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Sagot :
Sure, let’s solve the given mathematical expression step-by-step.
We need to solve 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]
1. Calculate the value inside the first parenthesis:
[tex]\[ 1 + \frac{1}{15} - \frac{9}{10} \][/tex]
To perform this calculation, we first convert each fraction to a common denominator, which is 30 in this case:
[tex]\[ \frac{1}{15} = \frac{2}{30} \][/tex]
[tex]\[ \frac{9}{10} = \frac{27}{30} \][/tex]
So we have:
[tex]\[ 1 + \frac{2}{30} - \frac{27}{30} \][/tex]
Now, let's add and subtract the fractions:
[tex]\[ 1 + \frac{2 - 27}{30} = 1 + \frac{-25}{30} = 1 - \frac{25}{30} \][/tex]
Simplifying the fraction:
[tex]\[ 1 - \frac{5}{6} \][/tex]
Converting [tex]\( 1 \)[/tex] to a fraction with a common denominator of 6:
[tex]\[ \frac{6}{6} - \frac{5}{6} = \frac{1}{6} \][/tex]
Thus, the value inside the first parenthesis is:
[tex]\[ \frac{1}{6} \][/tex]
2. Calculate the value inside the division in the second parenthesis:
[tex]\[ 1 + \frac{1}{4} \div 1 + \frac{1}{2} \][/tex]
We rewrite it to clarify the operation precedence:
[tex]\[ 1 + \left( \frac{1}{4} \div \left( 1 + \frac{1}{2} \right) \right) \][/tex]
Firstly, calculate [tex]\( 1 + \frac{1}{2} \)[/tex]:
[tex]\[ 1 + \frac{1}{2} = 1.5 \][/tex]
In fractional form:
[tex]\[ 1.5 = \frac{3}{2} \][/tex]
We now perform the division [tex]\( \frac{1}{4} \div \frac{3}{2} \)[/tex]:
[tex]\[ \frac{1}{4} \div \frac{3}{2} = \frac{1}{4} \times \frac{2}{3} = \frac{1 \times 2}{4 \times 3} = \frac{2}{12} = \frac{1}{6} \][/tex]
So the operation becomes:
[tex]\[ 1 + \frac{1}{6} \][/tex]
Converting 1 to a fraction with a common denominator of 6:
[tex]\[ \frac{6}{6} + \frac{1}{6} = \frac{7}{6} \][/tex]
Thus, the value inside the second parenthesis is:
[tex]\[ \frac{7}{6} \][/tex]
3. Perform the division:
Now we need to divide the value inside the first parenthesis by the value inside the second parenthesis:
[tex]\[ \frac{1}{6} \div \frac{7}{6} \][/tex]
Dividing two fractions is equivalent to multiplying by the reciprocal:
[tex]\[ \frac{1}{6} \times \frac{6}{7} = \frac{1 \times 6}{6 \times 7} = \frac{6}{42} = \frac{1}{7} \][/tex]
Thus, the result of the entire expression is approximately:
[tex]\[ 0.2 \][/tex]
Therefore, we have:
[tex]\[ 0.16666666666666663, 0.8333333333333334, 0.19999999999999996 \][/tex]
Hence, the value of [tex]\( \left(1 + \frac{1}{15} - \frac{9}{10}\right) \div \left(1 + \frac{1}{4} \div 1 + \frac{1}{2}\right) \)[/tex] is approximately [tex]\( 0.2 \)[/tex].
We need to solve 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]
1. Calculate the value inside the first parenthesis:
[tex]\[ 1 + \frac{1}{15} - \frac{9}{10} \][/tex]
To perform this calculation, we first convert each fraction to a common denominator, which is 30 in this case:
[tex]\[ \frac{1}{15} = \frac{2}{30} \][/tex]
[tex]\[ \frac{9}{10} = \frac{27}{30} \][/tex]
So we have:
[tex]\[ 1 + \frac{2}{30} - \frac{27}{30} \][/tex]
Now, let's add and subtract the fractions:
[tex]\[ 1 + \frac{2 - 27}{30} = 1 + \frac{-25}{30} = 1 - \frac{25}{30} \][/tex]
Simplifying the fraction:
[tex]\[ 1 - \frac{5}{6} \][/tex]
Converting [tex]\( 1 \)[/tex] to a fraction with a common denominator of 6:
[tex]\[ \frac{6}{6} - \frac{5}{6} = \frac{1}{6} \][/tex]
Thus, the value inside the first parenthesis is:
[tex]\[ \frac{1}{6} \][/tex]
2. Calculate the value inside the division in the second parenthesis:
[tex]\[ 1 + \frac{1}{4} \div 1 + \frac{1}{2} \][/tex]
We rewrite it to clarify the operation precedence:
[tex]\[ 1 + \left( \frac{1}{4} \div \left( 1 + \frac{1}{2} \right) \right) \][/tex]
Firstly, calculate [tex]\( 1 + \frac{1}{2} \)[/tex]:
[tex]\[ 1 + \frac{1}{2} = 1.5 \][/tex]
In fractional form:
[tex]\[ 1.5 = \frac{3}{2} \][/tex]
We now perform the division [tex]\( \frac{1}{4} \div \frac{3}{2} \)[/tex]:
[tex]\[ \frac{1}{4} \div \frac{3}{2} = \frac{1}{4} \times \frac{2}{3} = \frac{1 \times 2}{4 \times 3} = \frac{2}{12} = \frac{1}{6} \][/tex]
So the operation becomes:
[tex]\[ 1 + \frac{1}{6} \][/tex]
Converting 1 to a fraction with a common denominator of 6:
[tex]\[ \frac{6}{6} + \frac{1}{6} = \frac{7}{6} \][/tex]
Thus, the value inside the second parenthesis is:
[tex]\[ \frac{7}{6} \][/tex]
3. Perform the division:
Now we need to divide the value inside the first parenthesis by the value inside the second parenthesis:
[tex]\[ \frac{1}{6} \div \frac{7}{6} \][/tex]
Dividing two fractions is equivalent to multiplying by the reciprocal:
[tex]\[ \frac{1}{6} \times \frac{6}{7} = \frac{1 \times 6}{6 \times 7} = \frac{6}{42} = \frac{1}{7} \][/tex]
Thus, the result of the entire expression is approximately:
[tex]\[ 0.2 \][/tex]
Therefore, we have:
[tex]\[ 0.16666666666666663, 0.8333333333333334, 0.19999999999999996 \][/tex]
Hence, the value of [tex]\( \left(1 + \frac{1}{15} - \frac{9}{10}\right) \div \left(1 + \frac{1}{4} \div 1 + \frac{1}{2}\right) \)[/tex] is approximately [tex]\( 0.2 \)[/tex].
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