\(\left(-\frac{7}{4}\right) \div 4 \frac{2}{3}\)
Answer (2)
Convert the mixed number to an improper fraction: 4 3 2 = 3 14 .
Rewrite the division as multiplication by the reciprocal: ( − 4 7 ) ÷ 3 14 = ( − 4 7 ) × 14 3 .
Multiply the fractions: − 4 7 × 14 3 = − 56 21 .
Simplify the fraction: − 56 21 = − 8 3 . The final answer is − 8 3 .
Explanation
Understanding the Problem We are asked to evaluate the expression ( − 4 7 ) ÷ 4 3 2 . This involves dividing a negative fraction by a mixed number. To do this, we first need to convert the mixed number into an improper fraction, and then rewrite the division as multiplication by the reciprocal.
Converting Mixed Number to Improper Fraction First, convert the mixed number 4 3 2 to an improper fraction. To do this, we multiply the whole number part (4) by the denominator (3) and add the numerator (2), then put the result over the original denominator (3): 4 3 2 = 3 4 × 3 + 2 = 3 12 + 2 = 3 14 .
Rewriting Division as Multiplication Now, rewrite the division as multiplication by the reciprocal. The reciprocal of 3 14 is 14 3 . So, we have ( − 4 7 ) ÷ 3 14 = ( − 4 7 ) × 14 3 .
Multiplying the Fractions Multiply the fractions: ( − 4 7 ) × 14 3 = − 4 × 14 7 × 3 = − 56 21 .
Simplifying the Fraction Simplify the resulting fraction. Both 21 and 56 are divisible by 7: 56 21 = 56 ÷ 7 21 ÷ 7 = 8 3 . Therefore, the final result is − 56 21 = − 8 3 .
Final Answer Thus, ( − 4 7 ) ÷ 4 3 2 = − 8 3 .
Examples
Understanding fraction division is crucial in many real-life scenarios. For instance, if you're splitting a pizza among friends and need to determine how much each person gets, or if you're scaling down a recipe that calls for fractional amounts of ingredients, knowing how to divide fractions is essential. Imagine you have − 4 7 of a chocolate bar and want to divide it equally among 4 3 2 people. Each person would receive − 8 3 of the chocolate bar. This concept applies to various situations involving proportions and ratios.
To evaluate ( − 4 7 ) ÷ 4 3 2 , first convert the mixed number to an improper fraction, and then rewrite the division as multiplication by the reciprocal. After multiplying and simplifying, the final answer is − 8 3 .
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