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Chapter 1: Problem 360

Multiply. $$ -\frac{3}{4}\left(-\frac{4}{9}\right) $$

Short Answer

Expert verified
\(\frac{1}{3}\)

Step by step solution

01

Identify the Two Fractions

The first fraction is \(-\frac{3}{4}\) and the second fraction is \(-\frac{4}{9}\).
02

Determine the Product's Sign

Both fractions are negative. The product of two negative numbers is positive.
03

Multiply the Numerators

Multiply the numerators: \(-3\) and \(-4\). The result is: \(-3\times -4= 12\).
04

Multiply the Denominators

Multiply the denominators: \(4\) and \(9\). The result is: \(4 \times 9 = 36\).
05

Combine and Simplify the Fraction

Combine the results from Step 3 and Step 4 to get \(\frac{12}{36}\). Simplify \(\frac{12}{36}\) by dividing the numerator and the denominator by their greatest common divisor, which is 12. The simplified fraction is \(\frac{1}{3}\).

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Key Concepts

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

Negative Numbers
Let's start by understanding negative numbers. These are numbers less than zero, represented with a minus sign (-) in front of them. When you multiply two negative numbers together, the result is a positive number. For example, \(-1\times-1=+1\). This is because the negative signs cancel each other out. This rule is helpful when multiplying fractions, as seen in our exercise.
Numerators
In a fraction, the numerator is the top number. It tells you how many parts you have. For example, in the fraction \(\frac{3}{4}\), the numerator is 3. When multiplying fractions, you multiply the numerators together to get a new numerator. In our exercise, we have \(-3\) and \(-4\) as numerators. Following the multiplication steps: \(-3 \times -4 = 12\), which gives us the numerator for the resulting fraction.
Denominators
The denominator is the bottom number of a fraction. It indicates the total number of equal parts the whole is divided into. For instance, in \(\frac{3}{4}\), the denominator is 4. When multiplying fractions, you need to multiply the denominators together to get a new denominator. In our exercise, the denominators are 4 and 9. Thus, we multiply them: \4 \times 9 = 36\.
Simplifying Fractions
Simplifying fractions means making them as simple as possible. This involves finding the Greatest Common Divisor (GCD) of the numerator and the denominator and dividing both by this number. In our exercise, the fraction we get after multiplying is \(\frac{12}{36}\). The GCD of 12 and 36 is 12. Dividing both by 12, we get: \(\frac{12 \text{/} 12}{36 \text{/} 12} = \frac{1}{3}\). Therefore, \(\frac{12}{36}\) simplifies to \(\frac{1}{3}\). This gives us our final simplified fraction.

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