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Chapter 4: Problem 108

Multiply, and write the answer in simplified form. \(\frac{5}{12}\left(-\frac{8}{15}\right)\)

Short Answer

Expert verified
The simplified form is \(\frac{-2}{9}\).

Step by step solution

01

- Multiply the Numerators

Multiply the numerators of the fractions: \(5 \times -8 = -40\)
02

- Multiply the Denominators

Multiply the denominators of the fractions: \(12 \times 15 = 180\)
03

- Form the New Fraction

Combine the results from Steps 1 and 2 to create a new fraction: \(\frac{-40}{180}\)
04

- Simplify the Fraction

Find the greatest common divisor (GCD) of -40 and 180, which is 20.Divide both the numerator and the denominator by this GCD to simplify the fraction: \(\frac{-40 \div 20}{180 \div 20} = \frac{-2}{9}\)

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Key Concepts

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

Numerator and Denominator
Understanding the parts of a fraction is crucial before you start multiplying fractions. A fraction consists of two parts: the numerator and the denominator. The numerator is the number on the top, representing how many parts we have. The denominator is the number on the bottom, representing the total number of equal parts the whole is divided into.

For example, in the fraction \(\frac{5}{12}\), the numerator is 5 and the denominator is 12. When multiplying fractions, you multiply the numerators together and the denominators together.

Taking our example, \(\frac{5}{12} \) and \(\frac{-8}{15}\):
  • Multiply the numerators: 5 and -8 to get -40.
  • Multiply the denominators: 12 and 15 to get 180.

So the new fraction formed is \(\frac{-40}{180}\).
Simplifying Fractions
Simplifying fractions involves reducing them to their simplest form, where the numerator and the denominator are as small as possible, but still maintain the same value.

You'll need to find the Greatest Common Divisor (GCD) of the numerator and the denominator to simplify a fraction. The GCD is the largest number that evenly divides both numbers. Once you find it, you divide both the numerator and the denominator by this GCD.

For the fraction \(\frac{-40}{180}\):
  • The GCD of 40 and 180 is 20.
  • Divide both the numerator and the denominator by 20.
  • Simplified fraction: \(\frac{-40 \div 20}{180 \div 20}\) = \(\frac{-2}{9}\).

This method ensures that the fraction is as simple as possible.
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) plays a key role in simplifying fractions. It is the largest number that can perfectly divide both the numerator and the denominator without leaving a remainder.

To find the GCD, you can use several methods, such as listing factors or using the Euclidean algorithm.

For instance, to find the GCD of 40 and 180:
  • List out the factors of each number. For 40, factors are 1, 2, 4, 5, 8, 10, 20, 40. For 180, factors are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 60, 90, 180.
  • Identify the common factors. Here, the common factors are 1, 2, 5, 10, 20.
  • The greatest of these common factors is 20, which is the GCD.

Using the GCD to simplify \(\frac{-40}{180}\):
  • Divide the numerator and the denominator by 20.
  • \(\frac{-40 \div 20}{180 \div 20} \) simplifies to \(\frac{-2}{9}\).

This gives us the fraction in its simplest form.

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Most popular questions from this chapter

In the following exercises, simplify. $$ \frac{2}{d}+\frac{9}{d} $$

In the following exercises, add. $$\frac{15}{32}+\frac{9}{32}$$

In the following exercises, simplify. $$ -\frac{3}{10}+\left(-\frac{5}{8}\right) $$

In the following exercises, subtract. $$-\frac{31}{30}-\frac{7}{30}$$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(x-\frac{1}{3}\) when (a) \(x=\frac{2}{3} \quad\) (b) \(x=-\frac{2}{3}\)
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