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Chapter 4: Problem 103

Multiply, and write the answer in simplified form. \(-\frac{2}{3}\left(-\frac{3}{8}\right)\)

Short Answer

Expert verified
\( \frac{1}{4} \)

Step by step solution

01

Multiply the Numerators

Multiply the numerators of the two fractions together. The numerators are \(-2\) and \(-3\). Thus, the multiplication of the numerators is \(-2 \times -3 = 6\).
02

Multiply the Denominators

Multiply the denominators of the two fractions together. The denominators are \3\ and \8\. Thus, the multiplication of the denominators is \( 3 \times 8 = 24 \).
03

Form the New Fraction

Combine the results of steps 1 and 2 to form a new fraction with the numerator and denominator obtained. Thus, the new fraction is \( \frac{6}{24} \).
04

Simplify the Fraction

Simplify the fraction \( \frac{6}{24} \) by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 6 and 24 is 6, so \( \frac{6 \div 6}{24 \div 6} = \frac{1}{4} \). The simplified form of the fraction is \( \frac{1}{4} \).

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Key Concepts

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

Multiply Fractions
To multiply fractions, you follow a simple process. When you have two fractions \(\frac{a}{b} \times \frac{c}{d}\), you just multiply the numerators (top numbers) together and the denominators (bottom numbers) together.
So the product of \-\(\frac{2}{3}\) and \-\(\frac{3}{8}\) is:
Numerator: \(-2 \times -3 = 6\)
Denominator: \(3 \times 8 = 24\)
Therefore, the result is: \(\frac{6}{24}\).
Remember, always handle negative signs carefully. Multiplying two negative numbers, as shown here, gives a positive result.
Simplify Fractions
After multiplication, you typically need to simplify the fraction. Simplifying means reducing it to its lowest terms.
To simplify \(\frac{6}{24}\), you need to find a factor that both the numerator and denominator share. Here, it helps to know the Greatest Common Divisor (GCD).
Simplifying fractions involves dividing the numerator and the denominator by their GCD.
In this case, both 6 and 24 share a GCD of 6:
\(\frac{6 \text{ 梅 } 6}{24 \text{ 梅 } 6} = \frac{1}{4}\)
So, \(\frac{6}{24}\) simplifies to \(\frac{1}{4}\).
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) is the largest number that can exactly divide both the numerator and the denominator.
Finding the GCD can be done using the Euclidean algorithm, but for simple fractions, you can usually identify it through common sense or by listing out factors.
For \(\frac{6}{24}\), the factors of 6 are 1, 2, 3, and 6. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
The largest common factor is 6.
By dividing the numerator and denominator by this GCD, the fraction becomes simplified. Thus, GCD helps in keeping the final fraction in its simplest form, making it more understandable and easier to work with.

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Most popular questions from this chapter

In the following exercises, simplify. $$ \frac{9 a}{10} \div \frac{15 a}{8} $$

In the following exercises, find the least common denominator. $$\frac{3}{4}, \frac{1}{6}, \text { and } \frac{5}{10}$$

In the following exercises, simplify. $$ \frac{5}{20} $$

In the following exercises, perform the indicated operation. $$ 6 \frac{2}{5}+7 \frac{3}{5} $$

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