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Chapter 1: Problem 400

Simplify. $$ \frac{6 \cdot 6}{9 \cdot 2} $$

Short Answer

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2

Step by step solution

01

Write the Expression

The given expression is \( \frac{6 \times 6}{9 \times 2} \).
02

Perform Multiplication

Multiply the numbers in the numerator and the denominator. So, the expression becomes \( \frac{36}{18} \).
03

Simplify the Fraction

Find the greatest common divisor (GCD) of 36 and 18, which is 18. Divide both the numerator and the denominator by their GCD, giving \( \frac{36 \text{/} 18}{18 \text{/} 18} = \frac{2}{1} \).
04

Simplify the Result

Since \( \frac{2}{1} \) is simply 2, the simplified expression is 2.

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Key Concepts

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

Fractions
Understanding fractions is essential for solving many math problems, including the one in our exercise. A fraction represents a part of a whole. It has two parts: a numerator (the top part) and a denominator (the bottom part). For instance, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator.
Fractions can be simplified to make them easier to understand and work with. Simplifying a fraction means reducing it to its smallest form while keeping its value the same. This is often achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD).
Greatest Common Divisor
The greatest common divisor (GCD) is a key concept when simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. For example, the GCD of 8 and 12 is 4, because 4 is the highest number that can divide both evenly.
To find the GCD, you can use methods such as listing out the factors of each number and choosing the largest one they have in common. In our exercise, we needed to simplify the fraction \(\frac{36}{18}\). First, we find the GCD of 36 and 18, which is 18. Then we divide both the numerator and the denominator by 18 to simplify the fraction to \(\frac{2}{1}\), which is simply 2.
Multiplication
Multiplication involves combining equal groups to find a total. In the context of fractions, you multiply the numerators together and the denominators together. For example, when you have \(\frac{6 \times 6}{9 \times 2}\), you perform the multiplication step-by-step:
- Multiply the numerators: 6 * 6 = 36
- Multiply the denominators: 9 * 2 = 18
Thus, the fraction becomes \(\frac{36}{18}\).
.
After performing the multiplication, we then simplify the resultant fraction by finding the GCD of the numerator and the denominator, and dividing both by that GCD.

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