Question

for 1 - 16, find equivalent fractions to compare. then write >, <, or =.

1. $\frac{5}{6}$ $circ$ $\frac{2}{3}$
2. $\frac{1}{5}$ $circ$ $\frac{2}{8}$
3. $\frac{9}{10}$ $circ$ $\frac{3}{4}$
4. $\frac{3}{4}$ $circ$ $\frac{2}{8}$
5. $\frac{7}{8}$ $circ$ $\frac{1}{2}$
6. $\frac{2}{5}$ $circ$ $\frac{2}{6}$
7. $\frac{1}{3}$ $circ$ $\frac{3}{8}$
8. $\frac{2}{10}$ $circ$ $\frac{3}{5}$
9. $\frac{8}{10}$ $circ$ $\frac{3}{4}$
10. $\frac{3}{8}$ $circ$ $\frac{9}{12}$
11. $\frac{2}{3}$ $circ$ $\frac{10}{12}$
12. $\frac{7}{8}$ $circ$ $\frac{3}{4}$
13. $\frac{3}{4}$ $circ$ $\frac{7}{8}$
14. $\frac{2}{4}$ $circ$ $\frac{4}{8}$
15. $\frac{6}{8}$ $circ$ $\frac{8}{12}$
16. $\frac{1}{3}$ $circ$ $\frac{4}{8}$

Answer:

Let's solve these fraction comparison problems one by one. We'll use the method of finding equivalent fractions with a common denominator to compare them.

Problem 1: $\boldsymbol{\frac{5}{6} \circ \frac{2}{3}}$

Step 1: Find a common denominator

The denominators are 6 and 3. The least common denominator (LCD) is 6.

Step 2: Convert $\frac{2}{3}$ to an equivalent fraction with denominator 6

To convert $\frac{2}{3}$ to a fraction with denominator 6, we multiply the numerator and denominator by 2: $\frac{2 \times 2}{3 \times 2} = \frac{4}{6}$

Step 3: Compare the numerators

Now we compare $\frac{5}{6}$ and $\frac{4}{6}$. Since $5 > 4$, we have $\frac{5}{6} > \frac{4}{6}$, so $\frac{5}{6} > \frac{2}{3}$.

Problem 2: $\boldsymbol{\frac{1}{5} \circ \frac{2}{8}}$

Step 1: Simplify $\frac{2}{8}$

$\frac{2}{8} = \frac{1}{4}$ (dividing numerator and denominator by 2)

Step 2: Find a common denominator for $\frac{1}{5}$ and $\frac{1}{4}$

The LCD of 5 and 4 is 20.

Step 3: Convert both fractions to equivalent fractions with denominator 20

• $\frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20}$
• $\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$

Step 4: Compare the numerators

Since $4 < 5$, we have $\frac{4}{20} < \frac{5}{20}$, so $\frac{1}{5} < \frac{2}{8}$.

Problem 3: $\boldsymbol{\frac{9}{10} \circ \frac{3}{4}}$

Step 1: Find the LCD of 10 and 4

The prime factors of 10 are $2 \times 5$ and of 4 are $2 \times 2$. So the LCD is $2 \times 2 \times 5 = 20$.

Step 2: Convert both fractions to equivalent fractions with denominator 20

• $\frac{9}{10} = \frac{9 \times 2}{10 \times 2} = \frac{18}{20}$
• $\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$

Step 3: Compare the numerators

Since $18 > 15$, we have $\frac{18}{20} > \frac{15}{20}$, so $\frac{9}{10} > \frac{3}{4}$.

Problem 4: $\boldsymbol{\frac{3}{4} \circ \frac{2}{8}}$

Step 1: Simplify $\frac{2}{8}$

$\frac{2}{8} = \frac{1}{4}$ (dividing numerator and denominator by 2)

Step 2: Compare $\frac{3}{4}$ and $\frac{1}{4}$

Since the denominators are the same and $3 > 1$, we have $\frac{3}{4} > \frac{1}{4}$, so $\frac{3}{4} > \frac{2}{8}$.

Problem 5: $\boldsymbol{\frac{7}{8} \circ \frac{1}{2}}$

Step 1: Find the LCD of 8 and 2

The LCD is 8.

Step 2: Convert $\frac{1}{2}$ to an equivalent fraction with denominator 8

$\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$

Step 3: Compare the numerators

Since $7 > 4$, we have $\frac{7}{8} > \frac{4}{8}$, so $\frac{7}{8} > \frac{1}{2}$.

Problem 6: $\boldsymbol{\frac{2}{5} \circ \frac{2}{6}}$

Step 1: Analyze the fractions

Both fractions have the same numerator (2). When the numerators are the same, the fraction with the smaller denominator is larger.

Step 2: Compare the denominators

Since $5 < 6$, we have $\frac{2}{5} > \frac{2}{6}$.

Problem 7: $\boldsymbol{\frac{1}{3} \circ \frac{3}{8}}$

Step 1: Find the LCD of 3 and 8

The LCD is 24.

Step 2: Convert both fractions to equivalent fractions with denominator 24

• $\frac{1}{3} = \frac{1 \times 8}{3 \times 8} = \frac{8}{24}$
• $\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$

Step 3: Compare the numerators

Since $8 < 9$, we have $\frac{8}{24} < \frac{9}{24}$, so $\frac{1}{3} < \frac{3}{8}$.

Problem 8: $\boldsymbol{\frac{2}{10} \circ \frac{3}{5}}$

Step 1: Simplify $\frac{2}{10}$

$\frac{2}{10} = \frac{1}{5}$ (dividing numerator and denominator by 2)

Step 2: Compare $\frac{1}{5}$ and $\frac{3}{5}$

Since the denominators are the same and $1 < 3$, we have $\frac{1}{5} < \frac{3}{5}$, so $\frac{2}{10} < \frac{3}{5}$.

Problem 9: $\boldsymbol{\frac{8}{10} \circ \frac{3}{4}}$

Step 1: Simplify $\frac{8}{10}$

$\frac{8}{10} = \frac{4}{5}$ (dividing numerator and denominator by 2)

Step 2: Find the LCD of 5 and 4

The LCD is 20.

Step 3: Convert both fractions to equivalent fractions with denominator 20

• $\frac{4}{5} = \frac{4 \times 4}{5 \times 4} = \frac{16}{20}$
• $\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$

Step 4: Compare the numerators

Since $16 > 15$, we have $\frac{16}{20} > \frac{15}{20}$, so $\frac{8}{10} > \frac{3}{4}$.

Problem 10: $\boldsymbol{\frac{3}{8} \circ \frac{9}{12}}$

Step 1: Simplify $\frac{9}{12}$

$\frac{9}{12} = \frac{3}{4}$ (dividing numerator and denominator by 3)

Step 2: Find the LCD of 8 and 4

The LCD is 8.

Step 3: Convert $\frac{3}{4}$ to an equivalent fraction with denominator 8

$\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$

Step 4: Compare the numerators

Since $3 < 6$, we have $\frac{3}{8} < \frac{6}{8}$, so $\frac{3}{8} < \frac{9}{12}$.

Problem 11: $\boldsymbol{\frac{2}{3} \circ \frac{10}{12}}$

Step 1: Simplify $\frac{10}{12}$

$\frac{10}{12} = \frac{5}{6}$ (dividing numerator and denominator by 2)

Step 2: Find the LCD of 3 and 6

The LCD is 6.

Step 3: Convert $\frac{2}{3}$ to an equivalent fraction with denominator 6

$\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$

Step 4: Compare the numerators

Since $4 < 5$, we have $\frac{4}{6} < \frac{5}{6}$, so $\frac{2}{3} < \frac{10}{12}$.

Problem 12: $\boldsymbol{\frac{7}{8} \circ \frac{3}{4}}$

Step 1: Find the LCD of 8 and 4

The LCD is 8.

Step 2: Convert $\frac{3}{4}$ to an equivalent fraction with denominator 8

$\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$

Step 3: Compare the numerators

Since $7 > 6$, we have $\frac{7}{8} > \frac{6}{8}$, so $\frac{7}{8} > \frac{3}{4}$.

Problem 13: $\boldsymbol{\frac{3}{4} \circ \frac{7}{8}}$

Step 1: Find the LCD of 4 and 8

The LCD is 8.

Step 2: Convert $\frac{3}{4}$ to an equivalent fraction with denominator 8

$\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$

Step 3: Compare the numerators

Since $6 < 7$, we have $\frac{6}{8} < \frac{7}{8}$, so $\frac{3}{4} < \frac{7}{8}$.

Problem 14: $\boldsymbol{\frac{2}{4} \circ \frac{4}{8}}$

Step 1: Simplify both fractions

• $\frac{2}{4} = \frac{1}{2}$ (dividing numerator and denominator by 2)
• $\frac{4}{8} = \frac{1}{2}$ (dividing numerator and denominator by 4)

Step 2: Compare the fractions

Since both simplify to $\frac{1}{2}$, we have $\frac{2}{4} = \frac{4}{8}$.

Problem 15: $\boldsymbol{\frac{6}{8} \circ \frac{8}{12}}$

Step 1: Simplify both fractions

• $\frac{6}{8} = \frac{3}{4}$ (dividing numerator and denominator by 2)
• $\frac{8}{12} = \frac{2}{3}$ (dividing numerator and denominator by 4)

Step 2: Find the LCD of 4 and 3

The LCD is 12.

Step 3: Convert both fractions to equivalent fractions with denominator 12

• $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
• $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$

Step 4: Compare the numerators

Since $9 > 8$, we have $\frac{9}{12} > \frac{8}{12}$, so $\frac{6}{8} > \frac{8}{12}$.

Problem 16: $\boldsymbol{\frac{1}{3} \circ \frac{4}{8}}$

Step 1: Simplify $\frac{4}{8}$

$\frac{4}{8} = \frac{1}{2}$ (dividing numerator and denominator by 4)

Step 2: Find the LCD of 3 and 2

The LCD is 6.

Step 3: Convert both fractions to equivalent fractions with denominator 6

• $\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$
• $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

Step 4: Compare the numerators

Since $2 < 3$, we have $\frac{2}{6} < \frac{3}{6}$, so $\frac{1}{3} < \frac{4}{8}$.

Final Answers:

1. $\frac{5}{6} \boldsymbol{>} \frac{2}{3}$
2. $\frac{1}{5} \boldsymbol{<} \frac{2}{8}$
3. $\frac{9}{10} \boldsymbol{>} \frac{3}{4}$
4. $\frac{3}{4} \boldsymbol{>} \frac{2}{8}$
5. $\frac{7}{8} \boldsymbol{>} \frac{1}{2}$
6. $\frac{2}{5} \boldsymbol{>} \frac{2}{6}$
7. $\frac{1}{3} \boldsymbol{<} \frac{3}{8}$
8. $\frac{2}{10} \boldsymbol{<} \frac{3}{5}$
9. $\frac{8}{10} \boldsymbol{>} \frac{3}{4}$
10. $\frac{3}{8} \boldsymbol{<} \frac{9}{12}$
11. $\frac{2}{3} \boldsymbol{<} \frac{10}{12}$
12. $\frac{7}{8} \boldsymbol{>} \frac{3}{4}$
13. $\frac{3}{4} \boldsymbol{<} \frac{7}{8}$
14. $\frac{2}{4} \boldsymbol{=} \frac{4}{8}$
15. $\frac{6}{8} \boldsymbol{>} \frac{8}{12}$
16. $\frac{1}{3} \boldsymbol{<} \frac{4}{8}$