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Sagot :
Sure! Let's break down the problem step-by-step.
We need to evaluate the expression:
[tex]\[ \left(1 + \frac{1}{15} - \frac{9}{10}\right) \div \left(1 \frac{1}{4} \div 1 \frac{1}{2}\right) \][/tex]
Step 1: Simplify the numerator:
We begin with the expression inside the parentheses in the numerator:
[tex]\[ 1 + \frac{1}{15} - \frac{9}{10} \][/tex]
Convert all terms to a common denominator, which in this case is 30:
[tex]\[ 1 = \frac{30}{30}, \quad \frac{1}{15} = \frac{2}{30}, \quad \frac{9}{10} = \frac{27}{30} \][/tex]
Now we can write:
[tex]\[ 1 + \frac{1}{15} - \frac{9}{10} = \frac{30}{30} + \frac{2}{30} - \frac{27}{30} \][/tex]
Combine the fractions:
[tex]\[ \frac{30}{30} + \frac{2}{30} - \frac{27}{30} = \frac{30 + 2 - 27}{30} = \frac{5}{30} = \frac{1}{6} \][/tex]
So, the simplified numerator is:
[tex]\[ \frac{1}{6} \][/tex]
Step 2: Simplify the denominator:
We now need to simplify the expression inside the parentheses in the denominator:
[tex]\[ 1 \frac{1}{4} \div 1 \frac{1}{2} \][/tex]
First, convert the mixed numbers to improper fractions:
[tex]\[ 1 \frac{1}{4} = \frac{5}{4}, \quad 1 \frac{1}{2} = \frac{3}{2} \][/tex]
Now, perform the division by multiplying by the reciprocal:
[tex]\[ \frac{5}{4} \div \frac{3}{2} = \frac{5}{4} \times \frac{2}{3} = \frac{5 \cdot 2}{4 \cdot 3} = \frac{10}{12} = \frac{5}{6} \][/tex]
So, the simplified denominator is:
[tex]\[ \frac{5}{6} \][/tex]
Step 3: Divide the simplified numerator by the simplified denominator:
Now, we need to divide:
[tex]\[ \frac{1}{6} \div \frac{5}{6} \][/tex]
Perform the division by multiplying by the reciprocal:
[tex]\[ \frac{1}{6} \times \frac{6}{5} = \frac{1 \cdot 6}{6 \cdot 5} = \frac{6}{30} = \frac{1}{5} \][/tex]
Therefore, the final result of the given expression is:
[tex]\[ \frac{1}{5} = 0.2 \][/tex]
The detailed calculations show that the answer is [tex]\(0.2\)[/tex] or [tex]\(\frac{1}{5}\)[/tex].
We need to evaluate the expression:
[tex]\[ \left(1 + \frac{1}{15} - \frac{9}{10}\right) \div \left(1 \frac{1}{4} \div 1 \frac{1}{2}\right) \][/tex]
Step 1: Simplify the numerator:
We begin with the expression inside the parentheses in the numerator:
[tex]\[ 1 + \frac{1}{15} - \frac{9}{10} \][/tex]
Convert all terms to a common denominator, which in this case is 30:
[tex]\[ 1 = \frac{30}{30}, \quad \frac{1}{15} = \frac{2}{30}, \quad \frac{9}{10} = \frac{27}{30} \][/tex]
Now we can write:
[tex]\[ 1 + \frac{1}{15} - \frac{9}{10} = \frac{30}{30} + \frac{2}{30} - \frac{27}{30} \][/tex]
Combine the fractions:
[tex]\[ \frac{30}{30} + \frac{2}{30} - \frac{27}{30} = \frac{30 + 2 - 27}{30} = \frac{5}{30} = \frac{1}{6} \][/tex]
So, the simplified numerator is:
[tex]\[ \frac{1}{6} \][/tex]
Step 2: Simplify the denominator:
We now need to simplify the expression inside the parentheses in the denominator:
[tex]\[ 1 \frac{1}{4} \div 1 \frac{1}{2} \][/tex]
First, convert the mixed numbers to improper fractions:
[tex]\[ 1 \frac{1}{4} = \frac{5}{4}, \quad 1 \frac{1}{2} = \frac{3}{2} \][/tex]
Now, perform the division by multiplying by the reciprocal:
[tex]\[ \frac{5}{4} \div \frac{3}{2} = \frac{5}{4} \times \frac{2}{3} = \frac{5 \cdot 2}{4 \cdot 3} = \frac{10}{12} = \frac{5}{6} \][/tex]
So, the simplified denominator is:
[tex]\[ \frac{5}{6} \][/tex]
Step 3: Divide the simplified numerator by the simplified denominator:
Now, we need to divide:
[tex]\[ \frac{1}{6} \div \frac{5}{6} \][/tex]
Perform the division by multiplying by the reciprocal:
[tex]\[ \frac{1}{6} \times \frac{6}{5} = \frac{1 \cdot 6}{6 \cdot 5} = \frac{6}{30} = \frac{1}{5} \][/tex]
Therefore, the final result of the given expression is:
[tex]\[ \frac{1}{5} = 0.2 \][/tex]
The detailed calculations show that the answer is [tex]\(0.2\)[/tex] or [tex]\(\frac{1}{5}\)[/tex].
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