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Chapter 11: Problem 46

Simplify. $$\left(-\frac{3}{4}\right)\left(\frac{3 y}{-5}\right)$$

Short Answer

Expert verified
\( \frac{9y}{20} \)

Step by step solution

01

Multiply the Numerators

Multiply the numerators from both fractions, which are -3 and 3y. This will result in -9y.
02

Multiply the Denominators

Next, multiply the denominators from both fractions, which are 4 and -5. This will result in -20.
03

Simplify the Fraction

You should note that the product of two negatives is a positive. Therefore, even though you're left with -9y over -20, the two negative signs cancel out, resulting in a final answer of \( \frac{9y}{20} \).

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Key Concepts

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

Multiplying Fractions
When multiplying fractions, the process is straightforward and involves two main steps. First, you multiply the numerators, which are the top numbers in the fraction. Second, you multiply the denominators, which are the bottom numbers.

In this particular exercise, we start with the expression \(-\frac{3}{4}\) and \(\frac{3y}{-5}\). To find the product:
  • Multiply the numerators: \(-3\times 3y = -9y\)
  • Multiply the denominators: \(4\times -5 = -20\)
After performing these calculations, we end up with the new fraction \(\frac{-9y}{-20}\). The job isn't done yet, as we'll need to simplify it further in the next steps.
Numerators and Denominators
In any fraction, the numerator is the number above the fraction bar, and the denominator is the number below the fraction bar. They are key in performing operations with fractions.

For the expression \(-\frac{3}{4}\times\frac{3y}{-5}\), the numerators \(-3\) and \(3y\) are multiplied to give \(-9y\). Similarly, the denominators \(4\) and \(-5\) multiply to give \(-20\).

This showcases that regardless of the complexity of the algebraic expressions, you always follow the same rule: numerator times numerator and denominator times denominator. This consistent approach makes it easier to handle even more complex fraction problems in algebra.
Simplification Process
The simplification process involves making a fraction as simple as possible. Often, this means reducing it to its simplest form.

In multiplication of fractions, especially with variables, it is important to notice the sign of products. Here we have \(\frac{-9y}{-20}\), which initially has negative numbers for both numerator and denominator. Since a negative divided by a negative is a positive, they cancel out each other.

Thus, the simplified form of the fraction is \(\frac{9y}{20}\). Simplifying ensures that the expression is neat and easy to use in further calculations or evaluations.

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