Problem 26 - Basic College Mathematics

Question

Multiply. $$ \frac{21}{4} \cdot \frac{7}{5} $$

Step-by-Step Solution

Answer

\( \frac{147}{20} \)

Step 1 - Multiply the Numerators

To begin with, multiply the numerators of the given fractions. In this case, multiply 21 by 7. \[ 21 \times 7 = 147 \]

Step 2 - Multiply the Denominators

Next, multiply the denominators of the given fractions. Multiply 4 by 5. \[ 4 \times 5 = 20 \]

Step 3 - Form the New Fraction

Combine the results from the previous two steps to form a new fraction. \[ \frac{147}{20} \]

Step 4 - Simplify the Fraction

Check if the fraction can be simplified. In this case, 147 and 20 have no common factors other than 1, so the fraction \( \frac{147}{20} \) is already in simplest form.

Key Concepts

Numerator Multiplication

When you multiply fractions, you start with the numerators. The numerator is the top part of a fraction. To solve the problem \( \frac{21}{4} \cdot \frac{7}{5} \), begin by multiplying 21 and 7. This step is straightforward:

• Multiply 21 by 7

Doing the math: \[ 21 \times 7 = 147 \]
Now you have the numerator for your new fraction: 147.

Denominator Multiplication

Next, you need to multiply the denominators. The denominator is the bottom part of a fraction. Looking at the given fractions \( \frac{21}{4} \cdot \frac{7}{5} \), let's handle the denominators 4 and 5. Just like with the numerators, this step is simple:

• Multiply 4 by 5

Doing the math: \[ 4 \times 5 = 20 \]
Now you have the denominator for your new fraction: 20.

Basic Arithmetic Operations

Multiplying fractions is a basic arithmetic operation. It involves two primary steps:

• Numerator multiplication
• Denominator multiplication

Combining these operations forms a new fraction. After multiplying the numerators and the denominators, combine them:
\[ \frac{147}{20} \]
This shows the result of the multiplication.

Simplifying Fractions

The final step is to simplify the fraction if possible. To do this, find the greatest common divisor (GCD) of the numerator and the denominator. However, in this case, the numbers 147 and 20 have no common prime factors other than 1. Hence, \[ \frac{147}{20} \] is already simplified.
Here's a quick way to think about it:

• Check if both numbers have common factors.
• If they do, divide them by their GCD.
• If not, the fraction is already in simplest form.

So, \[ \frac{147}{20} \] remains unchanged because it's already as simple as it gets.