Simplify: $\frac{3}{4}$ of $\left(\frac{11}{15}-\overline{\frac{3}{5}-\frac{1}{3}}\right) \div 1 \frac{2}{5}$
Answer (2)
After simplifying the expression step by step, we find that the final result is 4 1 . This involves calculating the difference of fractions and adjusting for the mixed number before executing multiplication and division of fractions. The overall answer is expressed in a clear and simplified fraction format.
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Simplify the expression inside the parenthesis with the overline: 5 3 − 3 1 = 15 4 .
Subtract the fractions inside the parenthesis: 15 11 − 15 4 = 15 7 .
Convert the mixed number to an improper fraction: 1 5 2 = 5 7 .
Simplify the expression: 4 3 × 15 7 ÷ 5 7 = 4 3 × 15 7 × 7 5 = 4 1 .
4 1
Explanation
Understanding the Problem We are asked to simplify the expression 4 3 of ( 15 11 − 5 3 − 3 1 ) ÷ 1 5 2 . The overline indicates that we must first evaluate the expression 5 3 − 3 1 . The word 'of' means multiplication, and we will need to convert the mixed number 1 5 2 to an improper fraction.
Simplifying the Overline Expression First, we simplify the expression inside the parenthesis with the overline: 5 3 − 3 1 . To subtract these fractions, we need a common denominator, which is 15. So we have 5 3 − 3 1 = 5 × 3 3 × 3 − 3 × 5 1 × 5 = 15 9 − 15 5 = 15 9 − 5 = 15 4 .
Subtracting Fractions Now we substitute this result back into the original expression: 4 3 of ( 15 11 − 15 4 ) ÷ 1 5 2 . Next, we subtract the fractions inside the parenthesis: 15 11 − 15 4 = 15 11 − 4 = 15 7 .
Converting Mixed Number to Improper Fraction Now we have the expression: 4 3 of 15 7 ÷ 1 5 2 . We convert the mixed number to an improper fraction: 1 5 2 = 5 1 × 5 + 2 = 5 5 + 2 = 5 7 .
Rewriting the Expression Now we rewrite the expression as: 4 3 × 15 7 ÷ 5 7 . To divide by a fraction, we multiply by its reciprocal: 4 3 × 15 7 × 7 5 .
Simplifying the Expression Now we simplify the expression by canceling common factors: 4 3 × 15 7 × 7 5 = 4 3 × 3 × 5 7 × 7 5 = 4 × 3 × 5 × 7 3 × 7 × 5 . We can cancel the common factors of 3, 5, and 7: 4 × 3 × 5 × 7 3 × 7 × 5 = 4 1 .
Final Answer Therefore, the simplified expression is 4 1 .
Examples
Fractions are used in everyday life, such as when cooking, measuring ingredients, or splitting a bill with friends. Understanding how to simplify expressions involving fractions is essential for accurate calculations in these situations. For example, if you are following a recipe that calls for 4 3 of a cup of flour, but you only want to make half the recipe, you need to calculate 2 1 of 4 3 , which is 8 3 .
