Answer :
Alright, let's solve each expression step by step:
### Expression 1: [tex]\(2 \frac{1}{4} \times 1 \frac{3}{5} + 2 \frac{3}{4} \div \frac{1}{4} - 1 \frac{3}{4}\)[/tex]
1. Convert the mixed fractions to improper fractions:
[tex]\[ 2 \frac{1}{4} = \frac{9}{4}, \quad 1 \frac{3}{5} = \frac{8}{5}, \quad 2 \frac{3}{4} = \frac{11}{4}, \quad 1 \frac{3}{4} = \frac{7}{4} \][/tex]
2. Perform the multiplication:
[tex]\[ \frac{9}{4} \times \frac{8}{5} = \frac{72}{20} = \frac{18}{5} \][/tex]
3. Perform the division:
[tex]\[ \frac{11}{4} \div \frac{1}{4} = \frac{11}{4} \times 4 = 11 \][/tex]
4. Add the results from steps 2 and 3:
[tex]\[ \frac{18}{5} + 11 = \frac{18}{5} + \frac{55}{5} = \frac{73}{5} = 14.6 \][/tex]
5. Subtract the final term:
[tex]\[ 14.6 - \frac{7}{4} = 14.6 - 1.75 = 12.85 \][/tex]
So, the final result for the first expression is [tex]\(12.85\)[/tex].
### Expression 2: [tex]\(4 \frac{1}{6} \times 2 \frac{3}{38} - \frac{2}{7} + \frac{5}{6} \div \frac{1}{6}\)[/tex]
1. Convert the mixed fractions to improper fractions:
[tex]\[ 4 \frac{1}{6} = \frac{25}{6}, \quad 2 \frac{3}{38} = \frac{79}{38} \][/tex]
2. Perform the multiplication:
[tex]\[ \frac{25}{6} \times \frac{79}{38} = \frac{1975}{228} \approx 8.6623 \][/tex]
3. Perform the division:
[tex]\[ \frac{5}{6} \div \frac{1}{6} = \frac{5}{6} \times 6 = 5 \][/tex]
4. Subtract the [tex]\(\frac{2}{7}\)[/tex] term and add the result from the division:
[tex]\[ 8.6623 - \frac{2}{7} + 5 \approx 8.6623 - 0.2857 + 5 = 13.376566416040102 \][/tex]
So, the final result for the second expression is approximately [tex]\(13.376566416040102\)[/tex].
### Expression 3: [tex]\(3 \frac{1}{7} \times 1 \frac{4}{39} - \frac{1}{7} + \frac{6}{7} \div \frac{1}{7}\)[/tex]
1. Convert the mixed fractions to improper fractions:
[tex]\[ 3 \frac{1}{7} = \frac{22}{7}, \quad 1 \frac{4}{39} = \frac{43}{39} \][/tex]
2. Perform the multiplication:
[tex]\[ \frac{22}{7} \times \frac{43}{39} = \frac{946}{273} \approx 3.4652 \][/tex]
3. Perform the division:
[tex]\[ \frac{6}{7} \div \frac{1}{7} = \frac{6}{7} \times 7 = 6 \][/tex]
4. Subtract the [tex]\(\frac{1}{7}\)[/tex] term and add the result from the division:
[tex]\[ 3.4652 - \frac{1}{7} + 6 \approx 3.4652 - 0.1429 + 6 = 9.322344322344323 \][/tex]
So, the final result for the third expression is approximately [tex]\(9.322344322344323\)[/tex].
### Expression 4: [tex]\(6 \frac{1}{9} \times 2 \frac{1}{27} - 1 \frac{1}{6} + 3 \frac{1}{9} \div \frac{1}{3}\)[/tex]
1. Convert the mixed fractions to improper fractions:
[tex]\[ 6 \frac{1}{9} = \frac{55}{9}, \quad 2 \frac{1}{27} = \frac{55}{27}, \quad 1 \frac{1}{6} = \frac{7}{6}, \quad 3 \frac{1}{9} = \frac{28}{9} \][/tex]
2. Perform the multiplication:
[tex]\[ \frac{55}{9} \times \frac{55}{27} = \frac{3025}{243} \approx 12.4506 \][/tex]
3. Perform the division:
[tex]\[ \frac{28}{9} \div \frac{1}{3} = \frac{28}{9} \times 3 = \frac{84}{9} = 9.3333 \][/tex]
4. Subtract the [tex]\(\frac{7}{6}\)[/tex] term and add the result from the division:
[tex]\[ 12.4506 - \frac{7}{6} + 9.3333 \approx 12.4506 - 1.1667 + 9.3333 = 20.61522633744856 \][/tex]
So, the final result for the fourth expression is approximately [tex]\(20.61522633744856\)[/tex].
Thus, the final results for all four expressions are:
[tex]\[ \begin{array}{l} 1. \quad 12.85 \\ 2. \quad 13.376566416040102 \\ 3. \quad 9.322344322344323 \\ 4. \quad 20.61522633744856 \\ \end{array} \][/tex]
### Expression 1: [tex]\(2 \frac{1}{4} \times 1 \frac{3}{5} + 2 \frac{3}{4} \div \frac{1}{4} - 1 \frac{3}{4}\)[/tex]
1. Convert the mixed fractions to improper fractions:
[tex]\[ 2 \frac{1}{4} = \frac{9}{4}, \quad 1 \frac{3}{5} = \frac{8}{5}, \quad 2 \frac{3}{4} = \frac{11}{4}, \quad 1 \frac{3}{4} = \frac{7}{4} \][/tex]
2. Perform the multiplication:
[tex]\[ \frac{9}{4} \times \frac{8}{5} = \frac{72}{20} = \frac{18}{5} \][/tex]
3. Perform the division:
[tex]\[ \frac{11}{4} \div \frac{1}{4} = \frac{11}{4} \times 4 = 11 \][/tex]
4. Add the results from steps 2 and 3:
[tex]\[ \frac{18}{5} + 11 = \frac{18}{5} + \frac{55}{5} = \frac{73}{5} = 14.6 \][/tex]
5. Subtract the final term:
[tex]\[ 14.6 - \frac{7}{4} = 14.6 - 1.75 = 12.85 \][/tex]
So, the final result for the first expression is [tex]\(12.85\)[/tex].
### Expression 2: [tex]\(4 \frac{1}{6} \times 2 \frac{3}{38} - \frac{2}{7} + \frac{5}{6} \div \frac{1}{6}\)[/tex]
1. Convert the mixed fractions to improper fractions:
[tex]\[ 4 \frac{1}{6} = \frac{25}{6}, \quad 2 \frac{3}{38} = \frac{79}{38} \][/tex]
2. Perform the multiplication:
[tex]\[ \frac{25}{6} \times \frac{79}{38} = \frac{1975}{228} \approx 8.6623 \][/tex]
3. Perform the division:
[tex]\[ \frac{5}{6} \div \frac{1}{6} = \frac{5}{6} \times 6 = 5 \][/tex]
4. Subtract the [tex]\(\frac{2}{7}\)[/tex] term and add the result from the division:
[tex]\[ 8.6623 - \frac{2}{7} + 5 \approx 8.6623 - 0.2857 + 5 = 13.376566416040102 \][/tex]
So, the final result for the second expression is approximately [tex]\(13.376566416040102\)[/tex].
### Expression 3: [tex]\(3 \frac{1}{7} \times 1 \frac{4}{39} - \frac{1}{7} + \frac{6}{7} \div \frac{1}{7}\)[/tex]
1. Convert the mixed fractions to improper fractions:
[tex]\[ 3 \frac{1}{7} = \frac{22}{7}, \quad 1 \frac{4}{39} = \frac{43}{39} \][/tex]
2. Perform the multiplication:
[tex]\[ \frac{22}{7} \times \frac{43}{39} = \frac{946}{273} \approx 3.4652 \][/tex]
3. Perform the division:
[tex]\[ \frac{6}{7} \div \frac{1}{7} = \frac{6}{7} \times 7 = 6 \][/tex]
4. Subtract the [tex]\(\frac{1}{7}\)[/tex] term and add the result from the division:
[tex]\[ 3.4652 - \frac{1}{7} + 6 \approx 3.4652 - 0.1429 + 6 = 9.322344322344323 \][/tex]
So, the final result for the third expression is approximately [tex]\(9.322344322344323\)[/tex].
### Expression 4: [tex]\(6 \frac{1}{9} \times 2 \frac{1}{27} - 1 \frac{1}{6} + 3 \frac{1}{9} \div \frac{1}{3}\)[/tex]
1. Convert the mixed fractions to improper fractions:
[tex]\[ 6 \frac{1}{9} = \frac{55}{9}, \quad 2 \frac{1}{27} = \frac{55}{27}, \quad 1 \frac{1}{6} = \frac{7}{6}, \quad 3 \frac{1}{9} = \frac{28}{9} \][/tex]
2. Perform the multiplication:
[tex]\[ \frac{55}{9} \times \frac{55}{27} = \frac{3025}{243} \approx 12.4506 \][/tex]
3. Perform the division:
[tex]\[ \frac{28}{9} \div \frac{1}{3} = \frac{28}{9} \times 3 = \frac{84}{9} = 9.3333 \][/tex]
4. Subtract the [tex]\(\frac{7}{6}\)[/tex] term and add the result from the division:
[tex]\[ 12.4506 - \frac{7}{6} + 9.3333 \approx 12.4506 - 1.1667 + 9.3333 = 20.61522633744856 \][/tex]
So, the final result for the fourth expression is approximately [tex]\(20.61522633744856\)[/tex].
Thus, the final results for all four expressions are:
[tex]\[ \begin{array}{l} 1. \quad 12.85 \\ 2. \quad 13.376566416040102 \\ 3. \quad 9.322344322344323 \\ 4. \quad 20.61522633744856 \\ \end{array} \][/tex]
