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Chapter 4: Problem 594

In the following exercises, multiply. $$\frac{2}{9} \cdot\left(-\frac{45}{32}\right)$$

Short Answer

Expert verified
\(\frac{-5}{16}\)

Step by step solution

01

Multiply the numerators

Multiply the numerators of the fractions together: the numerator of the first fraction is 2, and the numerator of the second fraction is -45. Therefore, \(2 \times (-45) = -90\).
02

Multiply the denominators

Multiply the denominators of the fractions together: the denominator of the first fraction is 9, and the denominator of the second fraction is 32. Therefore, \(9 \times 32 = 288\).
03

Put the fractions together

Combine the results from Step 1 and Step 2 to form a new fraction: \(\frac{-90}{288}\).
04

Simplify the fraction

Simplify the fraction \(\frac{-90}{288}\) by finding the greatest common divisor (GCD) of 90 and 288. Let's find the GCD: Both 90 and 288 can be divided by 18.Therefore,\(\frac{-90 \text{梅} 18}{288 \text{梅} 18} = \frac{-5}{16}\).

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Key Concepts

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

Multiply Fractions
When multiplying fractions, the process is straightforward. You simply multiply the numerators (top numbers) together and then multiply the denominators (bottom numbers) together. This operation gives you a new fraction.

For example, in the exercise \(\frac{2}{9} \cdot \left(-\frac{45}{32}\right)\), we first multiply the numerators: 2 and -45.
So, \(2 \cdot (-45) = -90\).
Next, we multiply the denominators: 9 and 32.
The operation looks like this: \(9 \cdot 32 = 288\).

Combining these results gives us the new fraction \(\frac{-90}{288}\). This is the product of the two original fractions.
Simplify Fractions
Simplifying fractions means reducing them to their simplest form where the numerator and denominator are as small as possible. You do this by finding common factors of both the numerator and the denominator.

For our fraction \(\frac{-90}{288}\), we need to find a number that can evenly divide both 90 and 288. This will reduce the fraction.

Once we find this number, we divide both the numerator and the denominator by this common factor. In this case, we found the greatest common divisor (GCD) to be 18. Therefore, we divide both -90 and 288 by 18:
\(\frac{-90 \text{梅} 18}{288 \text{梅} 18} = \frac{-5}{16}\).

As a result, \(\frac{-5}{16}\) is the simplified form of \(\frac{-90}{288}\).
Greatest Common Divisor
The greatest common divisor (GCD), also known as the greatest common factor (GCF), of two numbers is the largest number that can evenly divide both numbers without leaving a remainder. It is crucial in simplifying fractions.

To find the GCD, you can use several methods, such as listing out all the factors of both numbers and finding the largest one common to both.
  • List the factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.
  • List the factors of 288: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288.

We see that the largest common factor between 90 and 288 is 18. Thus, the GCD is 18.

This method simplifies the fraction by the greatest possible factor, ensuring it's in the simplest form. Therefore, dividing both the numerator and the denominator by the GCD鈥�18鈥攇ives us a simplified answer.

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Most popular questions from this chapter

In the following exercises, multiply and write the answer in simplified form. $$\frac{15}{22} \cdot 3 \frac{3}{5}$$

In the following exercises, rewrite the improper fraction as a mixed number. $$\frac{28}{9}$$

Translate Sentences to Equations and Solve In the following exercises, translate to an algebraic equation and solve. \(m\) divided by 4 equals negative \(6 .\)

In the following exercises, add. $$\frac{3}{8}+\frac{2}{8}$$

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