac{24}{16} \)

\( \frac{24}{16}=\frac{3}{2} \) (divide numerator and denominator by 8).

Step3: Compare fractions

Since \( \frac{3}{2}=\frac{3}{2} \), we have \( 1\frac{1}{2}=\frac{24}{16} \).

Part e)

Step1: Simplify \( \frac{48}{10} \)

\( \frac{48}{10}=\frac{24}{5} \) (divide numerator and denominator by 2).

Step2: Compare fractions

Since \( \frac{24}{5}=\frac{24}{5} \), we have \( \frac{24}{5}=\frac{48}{10} \).

Part f)

Step1: Convert mixed number to improper fraction

\( 3\frac{4}{5}=\frac{3\times5 + 4}{5}=\frac{19}{5} \).

Step2: Find a common denominator

The denominators are 5 and 25. The LCD of 5 and 25 is 25.

Step3: Convert fraction

For \( \frac{19}{5} \), multiply numerator and denominator by 5: \( \frac{19\times5}{5\times5}=\frac{95}{25} \).

Step4: Compare numerators

Since \( 95>78 \), we have \( 3\frac{4}{5}>\frac{78}{25} \).

Answer:

a) \( \frac{11}{7}>\frac{10}{9} \)
b) \( \frac{21}{8}>\frac{31}{12} \)
c) \( \frac{17}{7}<2\frac{3}{4} \)
d) \( 1\frac{1}{2}=\frac{24}{16} \)
e) \( \frac{24}{5}=\frac{48}{10} \)
f) \( 3\frac{4}{5}>\frac{78}{25} \)