$\frac{9 \frac{1}{3} \times 27 \frac{1}{2}}{3 \frac{-1}{6} \times 3 \frac{1}{3}}$
Answer (2)
Convert mixed numbers to improper fractions.
Multiply the fractions in the numerator and denominator separately: 3 28 × 2 55 = 6 1540 and 6 17 × 3 10 = 18 170 .
Divide the numerator by the denominator: 18 170 6 1540 = 6 1540 × 170 18 .
Simplify the resulting fraction: 1020 27720 = 17 462 .
17 462
Explanation
Problem Analysis We are asked to evaluate the expression 3 6 − 1 × 3 3 1 9 3 1 × 27 2 1 . This involves mixed numbers, multiplication, and division. To simplify, we will first convert the mixed numbers to improper fractions.
Converting to Improper Fractions First, convert each mixed number to an improper fraction: 9 3 1 = 3 9 × 3 + 1 = 3 27 + 1 = 3 28 27 2 1 = 2 27 × 2 + 1 = 2 54 + 1 = 2 55 3 6 − 1 = 6 3 × 6 − 1 = 6 18 − 1 = 6 17 3 3 1 = 3 3 × 3 + 1 = 3 9 + 1 = 3 10
Substituting Improper Fractions Now, substitute these improper fractions into the original expression: 6 17 × 3 10 3 28 × 2 55
Multiplying Fractions Next, multiply the fractions in the numerator and the denominator: Numerator: 3 28 × 2 55 = 3 × 2 28 × 55 = 6 1540 Denominator: 6 17 × 3 10 = 6 × 3 17 × 10 = 18 170
Dividing Fractions Now, divide the numerator by the denominator: 18 170 6 1540 = 6 1540 ÷ 18 170 = 6 1540 × 170 18 = 6 × 170 1540 × 18 = 1020 27720
Simplifying the Fraction Simplify the resulting fraction: 1020 27720 = 102 2772 = 51 1386 = 17 462 Now, we can convert this improper fraction to a mixed number or a decimal. As a decimal, 17 462 ≈ 27.17647 .
Final Answer Therefore, the value of the expression is 17 462 .
Examples
Understanding how to simplify complex fractions is useful in many real-world scenarios, such as calculating scaling factors in recipes or determining gear ratios in mechanical systems. For example, if you need to increase a recipe by a factor of 27 2 1 but only have a measuring cup that measures 9 3 1 units, you can use fraction simplification to determine the exact number of times you need to fill the cup. Similarly, in mechanical engineering, simplifying gear ratios ensures that machines operate at the correct speeds and torques, optimizing performance and preventing damage.
To solve the expression, we convert the mixed numbers to improper fractions, multiply the fractions in the numerator and denominator, and finally divide and simplify to find that the result is 17 462 .
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