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Chapter 6: Problem 106

Multiply or divide as indicated. Write each answer in lowest terms. $$ \frac{3}{7} \cdot \frac{4}{5} $$

Short Answer

Expert verified
The simplified fraction is \( \frac{12}{35} \).

Step by step solution

01

Multiply the Numerators

Multiply the numerators of the two fractions: 3 and 4. The result is: \[ 3 \times 4 = 12 \]
02

Multiply the Denominators

Multiply the denominators of the two fractions: 7 and 5. The result is: \[ 7 \times 5 = 35 \]
03

Write the Result as a Fraction

Combine the results of the previous steps to write the fraction: \[ \frac{12}{35} \]
04

Simplify the Fraction

Check if the fraction \( \frac{12}{35} \) can be simplified. Since 12 and 35 have no common factors other than 1, the fraction is already in its lowest terms.

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Key Concepts

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

Numerator
When working with fractions, the numerator is the top number. It tells you how many parts of the whole you have. For example, in the fraction \(\frac{3}{7}\), the numerator is 3. Understanding the numerator is crucial when multiplying fractions because you need to multiply the numerators of the fractions to get the numerator of the product. In our example, we multiply the numerators: \(3 \times 4 = 12\). This gives us the numerator of the product fraction, which is 12.
Denominator
The denominator is the bottom number of a fraction. It tells you into how many equal parts the whole is divided. For instance, in the fraction \(\frac{3}{7}\), the denominator is 7. When performing fraction multiplication, you multiply the denominators to get the denominator of the product. In our example, we multiply the denominators: \(7 \times 5 = 35\). This gives us the denominator of the product fraction, which is 35.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). For example, if we had the fraction \(\frac{12}{36}\), we could simplify it by dividing both the numerator and the denominator by 12 (their GCD). This results in \(\frac{12 \text{ 梅 } 12}{36 \text{ 梅 } 12} = \frac{1}{3} \). When you multiply two fractions, always check if the resulting fraction can be simplified to make it easier to understand and work with.
Lowest Terms
A fraction is in its lowest terms when the numerator and denominator have no common factors other than 1. This makes the fraction as simple as possible. For example, the fraction \(\frac{12}{35}\) is already in its lowest terms because 12 and 35 have no common divisors other than 1. You can verify this by checking the factors of both the numerator and the denominator. For 12, the factors are 1, 2, 3, 4, 6, and 12. For 35, the factors are 1, 5, and 7. Since the only common factor is 1, \(\frac{12}{35}\) is in its simplest form.

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Most popular questions from this chapter

Identify the following as an expression or an equation. Then simplify the expression or solve the equation. \(\frac{3}{5} x-\frac{7}{10} x\)

Consider the complex fraction given in Exercise 1: \(\frac{\frac{1}{2}-\frac{4}{3}}{\frac{1}{6}-\frac{5}{12}}\). Answer each part, outlining Method 2 for simplifying this complex fraction. (a) We must determine the LCD of all the fractions within the complex fraction. What is this LCD? (b) Multiply every term in the complex fraction by the LCD found in part (a), but do not yet combine the terms in the numerator and the denominator. (c) Combine the terms from part (b) to obtain the simplified form of the complex fraction.

Multiply or divide. Write each answer in lowest terms. $$ \frac{r^{2}+r s-12 s^{2}}{r^{2}-r s-20 s^{2}} \div \frac{r^{2}-2 r s-3 s^{2}}{r^{2}+r s-30 s^{2}} $$

Find the LCD for the fractions in each list. $$ \frac{12}{m-3}, \frac{-4}{3-m} $$

Write each rational expression in lowest terms. $$ \frac{16 x+8}{14 x+7} $$
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