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Sagot :
Certainly! Let's solve the problem step-by-step.
We have the mixed numbers [tex]\(1 \frac{1}{9}\)[/tex] and [tex]\(2 \frac{3}{6}\)[/tex], and we need to multiply them together.
### Step 1: Convert Mixed Numbers to Improper Fractions
First, convert each mixed number to an improper fraction.
#### For [tex]\(1 \frac{1}{9}\)[/tex]:
1. Multiply the whole number by the denominator: [tex]\(1 \times 9 = 9\)[/tex].
2. Add the numerator to the result: [tex]\(9 + 1 = 10\)[/tex].
3. Place this result over the original denominator: [tex]\(\frac{10}{9}\)[/tex].
So, [tex]\(1 \frac{1}{9}\)[/tex] becomes [tex]\(\frac{10}{9}\)[/tex].
#### For [tex]\(2 \frac{3}{6}\)[/tex]:
1. Simplify the fraction [tex]\(\frac{3}{6}\)[/tex] to [tex]\(\frac{1}{2}\)[/tex], because [tex]\( \frac{3}{6} = \frac{1}{2} \)[/tex].
2. Multiply the whole number by the denominator: [tex]\(2 \times 2 = 4\)[/tex].
3. Add the numerator to the result: [tex]\(4 + 1 = 5\)[/tex].
4. Place this result over the original denominator: [tex]\(\frac{5}{2}\)[/tex].
So, [tex]\(2 \frac{1}{2}\)[/tex] becomes [tex]\(\frac{5}{2}\)[/tex].
### Step 2: Multiply the Fractions
Now multiply the two improper fractions:
[tex]\[ \frac{10}{9} \times \frac{5}{2} \][/tex]
### Step 3: Multiply the Numerators and Denominators
Multiply the numerators together and the denominators together:
[tex]\[ \text{Numerator: } 10 \times 5 = 50 \\ \text{Denominator: } 9 \times 2 = 18 \][/tex]
Therefore, the product of the fractions is:
[tex]\[ \frac{50}{18} \][/tex]
### Step 4: Simplify the Fraction (if possible)
Let's simplify [tex]\(\frac{50}{18}\)[/tex]:
1. Find the greatest common divisor (GCD) of 50 and 18. The GCD is 2.
2. Divide the numerator and the denominator by their GCD:
[tex]\[ \frac{50 \div 2}{18 \div 2} = \frac{25}{9} \][/tex]
So, [tex]\(\frac{50}{18}\)[/tex] simplifies to [tex]\(\frac{25}{9}\)[/tex].
### Step 5: Convert Back to Mixed Number (if needed)
If we want to convert [tex]\(\frac{25}{9}\)[/tex] back to a mixed number:
1. Divide the numerator by the denominator: [tex]\(25 \div 9 = 2\)[/tex] remainder 7.
2. So, [tex]\(\frac{25}{9} = 2 \frac{7}{9}\)[/tex].
Therefore, the result of multiplying [tex]\(1 \frac{1}{9} \times 2 \frac{3}{6}\)[/tex] is [tex]\(\frac{25}{9}\)[/tex] or [tex]\(2 \frac{7}{9}\)[/tex].
However, based on the numerical results from the provided data:
[tex]\[ 1.1111111111111112 \times 2.5 = 2.7777777777777777 \][/tex]
This verifies our solution.
So, the final product is approximately [tex]\(2.78\)[/tex] in decimal form or exactly [tex]\(\frac{25}{9}\)[/tex] or [tex]\(2 \frac{7}{9}\)[/tex].
We have the mixed numbers [tex]\(1 \frac{1}{9}\)[/tex] and [tex]\(2 \frac{3}{6}\)[/tex], and we need to multiply them together.
### Step 1: Convert Mixed Numbers to Improper Fractions
First, convert each mixed number to an improper fraction.
#### For [tex]\(1 \frac{1}{9}\)[/tex]:
1. Multiply the whole number by the denominator: [tex]\(1 \times 9 = 9\)[/tex].
2. Add the numerator to the result: [tex]\(9 + 1 = 10\)[/tex].
3. Place this result over the original denominator: [tex]\(\frac{10}{9}\)[/tex].
So, [tex]\(1 \frac{1}{9}\)[/tex] becomes [tex]\(\frac{10}{9}\)[/tex].
#### For [tex]\(2 \frac{3}{6}\)[/tex]:
1. Simplify the fraction [tex]\(\frac{3}{6}\)[/tex] to [tex]\(\frac{1}{2}\)[/tex], because [tex]\( \frac{3}{6} = \frac{1}{2} \)[/tex].
2. Multiply the whole number by the denominator: [tex]\(2 \times 2 = 4\)[/tex].
3. Add the numerator to the result: [tex]\(4 + 1 = 5\)[/tex].
4. Place this result over the original denominator: [tex]\(\frac{5}{2}\)[/tex].
So, [tex]\(2 \frac{1}{2}\)[/tex] becomes [tex]\(\frac{5}{2}\)[/tex].
### Step 2: Multiply the Fractions
Now multiply the two improper fractions:
[tex]\[ \frac{10}{9} \times \frac{5}{2} \][/tex]
### Step 3: Multiply the Numerators and Denominators
Multiply the numerators together and the denominators together:
[tex]\[ \text{Numerator: } 10 \times 5 = 50 \\ \text{Denominator: } 9 \times 2 = 18 \][/tex]
Therefore, the product of the fractions is:
[tex]\[ \frac{50}{18} \][/tex]
### Step 4: Simplify the Fraction (if possible)
Let's simplify [tex]\(\frac{50}{18}\)[/tex]:
1. Find the greatest common divisor (GCD) of 50 and 18. The GCD is 2.
2. Divide the numerator and the denominator by their GCD:
[tex]\[ \frac{50 \div 2}{18 \div 2} = \frac{25}{9} \][/tex]
So, [tex]\(\frac{50}{18}\)[/tex] simplifies to [tex]\(\frac{25}{9}\)[/tex].
### Step 5: Convert Back to Mixed Number (if needed)
If we want to convert [tex]\(\frac{25}{9}\)[/tex] back to a mixed number:
1. Divide the numerator by the denominator: [tex]\(25 \div 9 = 2\)[/tex] remainder 7.
2. So, [tex]\(\frac{25}{9} = 2 \frac{7}{9}\)[/tex].
Therefore, the result of multiplying [tex]\(1 \frac{1}{9} \times 2 \frac{3}{6}\)[/tex] is [tex]\(\frac{25}{9}\)[/tex] or [tex]\(2 \frac{7}{9}\)[/tex].
However, based on the numerical results from the provided data:
[tex]\[ 1.1111111111111112 \times 2.5 = 2.7777777777777777 \][/tex]
This verifies our solution.
So, the final product is approximately [tex]\(2.78\)[/tex] in decimal form or exactly [tex]\(\frac{25}{9}\)[/tex] or [tex]\(2 \frac{7}{9}\)[/tex].
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