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Chapter 5: Problem 36

Find the ratio of the first number to the second and simplify. $$ 232 \text { to } 116 $$

Short Answer

Expert verified
2:1

Step by step solution

01

Write the Ratio

The given numbers are 232 and 116. Write the ratio as a fraction: \ \ \[ \frac{232}{116} \]
02

Simplify the Fraction

To simplify the fraction, find the greatest common divisor (GCD) of 232 and 116. The GCD of 232 and 116 is 116.
03

Divide the Numerator and Denominator by the GCD

Divide both the numerator and denominator by 116: \ \ \[ \frac{232 \div 116}{116 \div 116} = \frac{2}{1} \]
04

Write the Simplified Ratio

The simplified ratio of 232 to 116 is 2:1.

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Key Concepts

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

Fractions
A fraction represents a part of a whole. It's usually written as \(\frac{a}{b}\), where \(a\) is the numerator (the top part) and \(b\) is the denominator (the bottom part). In the context of ratios, we can use fractions to compare two quantities.
To convert a ratio into a fraction, you simply place the first number as the numerator and the second number as the denominator. For example, the ratio 232 to 116 can be written as the fraction \(\frac{232}{116}\). This makes it easier to work with the numbers and to see if the fraction can be simplified.
If both the numerator and denominator share common factors, we can reduce the fraction, making it simpler and easier to understand.
Greatest Common Divisor
The greatest common divisor (GCD) of two numbers is the largest number that can divide both of them without leaving a remainder. Finding the GCD is essential in simplifying fractions.
To find the GCD of two numbers like 232 and 116, you can use various methods, including prime factorization, Euclidean algorithm, or simply listing the factors of both numbers. For example:
  • Prime factorization: Break down each number into prime factors and find the common ones.
  • Euclidean algorithm: This involves repeated division to find the GCD.
In our example, the GCD of 232 and 116 is 116. This means that 116 is the largest number that divides both 232 and 116 evenly.
Simplified Ratio
A simplified ratio is the easiest form of a ratio. This involves dividing both parts of the ratio by their greatest common divisor (GCD).
In our example, starting with the ratio 232 to 116, which we converted into the fraction \(\frac{232}{116}\), we found that the GCD of 232 and 116 is 116. Dividing both the numerator and the denominator by the GCD simplifies the fraction from \(\frac{232}{116}\) to \(\frac{2}{1}\).
This means the simplified ratio of 232 to 116 is 2:1. Always simplify ratios to make them easier to understand and compare.

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