Answer :
Sure, let's go through each problem step-by-step to understand how to find the product in simplest form, by canceling the common factors in the numerator and denominator.
### Problem 17: [tex]\(\frac{1}{2} \times \frac{4}{5}\)[/tex]
1. Multiply the numerators: [tex]\(1 \times 4 = 4\)[/tex]
2. Multiply the denominators: [tex]\(2 \times 5 = 10\)[/tex]
3. Simplify the fraction: [tex]\(\frac{4}{10} = \frac{2}{5}\)[/tex]
So, [tex]\(\frac{1}{2} \times \frac{4}{5} = \frac{2}{5}\)[/tex]
### Problem 21: [tex]\(12 \times \frac{9}{36}\)[/tex]
1. Express [tex]\(12\)[/tex] as a fraction: [tex]\(\frac{12}{1}\)[/tex]
2. Multiply the numerators: [tex]\(12 \times 9 = 108\)[/tex]
3. Multiply the denominators: [tex]\(1 \times 36 = 36\)[/tex]
4. Simplify the fraction: [tex]\(\frac{108}{36} = 3\)[/tex]
So, [tex]\(12 \times \frac{9}{36} = 3\)[/tex]
### Problem 18: [tex]\(\frac{8}{12} \times \frac{4}{6}\)[/tex]
1. Multiply the numerators: [tex]\(8 \times 4 = 32\)[/tex]
2. Multiply the denominators: [tex]\(12 \times 6 = 72\)[/tex]
3. Simplify the fraction by dividing by the greatest common divisor (GCD) which is 8: [tex]\(\frac{32}{72} = \frac{4}{9}\)[/tex]
So, [tex]\(\frac{8}{12} \times \frac{4}{6} = \frac{4}{9}\)[/tex]
### Problem 22: [tex]\(18 \times \frac{18}{54}\)[/tex]
1. Express [tex]\(18\)[/tex] as a fraction: [tex]\(\frac{18}{1}\)[/tex]
2. Multiply the numerators: [tex]\(18 \times 18 = 324\)[/tex]
3. Multiply the denominators: [tex]\(1 \times 54 = 54\)[/tex]
4. Simplify the fraction: [tex]\(\frac{324}{54} = 6\)[/tex]
So, [tex]\(18 \times \frac{18}{54} = 6\)[/tex]
### Problem 19: [tex]\(\frac{9}{15} \times \frac{5}{6}\)[/tex]
1. Multiply the numerators: [tex]\(9 \times 5 = 45\)[/tex]
2. Multiply the denominators: [tex]\(15 \times 6 = 90\)[/tex]
3. Simplify the fraction by dividing by GCD which is 45: [tex]\(\frac{45}{90} = \frac{1}{2}\)[/tex]
So, [tex]\(\frac{9}{15} \times \frac{5}{6} = \frac{1}{2}\)[/tex]
### Problem 23: [tex]\(21 \times \frac{5}{7} \times \frac{9}{35}\)[/tex]
1. Express [tex]\(21\)[/tex] as a fraction: [tex]\(\frac{21}{1}\)[/tex]
2. Multiply the numerators: [tex]\(21 \times 5 \times 9 = 945\)[/tex]
3. Multiply the denominators: [tex]\(1 \times 7 \times 35 = 245\)[/tex]
4. Simplify the fraction: [tex]\(\frac{945}{245} = 3.857142857142857\)[/tex]
So, [tex]\(21 \times \frac{5}{7} \times \frac{9}{35} = 3.857142857142857\)[/tex]
### Problem 20: [tex]\(\frac{10}{32} \times \frac{16}{20}\)[/tex]
1. Multiply the numerators: [tex]\(10 \times 16 = 160\)[/tex]
2. Multiply the denominators: [tex]\(32 \times 20 = 640\)[/tex]
3. Simplify the fraction by dividing by GCD which is 40: [tex]\(\frac{160}{640} = \frac{1}{4}\)[/tex]
So, [tex]\(\frac{10}{32} \times \frac{16}{20} = \frac{1}{4}\)[/tex]
### Problem 24: [tex]\(\frac{18}{27} \times \frac{18}{20} \times \frac{10}{6}\)[/tex]
1. Multiply the numerators: [tex]\(18 \times 18 \times 10 = 3240\)[/tex]
2. Multiply the denominators: [tex]\(27 \times 20 \times 6 = 3240\)[/tex]
3. Simplify the fraction: [tex]\(\frac{3240}{3240} = 1\)[/tex]
So, [tex]\(\frac{18}{27} \times \frac{18}{20} \times \frac{10}{6} = 1\)[/tex]
These are the simplified forms of each expression after canceling the common factors.
### Problem 17: [tex]\(\frac{1}{2} \times \frac{4}{5}\)[/tex]
1. Multiply the numerators: [tex]\(1 \times 4 = 4\)[/tex]
2. Multiply the denominators: [tex]\(2 \times 5 = 10\)[/tex]
3. Simplify the fraction: [tex]\(\frac{4}{10} = \frac{2}{5}\)[/tex]
So, [tex]\(\frac{1}{2} \times \frac{4}{5} = \frac{2}{5}\)[/tex]
### Problem 21: [tex]\(12 \times \frac{9}{36}\)[/tex]
1. Express [tex]\(12\)[/tex] as a fraction: [tex]\(\frac{12}{1}\)[/tex]
2. Multiply the numerators: [tex]\(12 \times 9 = 108\)[/tex]
3. Multiply the denominators: [tex]\(1 \times 36 = 36\)[/tex]
4. Simplify the fraction: [tex]\(\frac{108}{36} = 3\)[/tex]
So, [tex]\(12 \times \frac{9}{36} = 3\)[/tex]
### Problem 18: [tex]\(\frac{8}{12} \times \frac{4}{6}\)[/tex]
1. Multiply the numerators: [tex]\(8 \times 4 = 32\)[/tex]
2. Multiply the denominators: [tex]\(12 \times 6 = 72\)[/tex]
3. Simplify the fraction by dividing by the greatest common divisor (GCD) which is 8: [tex]\(\frac{32}{72} = \frac{4}{9}\)[/tex]
So, [tex]\(\frac{8}{12} \times \frac{4}{6} = \frac{4}{9}\)[/tex]
### Problem 22: [tex]\(18 \times \frac{18}{54}\)[/tex]
1. Express [tex]\(18\)[/tex] as a fraction: [tex]\(\frac{18}{1}\)[/tex]
2. Multiply the numerators: [tex]\(18 \times 18 = 324\)[/tex]
3. Multiply the denominators: [tex]\(1 \times 54 = 54\)[/tex]
4. Simplify the fraction: [tex]\(\frac{324}{54} = 6\)[/tex]
So, [tex]\(18 \times \frac{18}{54} = 6\)[/tex]
### Problem 19: [tex]\(\frac{9}{15} \times \frac{5}{6}\)[/tex]
1. Multiply the numerators: [tex]\(9 \times 5 = 45\)[/tex]
2. Multiply the denominators: [tex]\(15 \times 6 = 90\)[/tex]
3. Simplify the fraction by dividing by GCD which is 45: [tex]\(\frac{45}{90} = \frac{1}{2}\)[/tex]
So, [tex]\(\frac{9}{15} \times \frac{5}{6} = \frac{1}{2}\)[/tex]
### Problem 23: [tex]\(21 \times \frac{5}{7} \times \frac{9}{35}\)[/tex]
1. Express [tex]\(21\)[/tex] as a fraction: [tex]\(\frac{21}{1}\)[/tex]
2. Multiply the numerators: [tex]\(21 \times 5 \times 9 = 945\)[/tex]
3. Multiply the denominators: [tex]\(1 \times 7 \times 35 = 245\)[/tex]
4. Simplify the fraction: [tex]\(\frac{945}{245} = 3.857142857142857\)[/tex]
So, [tex]\(21 \times \frac{5}{7} \times \frac{9}{35} = 3.857142857142857\)[/tex]
### Problem 20: [tex]\(\frac{10}{32} \times \frac{16}{20}\)[/tex]
1. Multiply the numerators: [tex]\(10 \times 16 = 160\)[/tex]
2. Multiply the denominators: [tex]\(32 \times 20 = 640\)[/tex]
3. Simplify the fraction by dividing by GCD which is 40: [tex]\(\frac{160}{640} = \frac{1}{4}\)[/tex]
So, [tex]\(\frac{10}{32} \times \frac{16}{20} = \frac{1}{4}\)[/tex]
### Problem 24: [tex]\(\frac{18}{27} \times \frac{18}{20} \times \frac{10}{6}\)[/tex]
1. Multiply the numerators: [tex]\(18 \times 18 \times 10 = 3240\)[/tex]
2. Multiply the denominators: [tex]\(27 \times 20 \times 6 = 3240\)[/tex]
3. Simplify the fraction: [tex]\(\frac{3240}{3240} = 1\)[/tex]
So, [tex]\(\frac{18}{27} \times \frac{18}{20} \times \frac{10}{6} = 1\)[/tex]
These are the simplified forms of each expression after canceling the common factors.
