Simplify: $\frac{3}{4}$ of $\left(\frac{11}{15}-\overline{\frac{3}{5}-\frac{1}{3}}\right) \div 1 \frac{2}{5}$
Answer (2)
The expression simplifies to 4 1 after evaluating the components step by step: first calculating the difference of fractions, then multiplying by another fraction, and finally dividing by a mixed number.
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Evaluate the expression inside the parenthesis: 5 3 − 3 1 = 15 4 .
Subtract the repeating decimal from 15 11 : 15 11 − 15 4 = 15 7 .
Multiply the result by 4 3 : 4 3 × 15 7 = 20 7 .
Divide by the improper fraction 5 7 : 20 7 ÷ 5 7 = 4 1 .
The final answer is 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 . This involves fractions, a repeating decimal, and mixed numbers. Let's break it down step by step.
Evaluating the Innermost Parenthesis First, we need to evaluate the expression inside the parenthesis with the repeating decimal. We have 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 4 So, 5 3 − 3 1 = 15 4 .
Understanding Repeating Decimal Now we need to determine the repeating decimal representation of 15 4 . This means the fraction 15 4 repeats.
Subtracting the Repeating Decimal Next, we subtract the repeating decimal 15 4 from 15 11 . So we have 15 11 − 15 4 = 15 11 − 4 = 15 7
Multiplying by 3/4 Now, we multiply the result by 4 3 . So we have 4 3 × 15 7 = 4 × 15 3 × 7 = 60 21 = 20 7
Converting Mixed Number to Improper Fraction Now, we convert the mixed number 1 5 2 into an improper fraction. We have 1 5 2 = 5 1 × 5 + 2 = 5 7
Dividing by the Improper Fraction Finally, we divide the result from the previous step by the improper fraction. So we have 20 7 ÷ 5 7 = 20 7 × 7 5 = 20 × 7 7 × 5 = 140 35 = 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 complex expressions involving fractions can help you solve real-world problems more efficiently. For example, if you are trying to determine how much of an ingredient to use based on a recipe that calls for a fraction of a cup, simplifying the expression can help you find the exact amount needed. This skill is also useful in financial calculations, such as determining discounts or calculating interest rates.
