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Chapter 1: Problem 359

Multiply. $$ -\frac{2}{3}\left(-\frac{3}{8}\right) $$

Short Answer

Expert verified
\(-\frac{2}{3} \times -\frac{3}{8} = \frac{1}{4}\)

Step by step solution

01

Identify the numbers to be multiplied

Notice that the two numbers involved in the multiplication are \(-\frac{2}{3}\) and \(-\frac{3}{8}\).
02

Multiply the numerators

Multiply the numerators (the numbers on top of the fractions). \(-2 \times -3 = 6\). The product of the numerators is 6.
03

Multiply the denominators

Multiply the denominators (the numbers on the bottom of the fractions). \(3 \times 8 = 24\). The product of the denominators is 24.
04

Simplify the fraction

Now, put the products of the numerators and denominators together: \(\frac{6}{24}\). Simplify this fraction by finding the greatest common divisor (GCD) of 6 and 24, which is 6. \(\frac{6}{24} = \frac{6 \div 6}{24 \div 6} = \frac{1}{4}\). Therefore, the simplified fraction is \(\frac{1}{4}\).

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Key Concepts

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

Fractions
Fractions are a way to represent parts of a whole. They consist of two parts: a numerator (the top number) and a denominator (the bottom number). The numerator tells you how many parts you have, while the denominator tells you the total number of equal parts the whole is divided into. For example, in the fraction \(-\frac{2}{3}\), 2 is the numerator and 3 is the denominator. Understanding fractions is essential in many areas of math and science.
Multiplication
Multiplication of fractions involves multiplying the numerators together and the denominators together. In our example, to multiply \(-\frac{2}{3}\) by another fraction \(-\frac{3}{8}\), we do the following: \(-2 \times -3 = 6\) and \(3 \times 8 = 24\). Therefore, the product of these fractions is \(-\frac{2}{3} \times -\frac{3}{8} = \frac{6}{24}\). Remember, when you multiply two negative numbers, the result is a positive number.
Simplifying Fractions
Simplifying fractions refers to reducing them to their simplest form. When we have \(\frac{6}{24}\) from our multiplication, to simplify it, we need to find the greatest common divisor (GCD) of the numerator and denominator and divide both by that number. In this case, the GCD of 6 and 24 is 6. Thus, \(\frac{6}{24}\) simplifies to \(\frac{1}{4}\): \(\frac{6 \div 6}{24 \div 6} = \frac{1}{4}\). A simpler fraction is easier to understand and work with.
Greatest Common Divisor (GCD)
The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. To find the GCD of 6 and 24, we list the factors of both numbers. Factors of 6: 1, 2, 3, 6. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The largest common factor is 6. Using the GCD to simplify fractions reduces them to their simplest form. This is important for clarity and ease of further calculations.

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