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Sagot :
Let's solve each part of the question step-by-step:
### 4. [tex]\(2 \frac{1}{2} + \frac{2}{7} = 2 \frac{3}{a}\)[/tex]
First, convert the mixed number to an improper fraction:
[tex]\[ 2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \][/tex]
Then add the fractions:
[tex]\[ \frac{5}{2} + \frac{2}{7} \][/tex]
To add these fractions, find a common denominator (which is 14 in this case):
[tex]\[ \frac{5}{2} = \frac{5 \times 7}{2 \times 7} = \frac{35}{14} \][/tex]
[tex]\[ \frac{2}{7} = \frac{2 \times 2}{7 \times 2} = \frac{4}{14} \][/tex]
Now add the fractions:
[tex]\[ \frac{35}{14} + \frac{4}{14} = \frac{39}{14} \][/tex]
Expressing it as a mixed number:
[tex]\[ 2 \frac{3}{a} = \frac{39}{14} \][/tex]
[tex]\[ \frac{39}{14} = 2 + \frac{11}{14} \][/tex]
Thus:
[tex]\[ \frac{3}{a} = \frac{11}{14} \][/tex]
By solving for [tex]\( a \)[/tex]:
[tex]\[ a = \frac{3 \times 14}{11} = 3.818181818181819 \][/tex]
Result for 4. is:
[tex]\[ a = 3.818181818181819 \][/tex]
### 5. [tex]\(2 \frac{1}{3} - \frac{5}{12} \)[/tex]
First, convert the mixed number to an improper fraction:
[tex]\[ 2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \][/tex]
Then subtract the fractions:
[tex]\[ \frac{7}{3} - \frac{5}{12} \][/tex]
Find a common denominator (which is 12 in this case):
[tex]\[ \frac{7}{3} = \frac{7 \times 4}{3 \times 4} = \frac{28}{12} \][/tex]
Now subtract the fractions:
[tex]\[ \frac{28}{12} - \frac{5}{12} = \frac{23}{12} \][/tex]
Expressing it as a mixed number:
[tex]\[ \frac{23}{12} = 1 \frac{11}{12} \][/tex]
Result for 5. is:
[tex]\[ \frac{23}{12} = 1.9166666666666667 \][/tex]
### 6. [tex]\( \frac{5}{11} \div \frac{1}{2} \)[/tex]
Dividing fractions involves multiplying by the reciprocal:
[tex]\[ \frac{5}{11} \div \frac{1}{2} = \frac{5}{11} \times \frac{2}{1} = \frac{5 \times 2}{11 \times 1} = \frac{10}{11} \][/tex]
Result for 6. is:
[tex]\[ \frac{10}{11} = 0.9090909090909091 \][/tex]
### 7. [tex]\( 4 \frac{7}{12} - \frac{3}{4} \)[/tex]
First, convert the mixed number to an improper fraction:
[tex]\[ 4 \frac{7}{12} = 4 + \frac{7}{12} = \frac{48}{12} + \frac{7}{12} = \frac{55}{12} \][/tex]
Then subtract the fractions:
[tex]\[ \frac{55}{12} - \frac{3}{4} \][/tex]
Find a common denominator (which is 12 in this case):
[tex]\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \][/tex]
Now subtract the fractions:
[tex]\[ \frac{55}{12} - \frac{9}{12} = \frac{46}{12} \][/tex]
Expressing it as a mixed number:
[tex]\[ \frac{46}{12} = 3 \frac{10}{12} = 3 \frac{5}{6} \][/tex]
Result for 7. is:
[tex]\[ 3 \frac{10}{12} = 3.833333333333333 \][/tex]
### 8. [tex]\( 5 \frac{1}{4} - 1 \frac{7}{8} \)[/tex]
First, convert the mixed numbers to improper fractions:
[tex]\[ 5 \frac{1}{4} = 5 + \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4} \][/tex]
[tex]\[ 1 \frac{7}{8} = 1 + \frac{7}{8} = \frac{8}{8} + \frac{7}{8} = \frac{15}{8} \][/tex]
Then subtract the fractions:
[tex]\[ \frac{21}{4} - \frac{15}{8} \][/tex]
Find a common denominator (which is 8 in this case):
[tex]\[ \frac{21}{4} = \frac{21 \times 2}{4 \times 2} = \frac{42}{8} \][/tex]
Now subtract the fractions:
[tex]\[ \frac{42}{8} - \frac{15}{8} = \frac{27}{8} \][/tex]
Expressing it as a mixed number:
[tex]\[ \frac{27}{8} = 3 \frac{3}{8} \][/tex]
Result for 8. is:
[tex]\[ 3 \frac{3}{8} = 3.375 \][/tex]
### 9. [tex]\( \frac{9}{13} - \frac{3}{7} \)[/tex]
First, subtract the fractions by finding a common denominator (which is 91 in this case):
[tex]\[ \frac{9}{13} = \frac{9 \times 7}{13 \times 7} = \frac{63}{91} \][/tex]
[tex]\[ \frac{3}{7} = \frac{3 \times 13}{7 \times 13} = \frac{39}{91} \][/tex]
Now subtract the fractions:
[tex]\[ \frac{63}{91} - \frac{39}{91} = \frac{24}{91} \][/tex]
Result for 9. is:
[tex]\[ \frac{24}{91} = 0.26373626373626374 \][/tex]
### Comparison of Fractions:
10.
(No comparison provided)
11. [tex]\( \frac{3}{4} \)[/tex] and [tex]\( \frac{6}{8} \)[/tex]
[tex]\[ \frac{3}{4} = \frac{6}{8} \][/tex]
Result for 11. is:
[tex]\[ = \][/tex]
12. [tex]\( \frac{4}{6} \bigcirc \frac{1}{3} \)[/tex]
[tex]\[ \frac{4}{6} = \frac{2}{3} \][/tex]
[tex]\[ \frac{2}{3} > \frac{1}{3} \][/tex]
Result for 12. is:
[tex]\[ > \][/tex]
13. [tex]\( \frac{5}{9} \bigcirc \frac{5}{8} \)[/tex]
The denominators tell us that [tex]\( \frac{5}{9} < \frac{5}{8} \)[/tex]
Result for 13. is:
[tex]\[ < \][/tex]
14.
(No second fraction provided)
15. [tex]\( \frac{9}{9} \bigcirc \frac{8}{8} \)[/tex]
[tex]\[ \frac{9}{9} = 1 \][/tex]
[tex]\[ \frac{8}{8} = 1 \][/tex]
[tex]\[ 1 = 1 \][/tex]
Result for 15. is:
[tex]\[ = \][/tex]
16. [tex]\( \frac{1}{10} \bigcirc \frac{1}{5} \)[/tex]
[tex]\[ \frac{1}{10} < \frac{1}{5} \][/tex]
Result for 16. is:
[tex]\[ < \][/tex]
17. [tex]\( \frac{14}{20} \bigcirc \frac{9}{10} \)[/tex]
[tex]\[ \frac{14}{20} = \frac{7}{10} \][/tex]
[tex]\[ \frac{7}{10} < \frac{9}{10} \][/tex]
Result for 17. is:
[tex]\[ < \][/tex]
18. [tex]\( \frac{6}{12} \bigcirc \frac{1}{2} \)[/tex]
[tex]\[ \frac{6}{12} = \frac{1}{2} \][/tex]
Result for 18. is:
[tex]\[ = \][/tex]
These are the solutions and comparisons for the question.
### 4. [tex]\(2 \frac{1}{2} + \frac{2}{7} = 2 \frac{3}{a}\)[/tex]
First, convert the mixed number to an improper fraction:
[tex]\[ 2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \][/tex]
Then add the fractions:
[tex]\[ \frac{5}{2} + \frac{2}{7} \][/tex]
To add these fractions, find a common denominator (which is 14 in this case):
[tex]\[ \frac{5}{2} = \frac{5 \times 7}{2 \times 7} = \frac{35}{14} \][/tex]
[tex]\[ \frac{2}{7} = \frac{2 \times 2}{7 \times 2} = \frac{4}{14} \][/tex]
Now add the fractions:
[tex]\[ \frac{35}{14} + \frac{4}{14} = \frac{39}{14} \][/tex]
Expressing it as a mixed number:
[tex]\[ 2 \frac{3}{a} = \frac{39}{14} \][/tex]
[tex]\[ \frac{39}{14} = 2 + \frac{11}{14} \][/tex]
Thus:
[tex]\[ \frac{3}{a} = \frac{11}{14} \][/tex]
By solving for [tex]\( a \)[/tex]:
[tex]\[ a = \frac{3 \times 14}{11} = 3.818181818181819 \][/tex]
Result for 4. is:
[tex]\[ a = 3.818181818181819 \][/tex]
### 5. [tex]\(2 \frac{1}{3} - \frac{5}{12} \)[/tex]
First, convert the mixed number to an improper fraction:
[tex]\[ 2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \][/tex]
Then subtract the fractions:
[tex]\[ \frac{7}{3} - \frac{5}{12} \][/tex]
Find a common denominator (which is 12 in this case):
[tex]\[ \frac{7}{3} = \frac{7 \times 4}{3 \times 4} = \frac{28}{12} \][/tex]
Now subtract the fractions:
[tex]\[ \frac{28}{12} - \frac{5}{12} = \frac{23}{12} \][/tex]
Expressing it as a mixed number:
[tex]\[ \frac{23}{12} = 1 \frac{11}{12} \][/tex]
Result for 5. is:
[tex]\[ \frac{23}{12} = 1.9166666666666667 \][/tex]
### 6. [tex]\( \frac{5}{11} \div \frac{1}{2} \)[/tex]
Dividing fractions involves multiplying by the reciprocal:
[tex]\[ \frac{5}{11} \div \frac{1}{2} = \frac{5}{11} \times \frac{2}{1} = \frac{5 \times 2}{11 \times 1} = \frac{10}{11} \][/tex]
Result for 6. is:
[tex]\[ \frac{10}{11} = 0.9090909090909091 \][/tex]
### 7. [tex]\( 4 \frac{7}{12} - \frac{3}{4} \)[/tex]
First, convert the mixed number to an improper fraction:
[tex]\[ 4 \frac{7}{12} = 4 + \frac{7}{12} = \frac{48}{12} + \frac{7}{12} = \frac{55}{12} \][/tex]
Then subtract the fractions:
[tex]\[ \frac{55}{12} - \frac{3}{4} \][/tex]
Find a common denominator (which is 12 in this case):
[tex]\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \][/tex]
Now subtract the fractions:
[tex]\[ \frac{55}{12} - \frac{9}{12} = \frac{46}{12} \][/tex]
Expressing it as a mixed number:
[tex]\[ \frac{46}{12} = 3 \frac{10}{12} = 3 \frac{5}{6} \][/tex]
Result for 7. is:
[tex]\[ 3 \frac{10}{12} = 3.833333333333333 \][/tex]
### 8. [tex]\( 5 \frac{1}{4} - 1 \frac{7}{8} \)[/tex]
First, convert the mixed numbers to improper fractions:
[tex]\[ 5 \frac{1}{4} = 5 + \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4} \][/tex]
[tex]\[ 1 \frac{7}{8} = 1 + \frac{7}{8} = \frac{8}{8} + \frac{7}{8} = \frac{15}{8} \][/tex]
Then subtract the fractions:
[tex]\[ \frac{21}{4} - \frac{15}{8} \][/tex]
Find a common denominator (which is 8 in this case):
[tex]\[ \frac{21}{4} = \frac{21 \times 2}{4 \times 2} = \frac{42}{8} \][/tex]
Now subtract the fractions:
[tex]\[ \frac{42}{8} - \frac{15}{8} = \frac{27}{8} \][/tex]
Expressing it as a mixed number:
[tex]\[ \frac{27}{8} = 3 \frac{3}{8} \][/tex]
Result for 8. is:
[tex]\[ 3 \frac{3}{8} = 3.375 \][/tex]
### 9. [tex]\( \frac{9}{13} - \frac{3}{7} \)[/tex]
First, subtract the fractions by finding a common denominator (which is 91 in this case):
[tex]\[ \frac{9}{13} = \frac{9 \times 7}{13 \times 7} = \frac{63}{91} \][/tex]
[tex]\[ \frac{3}{7} = \frac{3 \times 13}{7 \times 13} = \frac{39}{91} \][/tex]
Now subtract the fractions:
[tex]\[ \frac{63}{91} - \frac{39}{91} = \frac{24}{91} \][/tex]
Result for 9. is:
[tex]\[ \frac{24}{91} = 0.26373626373626374 \][/tex]
### Comparison of Fractions:
10.
(No comparison provided)
11. [tex]\( \frac{3}{4} \)[/tex] and [tex]\( \frac{6}{8} \)[/tex]
[tex]\[ \frac{3}{4} = \frac{6}{8} \][/tex]
Result for 11. is:
[tex]\[ = \][/tex]
12. [tex]\( \frac{4}{6} \bigcirc \frac{1}{3} \)[/tex]
[tex]\[ \frac{4}{6} = \frac{2}{3} \][/tex]
[tex]\[ \frac{2}{3} > \frac{1}{3} \][/tex]
Result for 12. is:
[tex]\[ > \][/tex]
13. [tex]\( \frac{5}{9} \bigcirc \frac{5}{8} \)[/tex]
The denominators tell us that [tex]\( \frac{5}{9} < \frac{5}{8} \)[/tex]
Result for 13. is:
[tex]\[ < \][/tex]
14.
(No second fraction provided)
15. [tex]\( \frac{9}{9} \bigcirc \frac{8}{8} \)[/tex]
[tex]\[ \frac{9}{9} = 1 \][/tex]
[tex]\[ \frac{8}{8} = 1 \][/tex]
[tex]\[ 1 = 1 \][/tex]
Result for 15. is:
[tex]\[ = \][/tex]
16. [tex]\( \frac{1}{10} \bigcirc \frac{1}{5} \)[/tex]
[tex]\[ \frac{1}{10} < \frac{1}{5} \][/tex]
Result for 16. is:
[tex]\[ < \][/tex]
17. [tex]\( \frac{14}{20} \bigcirc \frac{9}{10} \)[/tex]
[tex]\[ \frac{14}{20} = \frac{7}{10} \][/tex]
[tex]\[ \frac{7}{10} < \frac{9}{10} \][/tex]
Result for 17. is:
[tex]\[ < \][/tex]
18. [tex]\( \frac{6}{12} \bigcirc \frac{1}{2} \)[/tex]
[tex]\[ \frac{6}{12} = \frac{1}{2} \][/tex]
Result for 18. is:
[tex]\[ = \][/tex]
These are the solutions and comparisons for the question.
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