Problem 122 Perform these computations. $$... [FREE SOLUTION]

Chapter 1: Problem 122

Perform these computations. $$-20 \div\left(-\frac{8}{3}\right)$$

Short Answer

Expert verified

$\frac{15}{2}$

Step by step solution

- Understand the Division of Negative Numbers and Fractions

When dividing by a fraction, it is equivalent to multiplying by its reciprocal. Additionally, dividing two negative numbers will result in a positive number.

- Write the Division as Multiplication

Transform the division into a multiplication by the reciprocal. Replace $-\frac{8}{3}$ with its reciprocal \left(-\frac{8}{3}\right)^{-1} = -\frac{3}{8}\. Thus, the problem becomes $-20 \times -\frac{3}{8}$.

- Multiply the Numbers

Multiply $-20$ by $-\frac{3}{8}$, which is $-20 \times -\frac{3}{8} = \frac{20 \times 3}{8} = \frac{60}{8}$.

- Simplify the Fraction

Reduce the fraction $\frac{60}{8}$. The greatest common divisor of 60 and 8 is 4, hence $\frac{60}{8} = \frac{60 \div 4}{8 \div 4} = \frac{15}{2}$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal of a Fraction

When you need to divide by a fraction, you actually multiply by the reciprocal of that fraction. The reciprocal of a fraction is what you get when you flip the numerator (top part) and the denominator (bottom part). For example, the reciprocal of $-\frac{8}{3}$ is $-\frac{3}{8}$. Remember, the signs do not change when finding the reciprocal. This flipping process turns a division problem into a multiplication problem, making it easier to solve.

So, for our problem $-20 \div \left(-\frac{8}{3}\right)$, instead of dividing, we multiply by $-\frac{3}{8}$.

Multiplication of Fractions

Multiplying fractions is straightforward. First, multiply the numerators together to get the new numerator. Then, multiply the denominators together to get the new denominator. For example, in the transformed problem $-20 \times -\frac{3}{8}$:

1. Multiply the numerators: $-20$ and $-3$
2. Multiply the denominators: $1$ and $8$

This gives us $\frac{-20 \times -3}{1 \times 8} = \frac{60}{8}$. Notice that multiplying two negative numbers results in a positive number ($20 \times 3 = 60$).

Simplifying Fractions

Once you have your fraction from the multiplication, you often need to simplify it. Simplifying makes the fraction easier to understand and work with. To simplify a fraction, divide the numerator and the denominator by their greatest common divisor (GCD).

For $\frac{60}{8}$, find the GCD of 60 and 8, which is 4:

Divide both the numerator and the denominator by 4.
This gives $\frac{60 \div 4}{8 \div 4} = \frac{15}{2}$.

The fraction $\frac{15}{2}$ is now in its simplest form.

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91影视

Perform these computations. $$-20 \div\left(-\frac{8}{3}\right)$$

Short Answer

Step by step solution

- Understand the Division of Negative Numbers and Fractions

- Write the Division as Multiplication

- Multiply the Numbers

- Simplify the Fraction

Key Concepts

Reciprocal of a Fraction

Multiplication of Fractions

Simplifying Fractions

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