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Chapter 5: Problem 2

Write ratio as a fraction and simplify. 18 to 54

Short Answer

Expert verified
The ratio 18 to 54 as a simplified fraction is \(\frac{1}{3}\).

Step by step solution

01

Understand the Ratio

The given ratio is 18 to 54. A ratio compares two quantities.
02

Write the Ratio as a Fraction

Convert the ratio 18 to 54 into a fraction: \ \[ \frac{18}{54} \]
03

Find the Greatest Common Divisor (GCD)

Identify the greatest common divisor of 18 and 54. The GCD of 18 and 54 is 18.
04

Simplify the Fraction

Divide the numerator and the denominator of the fraction by their GCD: \ \[ \frac{18 \div 18}{54 \div 18} = \frac{1}{3} \]

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Key Concepts

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

ratios in mathematics
A ratio is a way to compare two quantities. It shows how much of one thing there is compared to another. For example, the ratio 18 to 54 means that for every 18 units of one quantity, there are 54 units of another.
A key point to remember about ratios is that they can be written in different forms:
  • Using a colon, like 18:54
  • As a fraction, like \( \frac{18}{54} \)
When dealing with ratios, it's often helpful to simplify them to their simplest form. This makes them easier to understand and compare.
greatest common divisor
The greatest common divisor (GCD) is the largest number that can exactly divide two or more numbers without leaving a remainder. Finding the GCD is crucial in simplifying fractions, as you will see.
For example, in the ratio 18 to 54:
  • First, list the factors of each number.
  • Factors of 18: 1, 2, 3, 6, 9, 18
  • Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
The greatest common factor these two lists share is 18. This means 18 is the GCD of 18 and 54, essential when simplifying the fraction \( \frac{18}{54} \).
simplifying fractions
Simplifying fractions makes them easier to understand and work with. Simplifying involves dividing the numerator and the denominator by their greatest common divisor (GCD).
Let's simplify \( \frac{18}{54} \):
  • First, identify the GCD, which in this case is 18.
  • Next, divide both the numerator and the denominator by 18:
\[ \frac{18 \div 18}{54 \div 18} = \frac{1}{3} \]
After simplifying, the fraction \( \frac{18}{54} \) becomes \( \frac{1}{3} \). This means that 18 to 54 in its simplest form is 1 to 3. Simplifying ratios through fractions helps to compare quantities easily.

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