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Chapter 6: Problem 52

Write each fraction in simplest form. $$\frac{27}{18}$$

Short Answer

Expert verified
The simplest form of \(\frac{27}{18}\) is \(\frac{3}{2}\).

Step by step solution

01

Find the GCF

To simplify a fraction, we need to find the greatest common factor (GCF) of the numerator (27) and the denominator (18). We do this by listing the factors of each number. Factors of 27 are 1, 3, 9, and 27. Factors of 18 are 1, 2, 3, 6, 9, and 18. The largest common factor is 9.
02

Divide by the GCF

Now that we have the GCF, which is 9, divide both the numerator and the denominator by the GCF.\[\frac{27}{18} = \frac{27 \div 9}{18 \div 9} = \frac{3}{2}\]
03

Confirm Fraction is Simplified

To ensure the fraction \(\frac{3}{2}\) is in simplest form, check if there are any common factors between 3 and 2 besides 1. Since there are none, \(\frac{3}{2}\) is already in its simplest form.

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Key Concepts

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

Greatest Common Factor (GCF)
When simplifying fractions, the first step is often to find the greatest common factor, or GCF. The GCF is the largest number that can divide two numbers without leaving a remainder. Identifying the GCF is crucial because it helps us reduce fractions to their simplest form.

For example, if we look at the fraction \(\frac{27}{18}\), we begin by listing the factors:
  • Factors of 27: 1, 3, 9, 27
  • Factors of 18: 1, 2, 3, 6, 9, 18
When we compare these factor lists, the largest common factor is 9. This means 9 is the greatest number that evenly divides both 27 and 18.

Once you find the GCF, you use it to simplify the fraction by dividing both the numerator and the denominator by this factor.
Numerator and Denominator
A fraction consists of two key parts: the numerator and the denominator. Understanding these components is essential. So what do they mean?

The **numerator** is the top number in a fraction. It signifies how many parts of the whole are being considered. In our example \(\frac{27}{18}\), the numerator is 27, indicating there are 27 parts being looked at.

The **denominator**, on the other hand, is the bottom number in a fraction. It defines how many equal parts the whole is divided into. In \(\frac{27}{18}\), the denominator is 18, suggesting that the whole is divided into 18 parts.

To simplify a fraction, we need to divide both the numerator and denominator by the GCF, which reduces the fraction to a simpler, but equivalent, form.
Simplest Form
After finding the GCF and dividing both parts of the fraction by it, you'll achieve what is called the simplest form. A fraction is in its simplest form when the numerator and the denominator can no longer be divided by any common factor except for 1.

When we simplify \(\frac{27}{18}\) by dividing both numbers by their GCF, 9, we get \(\frac{3}{2}\). This fraction is in simplest form because there are no common factors between 3 and 2 other than 1.

Being in simplest form makes it easier to understand and work with fractions. Simplified fractions are clear, reduce calculation errors, and provide an accurate representation of the portion of the whole being described.

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Most popular questions from this chapter

The length of a rectangle is 7 centimeters less than twice its width. Its area is 30 square centimeters. Find the dimensions of the rectangle.

Factor completely. See Examples 1 through 7 . \(12 x^{3}+11 x^{2}+2 x\)

The equation \(D=\frac{1}{2} n(n-3)\) gives the number of diagonals \(D\) for a polygon with \(n\) sides. For example, a polygon with 6 sides has \(D=\frac{1}{2} \cdot 6(6-3)\) or \(D=9\) diagonals. (See if you can count all 9 diagonals. Some are shown in the figure.) Use this equation, \(D=\frac{1}{2} n(n-3).\) (GRAPH CANNOT COPY) Find the number of diagonals for a polygon that has 12 sides.

Solve. A hang glider pilot accidentally drops her compass from the top of a 400 -foot cliff. The height \(h\) of the compass after \(t\) seconds is given by the quadratic equation \(h=-16 t^{2}+400\) When will the compass hit the ground? (IMAGE CANNOT COPY)

Solve each equation. First, multiply the binomial(s). To solve \((x-6)(2 x-3)=(x+2)(x+9),\) see below. $$\begin{aligned} (x-6)(2 x-3) &=(x+2)(x+9) \\ 2 x^{2}-15 x+18 &=x^{2}+11 x+18 \\ x^{2}-26 x &=0 \\ x(x-26) &=0 \\ x=0 \quad \text { or } \quad x-26 &=0 \\ x &=26 \end{aligned}$$ $$ (2 x-3)(x+6)=(x-9)(x+2) $$
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