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Chapter 4: Problem 230

In the following exercises, simplify. $$\frac{4 \cdot 3}{6 \cdot 6}$$

Short Answer

Expert verified
The simplified form of \(\frac{4 \cdot 3}{6 \cdot 6}\) is \(\frac{1}{3}\).

Step by step solution

01

Multiply the Numerator

Multiply the two numbers in the numerator: \(4 \times 3 = 12\)
02

Multiply the Denominator

Multiply the two numbers in the denominator: \(6 \times 6 = 36\)
03

Create a Fraction and Simplify

Now replace the numerator and the denominator with the results from Steps 1 and 2: \(\frac{12}{36}\). Next, simplify the fraction. Find the greatest common divisor (GCD) of 12 and 36, which is 12. Then, divide both the numerator and the denominator by their GCD:\[\frac{12 \div 12}{36 \div 12} = \frac{1}{3}\]

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Key Concepts

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

multiplication
Multiplication is a basic arithmetic operation that combines two numbers to form a single product. It's one of the first math concepts students learn and is fundamental to understanding more complex problems.

In the given exercise, we perform two separate multiplications:
  • First, we multiply the numbers in the numerator: 4 and 3.
  • Secondly, we multiply the numbers in the denominator: 6 and 6.

Calculating these, we get:

  • \(4 \times 3 = 12\)
  • \(6 \times 6 = 36\)

This results in the fraction \(\frac{12}{36}\). Multiplication here helps us combine numbers to simplify later.
greatest common divisor
The Greatest Common Divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. It is crucial for simplifying fractions effectively.

To find the GCD of two numbers:
  • List out the factors of each number.
  • Identify the largest factor that appears in both lists.

For the example \(\frac{12}{36}\), the steps are:
  • Factors of 12: 1, 2, 3, 4, 6, 12
  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

So, the GCD of 12 and 36 is 12, as it is the largest number that appears in both lists.

This GCD helps reduce the fraction \(\frac{12}{36}\) to its simplest form.
fraction simplification
Fraction simplification is the process of reducing a fraction to its simplest form where the numerator and the denominator have no common factors other than 1.
Here are the steps:
  • Determine the GCD of the numerator and the denominator.
  • Divide both the numerator and the denominator by their GCD.

For the fraction \(\frac{12}{36}\), the GCD is 12. We divide both parts by 12:
\ \[\frac{12 \div 12}{36 \div 12} = \frac{1}{3}\ \]
So, \(\frac{12}{36}\) simplifies to \(\frac{1}{3}\). This means the fraction鈥檚 value stays the same but is now in the simplest form.

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Most popular questions from this chapter

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(x+\frac{2}{3}\) when (a)\(x=-\frac{1}{6} \quad\) (b) \(x=-\frac{5}{3}\)

In the following exercises, change to equivalent fractions using the given LCD. $$\frac{3}{8} \text { and } \frac{5}{6}, \quad L C D=24$$

In the following exercises, perform the indicated operation. $$ 5 \frac{8}{11}+2 \frac{4}{11} $$

In the following exercises, translate and solve. The difference of \(q\) and one-tenth is \(\frac{1}{2}\) .

In the following exercises, solve the equation. $$ h-\left(-\frac{7}{8}\right)=-\frac{2}{5} $$
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