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Chapter 4: Problem 237

In the following exercises, simplify. $$\frac{12 \cdot 9-3^{2}}{3 \cdot 18}$$

Short Answer

Expert verified
\(\frac{11}{6}\).

Step by step solution

01

- Simplify the numerator

Calculate the operations in the numerator: First, calculate the power \(3^2 = 9\). Next, perform the multiplication \(12 \cdot 9 = 108\). Finally, subtract \(108 - 9 = 99\). So, the numerator simplifies to 99.
02

- Simplify the denominator

Calculate the operations in the denominator: Perform the multiplication \(3 \cdot 18 = 54\). So, the denominator simplifies to 54.
03

- Simplify the fraction

Combine the simplified numerator and denominator: \(\frac{99}{54}\). Next, find the greatest common divisor (GCD) of 99 and 54, which is 9. Divide both the numerator and the denominator by 9: \(\frac{99 \div 9}{54 \div 9} = \frac{11}{6}\). Therefore, the simplified fraction is \(\frac{11}{6}\).

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Key Concepts

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

Numerator and Denominator
A fraction is made up of two parts, the numerator and the denominator.
The **numerator** is the top number of the fraction and represents how many parts we have.
The **denominator** is the bottom number of the fraction and shows into how many parts the whole is divided.
For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator.
In our exercise, after simplifying the original expression, the fraction becomes \(\frac{99}{54}\), where 99 is the numerator, and 54 is the denominator.
Understanding the roles of the numerator and the denominator is crucial for simplifying fractions.
Greatest Common Divisor (GCD)
To simplify a fraction, it's useful to understand the **Greatest Common Divisor (GCD)**.
The GCD of two numbers is the largest number that divides both of them without leaving a remainder.
Finding the GCD helps us reduce fractions to their simplest form.
For example, the numbers 99 and 54 both share several common divisors, but their greatest is 9.
Here鈥檚 a quick way to find the GCD using the Euclidean algorithm:
  • Divide the larger number by the smaller number.
  • Take the remainder and divide the previous smaller number by this remainder.
  • Repeat the process until you get a remainder of 0.
  • What you get as the last divisor before the remainder turns 0 is the GCD.
For our exercise, we computed that the GCD of 99 and 54 is 9.
Dividing both the numerator and the denominator by their GCD simplifies the fraction.
\[ \frac{99}{54} = \frac{99 \div 9}{54 \div 9} = \frac{11}{6} \]
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form where the numerator and the denominator are as small as possible.
It makes fractions easier to understand and work with.
Let鈥檚 simplify \(\frac{99}{54}\) step by step:
  1. Identify the numerator (99) and the denominator (54).
  2. Find the GCD of 99 and 54, which is 9.
  3. Divide both the numerator and the denominator by the GCD.
Dividing both numbers by their GCD: \[ \frac{99}{54} = \frac{99 \div 9}{54 \div 9} = \frac{11}{6} \] So, \(\frac{11}{6}\) is the simplified form of \(\frac{99}{54}\).
Simplifying makes fractions less complex and easier to use in further math problems or real-life applications.

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Most popular questions from this chapter

In the following exercises, simplify. $$12 \left(\frac{9}{20}-\frac{4}{15}\right)$$

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