Problem 17 - Basic College Mathematics

Question

Multiply and simplify. $$ \frac{7}{10} \cdot \frac{10}{7} $$

Step-by-Step Solution

Answer

1

Step 1: Write Down the Problem

Start by writing the given problem: \[ \frac{7}{10} \times \frac{10}{7} \]

Step 2: Multiply the Numerators

Multiply the numerators of both fractions together: \[ 7 \times 10 = 70 \]

Step 3: Multiply the Denominators

Multiply the denominators of both fractions together: \[ 10 \times 7 = 70 \]

Step 4: Form the Fraction

Combine the results of the numerators and denominators to form a single fraction: \[ \frac{70}{70} \]

Step 5: Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (which is 70 in this case): \[ \frac{70 \div 70}{70 \div 70} = \frac{1}{1} = 1 \]

Key Concepts

fraction multiplication

When we talk about multiplying fractions, the process is straightforward. Multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For example, in the given exercise, we have: \(\frac{7}{10} \times \frac{10}{7}\) To multiply, you perform the following steps:

• Multiply the numerators: 7 and 10. So, \( 7 \times 10 = 70 \)
• Multiply the denominators: 10 and 7. So, \( 10 \times 7 = 70 \)

Now you’ve got \( \frac{70}{70} \).

simplifying fractions

Simplifying fractions means reducing the fraction to its simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). In our example: When we have \( \frac{70}{70} \), we notice that both numerator and denominator are the same. That means their GCD is 70. So, we divide both by this number: \(\frac{70}{70} \div 70 = \frac{1}{1} = 1\) Fractions that have the same numerator and denominator simplify to 1. That’s because any number divided by itself is 1.

greatest common divisor (GCD)

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving any remainder. To find the GCD of 70 and 70, for instance, consider the following:

• The factors of 70 are 1, 2, 5, 7, 10, 14, 35, 70.
• Since both numbers are the same, their GCD is 70.

By dividing both the numerator and the denominator of \( \frac{70}{70} \) by 70, we simplify the fraction to \( \frac{1}{1} \) which equals 1.

basic arithmetic

Basic arithmetic skills are essential when working with fractions. You need to be comfortable with multiplication and division for both simplifying fractions and finding the GCD. Here’s a rundown:

• Multiplication: Used to multiply numerators and denominators.
• Division: Used to simplify fractions by dividing by the GCD.

By mastering these basic operations, you will find working with fractions simpler and more intuitive.