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Chapter 4: Problem 105

Multiply, and write the answer in simplified form. \(-\frac{5}{9} \cdot \frac{3}{10}\)

Short Answer

Expert verified
-\frac{1}{6}.

Step by step solution

01

- Multiply the Numerators

To multiply two fractions, start by multiplying their numerators. The numerator of the first fraction \(-\frac{5}{9}\) is -5, and the numerator of the second fraction \(\frac{3}{10}\) is 3. Multiply these numerators together: \(-5 \cdot 3 = -15\).
02

- Multiply the Denominators

Next, multiply the denominators of the two fractions. The denominator of the first fraction \(-\frac{5}{9}\) is 9, and the denominator of the second fraction \(\frac{3}{10}\) is 10. Multiply these denominators together: \(\frac{9 }{\times 10}\rightarrow 90\).
03

- Form the New Fraction

Create a new fraction using the results from Steps 1 and 2. The new fraction will have the numerator -15 and the denominator 90: \(-\frac{15}{90}\).
04

- Simplify the Fraction

To simplify the fraction \(-\frac{15}{90}\), divide both the numerator and denominator by their greatest common divisor (GCD). The GCD of 15 and 90 is 15. Dividing both by 15 yields: \(-\frac{15 \div 15}{90 \div 15} = -\frac{1}{6}\).

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Key Concepts

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

Numerators
When working with fractions, the **numerator** is the number above the fraction line. In a fraction like \(-\frac{5}{9}\), -5 is the numerator. It tells us how many parts of the denominator we are dealing with. For instance, in the exercise problem, we need to multiply the numerators of two fractions together. In this step, the numerators are -5 and 3. To multiply them, you simply do \(-5 \times 3 = -15\). This result gives us the numerator for our new fraction.
Denominators
The **denominator** is the number below the fraction line. It shows us into how many equal parts the whole is divided. For instance, in \(\frac{3}{10}\), 10 is the denominator. In the given exercise, we multiply the denominators of the two fractions. Here, the denominators are 9 and 10. By multiplying these values, you get \((9 \times 10 = 90)\). This gives us the new fraction's denominator. Combining the new numerator and the new denominator from Step 1, we get \(-\frac{15}{90}\).
Simplifying Fractions
Once you have your new fraction, you'll often need to **simplify** it. This means making it as simple as possible. To do this, you divide both the numerator and the denominator by their greatest common divisor (**GCD**). Simplifying helps in making the fraction easier to understand and work with. In our exercise, the fraction \(-\frac{15}{90}\) needs to be simplified. The GCD of 15 and 90 is 15. You divide the numerator and the denominator by 15: \(\frac{15 \text{ (numerator) 梅 15}}{90 \text{ (denominator) 梅 15}} = -\frac{1}{6}\). Now, our fraction is simplified to \(-\frac{1}{6}\). Always remember to look for the GCD to simplify any fraction you encounter.

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Most popular questions from this chapter

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(x+\frac{1}{2}\) when (a) \(x=-\frac{1}{8} \quad\) ( b) \(x=-\frac{1}{2}\)

In the following exercises, perform the indicated operations and simplify. $$\frac{4}{11} \div \frac{2}{7 d}$$

In the following exercises, solve the equation. $$ y+\frac{3}{5}=\frac{7}{5} $$

In the following exercises, perform the indicated operations and simplify. $$-\frac{9}{10}-\frac{3}{4}$$

In the following exercises, add. $$\frac{2}{5}+\frac{1}{5}$$
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