Problem 45 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$9 \cdot \frac{4}{7}$$

Step-by-Step Solution

Answer

The simplified answer is \( \frac{36}{7}\)

Step 1: Perform the Multiplication

Start by performing the multiplication operation as instructed. Multiply the whole number 9 and the numerator of the fraction (which is 4). This acts the same as \(9 \cdot \frac{4}{1}\) basically multiplying both numerator and denominator by 9. So the initial result becomes \( \frac{9 \cdot 4}{7} = \frac{36}{7}\)

Step 2: Simplify the Fraction

Once the multiplication has been performed, check if the result can be simplified. In this case, the final answer \( \frac{36}{7}\) is already in its simplest form. Thus, no more simplification needed

Key Concepts

Fraction Multiplication

When multiplying fractions, your goal is to multiply the numerators together and the denominators together. This process is straightforward and involves only a few simple steps. First, align the fractions side by side. Then, multiply the numerators (top numbers) with each other to get the new numerator. Similarly, multiply the denominators (bottom numbers) to get the new denominator.
For example, when multiplying \( \frac{4}{7} \times \frac{3}{5} \), multiply 4 and 3 to get 12, and 7 and 5 to get 35, resulting in the fraction \( \frac{12}{35} \). There's no need to find a common denominator, unlike fraction addition or subtraction, which makes fraction multiplication a bit simpler.

• Step 1: Multiply numerators.
• Step 2: Multiply denominators.

Remember, if you’re dealing with a whole number, like 9 in \( 9 \times \frac{4}{7} \), treat it as \( \frac{9}{1} \) to simplify your multiplication.

Simplifying Fractions

After you multiply fractions, it's a good practice to simplify the result whenever possible. Simplifying a fraction means finding an equivalent fraction where the numerator and denominator have no common factors other than 1. You can simplify by dividing both the numerator and the denominator by their greatest common divisor (GCD).
Imagine you have \( \frac{24}{36} \). The GCD of 24 and 36 is 12. So, divide both the numerator and the denominator by 12 to get \( \frac{2}{3} \). Ensure that both numbers are the smallest possible integers to represent that fraction.
In some cases, like \( \frac{36}{7} \), the fraction is already in its simplest form because 36 and 7 share no common factors. You know you're done simplifying when no number except 1 can evenly divide both the numerator and the denominator.

• Identify the GCD of both numbers.
• Divide the numerator and the denominator by the GCD.

Multiplying Whole Numbers with Fractions

When you multiply a whole number by a fraction, it's just as simple as multiplying two fractions. Treat the whole number as a fraction where the denominator is 1. This makes it easier to visualize and carry out the multiplication.
For example, to multiply 9 and \( \frac{4}{7} \), you can think of 9 as \( \frac{9}{1} \). Now multiply the numerators to get 36, and the denominators to get 7, resulting in \( \frac{36}{7} \).

• Rewrite the whole number as a fraction with a denominator of 1.
• Multiply across numerators and across denominators, just like with any fraction multiplication.
• Check to see if the resulting fraction can be simplified.

This method not only makes calculations simpler but also helps in understanding how numbers can change form without altering their value. Keep practicing, and soon it'll become second nature!