Answer :
Sure! Let's go through each problem step-by-step to understand how we multiply and simplify fractions.
### 1) [tex]\(\frac{3}{8} \cdot \frac{4}{9} \cdot \frac{3}{8} : \frac{4}{9} \frac{12}{72}\)[/tex]
First, calculate the product of the fractions:
Multiply the numerators: [tex]\(3 \times 4 \times 3 = 36\)[/tex]
Multiply the denominators: [tex]\(8 \times 9 \times 8 = 576\)[/tex]
Now simplify the fraction [tex]\(\frac{36}{576}\)[/tex]:
The greatest common divisor (GCD) of 36 and 576 is 36.
[tex]\(\frac{36}{576} = \frac{1}{16}\)[/tex]
Then divide this by the fraction [tex]\(\frac{4}{9}\)[/tex]:
Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\(\frac{1}{16} \cdot \frac{9}{4} = \frac{9}{64}\)[/tex]
Now apply the operation with [tex]\(\frac{12}{72}\)[/tex]:
Since you are dividing again, multiply by the reciprocal:
[tex]\(\frac{9}{64} \cdot \frac{72}{12} = \frac{648}{768}\)[/tex]
Simplify [tex]\(\frac{648}{768}\)[/tex]:
The GCD of 648 and 768 is 64.
So, [tex]\(\frac{648}{768} = \frac{1}{124416}\)[/tex]
### 2) [tex]\(\frac{2}{9} \cdot \frac{1}{4} \cdot \frac{2}{3612}\)[/tex]
Multiply the numerators: [tex]\(2 \times 1 \times 2 = 4\)[/tex]
Multiply the denominators: [tex]\(9 \times 4 \times 3612 = 130608\)[/tex]
Simplify [tex]\(\frac{4}{130608}\)[/tex]:
The GCD of 4 and 130608 is 4.
[tex]\(\frac{4}{130608} = \frac{1}{32508}\)[/tex]
### 3) [tex]\(\frac{5}{6} \cdot \frac{12}{13} \cdot \frac{60}{78}\)[/tex]
Multiply the numerators: [tex]\(5 \times 12 \times 60 = 3600\)[/tex]
Multiply the denominators: [tex]\(6 \times 13 \times 78 = 6084\)[/tex]
Simplify [tex]\(\frac{3600}{6084}\)[/tex]:
Divide both the numerator and denominator by their GCD, 36:
[tex]\(\frac{3600}{6084} = \frac{100}{169}\)[/tex]
### 4) [tex]\(\frac{5}{6} \cdot \frac{4}{9}\)[/tex]
Multiply the numerators: [tex]\(5 \times 4 = 20\)[/tex]
Multiply the denominators: [tex]\(6 \times 9 = 54\)[/tex]
Simplify [tex]\(\frac{20}{54}\)[/tex]:
The GCD of 20 and 54 is 2.
[tex]\(\frac{20}{54} = \frac{10}{27}\)[/tex]
### 5) [tex]\(\frac{18}{20} \cdot \frac{5}{9}\)[/tex]
Multiply the numerators: [tex]\(18 \times 5 = 90\)[/tex]
Multiply the denominators: [tex]\(20 \times 9 = 180\)[/tex]
Simplify [tex]\(\frac{90}{180}\)[/tex]:
The GCD of 90 and 180 is 90.
[tex]\(\frac{90}{180} = \frac{1}{2}\)[/tex]
### 6) [tex]\(\frac{3}{28} \cdot \frac{4}{15}\)[/tex]
Multiply the numerators: [tex]\(3 \times 4 = 12\)[/tex]
Multiply the denominators: [tex]\(28 \times 15 = 420\)[/tex]
Simplify [tex]\(\frac{12}{420}\)[/tex]:
The GCD of 12 and 420 is 12.
[tex]\(\frac{12}{420} = \frac{1}{35}\)[/tex]
These are the simplified products of the fractions for each question.
### 1) [tex]\(\frac{3}{8} \cdot \frac{4}{9} \cdot \frac{3}{8} : \frac{4}{9} \frac{12}{72}\)[/tex]
First, calculate the product of the fractions:
Multiply the numerators: [tex]\(3 \times 4 \times 3 = 36\)[/tex]
Multiply the denominators: [tex]\(8 \times 9 \times 8 = 576\)[/tex]
Now simplify the fraction [tex]\(\frac{36}{576}\)[/tex]:
The greatest common divisor (GCD) of 36 and 576 is 36.
[tex]\(\frac{36}{576} = \frac{1}{16}\)[/tex]
Then divide this by the fraction [tex]\(\frac{4}{9}\)[/tex]:
Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\(\frac{1}{16} \cdot \frac{9}{4} = \frac{9}{64}\)[/tex]
Now apply the operation with [tex]\(\frac{12}{72}\)[/tex]:
Since you are dividing again, multiply by the reciprocal:
[tex]\(\frac{9}{64} \cdot \frac{72}{12} = \frac{648}{768}\)[/tex]
Simplify [tex]\(\frac{648}{768}\)[/tex]:
The GCD of 648 and 768 is 64.
So, [tex]\(\frac{648}{768} = \frac{1}{124416}\)[/tex]
### 2) [tex]\(\frac{2}{9} \cdot \frac{1}{4} \cdot \frac{2}{3612}\)[/tex]
Multiply the numerators: [tex]\(2 \times 1 \times 2 = 4\)[/tex]
Multiply the denominators: [tex]\(9 \times 4 \times 3612 = 130608\)[/tex]
Simplify [tex]\(\frac{4}{130608}\)[/tex]:
The GCD of 4 and 130608 is 4.
[tex]\(\frac{4}{130608} = \frac{1}{32508}\)[/tex]
### 3) [tex]\(\frac{5}{6} \cdot \frac{12}{13} \cdot \frac{60}{78}\)[/tex]
Multiply the numerators: [tex]\(5 \times 12 \times 60 = 3600\)[/tex]
Multiply the denominators: [tex]\(6 \times 13 \times 78 = 6084\)[/tex]
Simplify [tex]\(\frac{3600}{6084}\)[/tex]:
Divide both the numerator and denominator by their GCD, 36:
[tex]\(\frac{3600}{6084} = \frac{100}{169}\)[/tex]
### 4) [tex]\(\frac{5}{6} \cdot \frac{4}{9}\)[/tex]
Multiply the numerators: [tex]\(5 \times 4 = 20\)[/tex]
Multiply the denominators: [tex]\(6 \times 9 = 54\)[/tex]
Simplify [tex]\(\frac{20}{54}\)[/tex]:
The GCD of 20 and 54 is 2.
[tex]\(\frac{20}{54} = \frac{10}{27}\)[/tex]
### 5) [tex]\(\frac{18}{20} \cdot \frac{5}{9}\)[/tex]
Multiply the numerators: [tex]\(18 \times 5 = 90\)[/tex]
Multiply the denominators: [tex]\(20 \times 9 = 180\)[/tex]
Simplify [tex]\(\frac{90}{180}\)[/tex]:
The GCD of 90 and 180 is 90.
[tex]\(\frac{90}{180} = \frac{1}{2}\)[/tex]
### 6) [tex]\(\frac{3}{28} \cdot \frac{4}{15}\)[/tex]
Multiply the numerators: [tex]\(3 \times 4 = 12\)[/tex]
Multiply the denominators: [tex]\(28 \times 15 = 420\)[/tex]
Simplify [tex]\(\frac{12}{420}\)[/tex]:
The GCD of 12 and 420 is 12.
[tex]\(\frac{12}{420} = \frac{1}{35}\)[/tex]
These are the simplified products of the fractions for each question.
