$(-5) \div\left(-3 \frac{1}{3}\right) \div\left(-\frac{1}{8}\right)$
Answer (2)
Convert the mixed number to an improper fraction: − 3 3 1 = − 3 10 .
Rewrite the division as multiplication by the reciprocal: ( − 5 ) × ( − 10 3 ) × ( − 8 ) .
Multiply the first two numbers: ( − 5 ) × ( − 10 3 ) = 2 3 .
Multiply the result by the last number: 2 3 × ( − 8 ) = − 12 .
The final answer is − 12 .
Explanation
Understanding the Problem We are asked to evaluate the expression ( − 5 ) ÷ ( − 3 3 1 ) ÷ ( − 8 1 ) . This involves division of negative numbers and a mixed number. We will proceed step by step, converting the mixed number to an improper fraction and then performing the divisions from left to right.
Converting to Improper Fraction First, we convert the mixed number − 3 3 1 to an improper fraction. We have − 3 3 1 = − ( 3 + 3 1 ) = − ( 3 9 + 3 1 ) = − 3 10 .
Rewriting the Expression Now we rewrite the original expression using the improper fraction: ( − 5 ) ÷ ( − 3 10 ) ÷ ( − 8 1 ) .
Converting Division to Multiplication Next, we replace the division operations with multiplication by the reciprocal. Remember that dividing by a fraction is the same as multiplying by its reciprocal. So, we have ( − 5 ) × ( − 10 3 ) × ( − 8 ) .
First Multiplication Now we perform the multiplication from left to right. First, we multiply ( − 5 ) by ( − 10 3 ) :
( − 5 ) × ( − 10 3 ) = 10 ( − 5 ) × ( − 3 ) = 10 15 = 2 3 .
Second Multiplication Now we multiply the result 2 3 by ( − 8 ) :
2 3 × ( − 8 ) = 2 3 × ( − 8 ) = 2 − 24 = − 12.
Final Answer Therefore, the final result is − 12 .
Examples
Understanding how to divide negative numbers and fractions is essential in many real-world scenarios, such as calculating financial losses, determining rates of change, or understanding proportions in recipes. For instance, if a company's losses are distributed over several periods, dividing negative numbers helps determine the average loss per period. Similarly, in cooking, if you need to halve a recipe that already uses fractional quantities, these skills become invaluable.
To solve the expression ( − 5 ) ÷ ( − 3 3 1 ) ÷ ( − 8 1 ) , we convert the mixed number to an improper fraction and then use the reciprocals to multiply instead of divide. After performing the calculations through multiplication, we find the final result is − 12.
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