Answer :
Let's solve each part of the question step-by-step:
a) [tex]\(\frac{25}{36} \times \frac{18}{20}\)[/tex]
To multiply fractions, multiply the numerators and the denominators:
[tex]\[
\frac{25 \times 18}{36 \times 20} = \frac{450}{720}
\][/tex]
Now simplify [tex]\(\frac{450}{720}\)[/tex]:
- The greatest common divisor (GCD) of 450 and 720 is 90.
- Divide both the numerator and the denominator by 90:
[tex]\[
\frac{450 \div 90}{720 \div 90} = \frac{5}{8}
\][/tex]
b) [tex]\(\frac{4}{5} \text{ of } 55\)[/tex]
This means [tex]\(\frac{4}{5} \times 55\)[/tex]:
[tex]\[
\frac{4 \times 55}{5} = \frac{220}{5} = 44
\][/tex]
c) [tex]\(\frac{7}{9} \text{ of } \frac{18}{24}\)[/tex]
Multiply the fractions:
[tex]\[
\frac{7 \times 18}{9 \times 24} = \frac{126}{216}
\][/tex]
Simplify [tex]\(\frac{126}{216}\)[/tex]:
- The GCD of 126 and 216 is 18.
- Divide both the numerator and the denominator by 18:
[tex]\[
\frac{126 \div 18}{216 \div 18} = \frac{7}{12}
\][/tex]
d) [tex]\(\frac{4}{9} \times 2 \frac{1}{4} \times \frac{3}{5}\)[/tex]
First, convert [tex]\(2 \frac{1}{4}\)[/tex] to an improper fraction:
[tex]\[
2 \frac{1}{4} = \frac{9}{4}
\][/tex]
Now multiply:
[tex]\[
\frac{4}{9} \times \frac{9}{4} \times \frac{3}{5} = \frac{4 \times 9 \times 3}{9 \times 4 \times 5} = \frac{108}{180}
\][/tex]
Simplify [tex]\(\frac{108}{180}\)[/tex]:
- The GCD of 108 and 180 is 36.
- Divide both by 36:
[tex]\[
\frac{108 \div 36}{180 \div 36} = \frac{3}{5}
\][/tex]
e) [tex]\(\frac{9}{10} + 4 \frac{1}{6}\)[/tex]
Convert [tex]\(4 \frac{1}{6}\)[/tex] to an improper fraction:
[tex]\[
4 \frac{1}{6} = \frac{25}{6}
\][/tex]
To add [tex]\(\frac{9}{10}\)[/tex] and [tex]\(\frac{25}{6}\)[/tex], find a common denominator, which is 30:
[tex]\[
\frac{9}{10} = \frac{27}{30}, \quad \frac{25}{6} = \frac{125}{30}
\][/tex]
Add the fractions:
[tex]\[
\frac{27}{30} + \frac{125}{30} = \frac{152}{30}
\][/tex]
Simplify [tex]\(\frac{152}{30}\)[/tex]:
- The GCD of 152 and 30 is 2.
- Divide both by 2:
[tex]\[
\frac{152 \div 2}{30 \div 2} = \frac{76}{15}
\][/tex]
f) [tex]\(\frac{5}{14} \times \left(\frac{3}{10} + \frac{2}{5}\right)\)[/tex]
Add the fractions inside the parentheses:
[tex]\[
\frac{3}{10} + \frac{2}{5} = \frac{3}{10} + \frac{4}{10} = \frac{7}{10}
\][/tex]
Now multiply:
[tex]\[
\frac{5}{14} \times \frac{7}{10} = \frac{5 \times 7}{14 \times 10} = \frac{35}{140}
\][/tex]
Simplify [tex]\(\frac{35}{140}\)[/tex]:
- The GCD of 35 and 140 is 35.
- Divide both by 35:
[tex]\[
\frac{35 \div 35}{140 \div 35} = \frac{1}{4}
\][/tex]
g) [tex]\(7 - 2 \frac{1}{4} + 1 \frac{1}{2}\)[/tex]
Convert to improper fractions:
- [tex]\(2 \frac{1}{4} = \frac{9}{4}\)[/tex]
- [tex]\(1 \frac{1}{2} = \frac{3}{2}\)[/tex]
Now perform the operations:
[tex]\[
7 = \frac{28}{4}, \quad \frac{3}{2} = \frac{6}{4}
\][/tex]
[tex]\[
\frac{28}{4} - \frac{9}{4} + \frac{6}{4} = \frac{28 - 9 + 6}{4} = \frac{25}{4} = 6 \frac{1}{4}
\][/tex]
h) [tex]\(\frac{1}{2} \div \frac{6^2}{4}\)[/tex]
Calculate [tex]\(6^2 = 36\)[/tex]:
[tex]\[
\frac{6^2}{4} = \frac{36}{4} = 9
\][/tex]
Now divide:
[tex]\[
\frac{1}{2} \div 9 = \frac{1}{2} \div \frac{9}{1} = \frac{1}{2} \times \frac{1}{9} = \frac{1}{18}
\][/tex]
i) [tex]\(\frac{6}{19} \times \left(\frac{2}{4} + \frac{3}{9}\right)\)[/tex]
Add the fractions:
[tex]\[
\frac{2}{4} = \frac{1}{2}, \quad \frac{3}{9} = \frac{1}{3}
\][/tex]
Find a common denominator (6):
[tex]\[
\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}
\][/tex]
Add them:
[tex]\[
\frac{3}{6} + \frac{2}{6} = \frac{5}{6}
\][/tex]
Now multiply:
[tex]\[
\frac{6}{19} \times \frac{5}{6} = \frac{6 \times 5}{19 \times 6} = \frac{30}{114}
\][/tex]
Simplify [tex]\(\frac{30}{114}\)[/tex]:
- The GCD of 30 and 114 is 6.
- Divide both by 6:
[tex]\[
\frac{30 \div 6}{114 \div 6} = \frac{5}{19}
\][/tex]
These are the step-by-step simplified results for each part of the problem.
a) [tex]\(\frac{25}{36} \times \frac{18}{20}\)[/tex]
To multiply fractions, multiply the numerators and the denominators:
[tex]\[
\frac{25 \times 18}{36 \times 20} = \frac{450}{720}
\][/tex]
Now simplify [tex]\(\frac{450}{720}\)[/tex]:
- The greatest common divisor (GCD) of 450 and 720 is 90.
- Divide both the numerator and the denominator by 90:
[tex]\[
\frac{450 \div 90}{720 \div 90} = \frac{5}{8}
\][/tex]
b) [tex]\(\frac{4}{5} \text{ of } 55\)[/tex]
This means [tex]\(\frac{4}{5} \times 55\)[/tex]:
[tex]\[
\frac{4 \times 55}{5} = \frac{220}{5} = 44
\][/tex]
c) [tex]\(\frac{7}{9} \text{ of } \frac{18}{24}\)[/tex]
Multiply the fractions:
[tex]\[
\frac{7 \times 18}{9 \times 24} = \frac{126}{216}
\][/tex]
Simplify [tex]\(\frac{126}{216}\)[/tex]:
- The GCD of 126 and 216 is 18.
- Divide both the numerator and the denominator by 18:
[tex]\[
\frac{126 \div 18}{216 \div 18} = \frac{7}{12}
\][/tex]
d) [tex]\(\frac{4}{9} \times 2 \frac{1}{4} \times \frac{3}{5}\)[/tex]
First, convert [tex]\(2 \frac{1}{4}\)[/tex] to an improper fraction:
[tex]\[
2 \frac{1}{4} = \frac{9}{4}
\][/tex]
Now multiply:
[tex]\[
\frac{4}{9} \times \frac{9}{4} \times \frac{3}{5} = \frac{4 \times 9 \times 3}{9 \times 4 \times 5} = \frac{108}{180}
\][/tex]
Simplify [tex]\(\frac{108}{180}\)[/tex]:
- The GCD of 108 and 180 is 36.
- Divide both by 36:
[tex]\[
\frac{108 \div 36}{180 \div 36} = \frac{3}{5}
\][/tex]
e) [tex]\(\frac{9}{10} + 4 \frac{1}{6}\)[/tex]
Convert [tex]\(4 \frac{1}{6}\)[/tex] to an improper fraction:
[tex]\[
4 \frac{1}{6} = \frac{25}{6}
\][/tex]
To add [tex]\(\frac{9}{10}\)[/tex] and [tex]\(\frac{25}{6}\)[/tex], find a common denominator, which is 30:
[tex]\[
\frac{9}{10} = \frac{27}{30}, \quad \frac{25}{6} = \frac{125}{30}
\][/tex]
Add the fractions:
[tex]\[
\frac{27}{30} + \frac{125}{30} = \frac{152}{30}
\][/tex]
Simplify [tex]\(\frac{152}{30}\)[/tex]:
- The GCD of 152 and 30 is 2.
- Divide both by 2:
[tex]\[
\frac{152 \div 2}{30 \div 2} = \frac{76}{15}
\][/tex]
f) [tex]\(\frac{5}{14} \times \left(\frac{3}{10} + \frac{2}{5}\right)\)[/tex]
Add the fractions inside the parentheses:
[tex]\[
\frac{3}{10} + \frac{2}{5} = \frac{3}{10} + \frac{4}{10} = \frac{7}{10}
\][/tex]
Now multiply:
[tex]\[
\frac{5}{14} \times \frac{7}{10} = \frac{5 \times 7}{14 \times 10} = \frac{35}{140}
\][/tex]
Simplify [tex]\(\frac{35}{140}\)[/tex]:
- The GCD of 35 and 140 is 35.
- Divide both by 35:
[tex]\[
\frac{35 \div 35}{140 \div 35} = \frac{1}{4}
\][/tex]
g) [tex]\(7 - 2 \frac{1}{4} + 1 \frac{1}{2}\)[/tex]
Convert to improper fractions:
- [tex]\(2 \frac{1}{4} = \frac{9}{4}\)[/tex]
- [tex]\(1 \frac{1}{2} = \frac{3}{2}\)[/tex]
Now perform the operations:
[tex]\[
7 = \frac{28}{4}, \quad \frac{3}{2} = \frac{6}{4}
\][/tex]
[tex]\[
\frac{28}{4} - \frac{9}{4} + \frac{6}{4} = \frac{28 - 9 + 6}{4} = \frac{25}{4} = 6 \frac{1}{4}
\][/tex]
h) [tex]\(\frac{1}{2} \div \frac{6^2}{4}\)[/tex]
Calculate [tex]\(6^2 = 36\)[/tex]:
[tex]\[
\frac{6^2}{4} = \frac{36}{4} = 9
\][/tex]
Now divide:
[tex]\[
\frac{1}{2} \div 9 = \frac{1}{2} \div \frac{9}{1} = \frac{1}{2} \times \frac{1}{9} = \frac{1}{18}
\][/tex]
i) [tex]\(\frac{6}{19} \times \left(\frac{2}{4} + \frac{3}{9}\right)\)[/tex]
Add the fractions:
[tex]\[
\frac{2}{4} = \frac{1}{2}, \quad \frac{3}{9} = \frac{1}{3}
\][/tex]
Find a common denominator (6):
[tex]\[
\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}
\][/tex]
Add them:
[tex]\[
\frac{3}{6} + \frac{2}{6} = \frac{5}{6}
\][/tex]
Now multiply:
[tex]\[
\frac{6}{19} \times \frac{5}{6} = \frac{6 \times 5}{19 \times 6} = \frac{30}{114}
\][/tex]
Simplify [tex]\(\frac{30}{114}\)[/tex]:
- The GCD of 30 and 114 is 6.
- Divide both by 6:
[tex]\[
\frac{30 \div 6}{114 \div 6} = \frac{5}{19}
\][/tex]
These are the step-by-step simplified results for each part of the problem.
