Answer :
Sure, let's solve each problem step by step:
### Part a: [tex]\( \frac{13}{41} + \frac{27}{82} \)[/tex]
1. Find a common denominator for the fractions, which in this case is [tex]\(82\)[/tex].
2. Convert [tex]\(\frac{13}{41}\)[/tex] to have a denominator of [tex]\(82\)[/tex]:
[tex]\[ \frac{13}{41} = \frac{13 \times 2}{41 \times 2} = \frac{26}{82} \][/tex]
3. Now add the fractions:
[tex]\[ \frac{26}{82} + \frac{27}{82} = \frac{26 + 27}{82} = \frac{53}{82} \][/tex]
4. Convert the result to decimal form:
[tex]\[ \frac{53}{82} \approx 0.6463414634146342 \][/tex]
### Part b: [tex]\( 3 \frac{5}{24} + 6 \frac{7}{24} + 4 \frac{9}{24} \)[/tex]
1. Convert mixed numbers to improper fractions:
[tex]\[ 3 \frac{5}{24} = 3 + \frac{5}{24} = \frac{72}{24} + \frac{5}{24} = \frac{77}{24} \][/tex]
[tex]\[ 6 \frac{7}{24} = 6 + \frac{7}{24} = \frac{144}{24} + \frac{7}{24} = \frac{151}{24} \][/tex]
[tex]\[ 4 \frac{9}{24} = 4 + \frac{9}{24} = \frac{96}{24} + \frac{9}{24} = \frac{105}{24} \][/tex]
2. Add the fractions:
[tex]\[ \frac{77}{24} + \frac{151}{24} + \frac{105}{24} = \frac{77 + 151 + 105}{24} = \frac{333}{24} \][/tex]
3. Convert the result to a mixed number or decimal form:
[tex]\[ \frac{333}{24} = 13.875 \][/tex]
### Part c: [tex]\( 5 \frac{2}{3} + \frac{29}{69} + 6 \frac{21}{23} \)[/tex]
1. Convert mixed numbers and then to a common denominator.
[tex]\[ 5 \frac{2}{3} = 5 + \frac{2}{3} = 5 + 0.6667 \approx 5.6667 \][/tex]
[tex]\[ 6 \frac{21}{23} = 6 + \frac{21}{23} = 6 + 0.913 \approx 6.913 \][/tex]
2. Add the fractions:
[tex]\[ 5.6667 + \frac{29}{69} + 6.913 = 5.6667 + 0.4203 + 6.913 \approx 13.0 \][/tex]
### Part d: [tex]\( \frac{39}{10} + \frac{4}{9} + \frac{7}{45} + 4 \)[/tex]
1. Calculate each fraction separately:
[tex]\[ \frac{39}{10} = 3.9 \][/tex]
[tex]\[ \frac{4}{9} \approx 0.4444 \][/tex]
[tex]\[ \frac{7}{45} \approx 0.1556 \][/tex]
2. Add the fractions:
[tex]\[ 3.9 + 0.4444 + 0.1556 + 4 = 8.5 \][/tex]
### Part e: [tex]\( 6 - \frac{7}{15} \)[/tex]
1. Calculate the fraction:
[tex]\[ \frac{7}{15} \approx 0.4667 \][/tex]
2. Subtract the fraction:
[tex]\[ 6 - 0.4667 = 5.533333333333333 \][/tex]
### Part f: [tex]\( \frac{113}{8} - \frac{7}{8} \)[/tex]
1. Subtract the fractions with the same denominator directly:
[tex]\[ \frac{113 - 7}{8} = \frac{106}{8} = 13.25 \][/tex]
### Part g: [tex]\( \frac{71}{6} - \frac{34}{9} \)[/tex]
1. Convert both fractions to decimals:
[tex]\[ \frac{71}{6} \approx 11.8333 \][/tex]
[tex]\[ \frac{34}{9} \approx 3.7778 \][/tex]
2. Subtract the fractions:
[tex]\[ 11.8333 - 3.7778 = 8.055555555555557 \][/tex]
### Part h: [tex]\( 5 \frac{3}{8} - 3 \frac{2}{5} \)[/tex]
1. Convert mixed numbers to improper fractions and then decimals:
[tex]\[ 5 \frac{3}{8} = 5 + \frac{3}{8} = 5 + 0.375 = 5.375 \][/tex]
[tex]\[ 3 \frac{2}{5} = 3 + \frac{2}{5} = 3 + 0.4 = 3.4 \][/tex]
2. Subtract the fractions:
[tex]\[ 5.375 - 3.4 = 1.975 \][/tex]
The final answers are:
- a. [tex]\( \approx 0.6463414634146342 \)[/tex]
- b. [tex]\( 13.875 \)[/tex]
- c. [tex]\( 13.0 \)[/tex]
- d. [tex]\( 8.5 \)[/tex]
- e. [tex]\( \approx 5.533333333333333 \)[/tex]
- f. [tex]\( 13.25 \)[/tex]
- g. [tex]\( \approx 8.055555555555557 \)[/tex]
- h. [tex]\( 1.975 \)[/tex]
### Part a: [tex]\( \frac{13}{41} + \frac{27}{82} \)[/tex]
1. Find a common denominator for the fractions, which in this case is [tex]\(82\)[/tex].
2. Convert [tex]\(\frac{13}{41}\)[/tex] to have a denominator of [tex]\(82\)[/tex]:
[tex]\[ \frac{13}{41} = \frac{13 \times 2}{41 \times 2} = \frac{26}{82} \][/tex]
3. Now add the fractions:
[tex]\[ \frac{26}{82} + \frac{27}{82} = \frac{26 + 27}{82} = \frac{53}{82} \][/tex]
4. Convert the result to decimal form:
[tex]\[ \frac{53}{82} \approx 0.6463414634146342 \][/tex]
### Part b: [tex]\( 3 \frac{5}{24} + 6 \frac{7}{24} + 4 \frac{9}{24} \)[/tex]
1. Convert mixed numbers to improper fractions:
[tex]\[ 3 \frac{5}{24} = 3 + \frac{5}{24} = \frac{72}{24} + \frac{5}{24} = \frac{77}{24} \][/tex]
[tex]\[ 6 \frac{7}{24} = 6 + \frac{7}{24} = \frac{144}{24} + \frac{7}{24} = \frac{151}{24} \][/tex]
[tex]\[ 4 \frac{9}{24} = 4 + \frac{9}{24} = \frac{96}{24} + \frac{9}{24} = \frac{105}{24} \][/tex]
2. Add the fractions:
[tex]\[ \frac{77}{24} + \frac{151}{24} + \frac{105}{24} = \frac{77 + 151 + 105}{24} = \frac{333}{24} \][/tex]
3. Convert the result to a mixed number or decimal form:
[tex]\[ \frac{333}{24} = 13.875 \][/tex]
### Part c: [tex]\( 5 \frac{2}{3} + \frac{29}{69} + 6 \frac{21}{23} \)[/tex]
1. Convert mixed numbers and then to a common denominator.
[tex]\[ 5 \frac{2}{3} = 5 + \frac{2}{3} = 5 + 0.6667 \approx 5.6667 \][/tex]
[tex]\[ 6 \frac{21}{23} = 6 + \frac{21}{23} = 6 + 0.913 \approx 6.913 \][/tex]
2. Add the fractions:
[tex]\[ 5.6667 + \frac{29}{69} + 6.913 = 5.6667 + 0.4203 + 6.913 \approx 13.0 \][/tex]
### Part d: [tex]\( \frac{39}{10} + \frac{4}{9} + \frac{7}{45} + 4 \)[/tex]
1. Calculate each fraction separately:
[tex]\[ \frac{39}{10} = 3.9 \][/tex]
[tex]\[ \frac{4}{9} \approx 0.4444 \][/tex]
[tex]\[ \frac{7}{45} \approx 0.1556 \][/tex]
2. Add the fractions:
[tex]\[ 3.9 + 0.4444 + 0.1556 + 4 = 8.5 \][/tex]
### Part e: [tex]\( 6 - \frac{7}{15} \)[/tex]
1. Calculate the fraction:
[tex]\[ \frac{7}{15} \approx 0.4667 \][/tex]
2. Subtract the fraction:
[tex]\[ 6 - 0.4667 = 5.533333333333333 \][/tex]
### Part f: [tex]\( \frac{113}{8} - \frac{7}{8} \)[/tex]
1. Subtract the fractions with the same denominator directly:
[tex]\[ \frac{113 - 7}{8} = \frac{106}{8} = 13.25 \][/tex]
### Part g: [tex]\( \frac{71}{6} - \frac{34}{9} \)[/tex]
1. Convert both fractions to decimals:
[tex]\[ \frac{71}{6} \approx 11.8333 \][/tex]
[tex]\[ \frac{34}{9} \approx 3.7778 \][/tex]
2. Subtract the fractions:
[tex]\[ 11.8333 - 3.7778 = 8.055555555555557 \][/tex]
### Part h: [tex]\( 5 \frac{3}{8} - 3 \frac{2}{5} \)[/tex]
1. Convert mixed numbers to improper fractions and then decimals:
[tex]\[ 5 \frac{3}{8} = 5 + \frac{3}{8} = 5 + 0.375 = 5.375 \][/tex]
[tex]\[ 3 \frac{2}{5} = 3 + \frac{2}{5} = 3 + 0.4 = 3.4 \][/tex]
2. Subtract the fractions:
[tex]\[ 5.375 - 3.4 = 1.975 \][/tex]
The final answers are:
- a. [tex]\( \approx 0.6463414634146342 \)[/tex]
- b. [tex]\( 13.875 \)[/tex]
- c. [tex]\( 13.0 \)[/tex]
- d. [tex]\( 8.5 \)[/tex]
- e. [tex]\( \approx 5.533333333333333 \)[/tex]
- f. [tex]\( 13.25 \)[/tex]
- g. [tex]\( \approx 8.055555555555557 \)[/tex]
- h. [tex]\( 1.975 \)[/tex]
