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Chapter 5: Problem 26

Find the ratio of the first number to the second and simplify. $$ 28 \text { to } 36 $$

Short Answer

Expert verified
The simplified ratio is 7 to 9.

Step by step solution

01

Write the numbers as a ratio

Represent the given numbers as a ratio. So, the ratio of 28 to 36 can be written as \[ \frac{28}{36} \]
02

Find the Greatest Common Divisor (GCD)

Determine the greatest common divisor (GCD) of the two numbers. The GCD of 28 and 36 is 4.
03

Simplify the ratio by dividing by the GCD

Divide both the numerator and the denominator by the GCD. \[ \frac{28 \div 4}{36 \div 4} = \frac{7}{9} \]
04

Write the simplified ratio

The simplified ratio of 28 to 36 is 7 to 9 or \[ \frac{7}{9} \]

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Key Concepts

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

Greatest Common Divisor (GCD)
To simplify a ratio, one crucial step is to find the Greatest Common Divisor (GCD). The GCD is the largest number that can evenly divide both given numbers. For example, in the ratio 28 to 36, we need to find the GCD of 28 and 36.
The factors of 28 are 1, 2, 4, 7, 14, and 28.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The common factors of 28 and 36 are 1, 2, and 4. The greatest of these common factors is 4, making it the GCD.
Using the GCD makes the process of simplifying fractions easier and helps to reduce the numbers to their simplest form.
Numerator and Denominator
In any ratio or fraction, the numerator is the top number, and the denominator is the bottom number. For the ratio 28 to 36, you can represent it as the fraction \(\frac{28}{36}\). Here, 28 is the numerator, and 36 is the denominator.
It is essential to understand these terms because simplifying a fraction involves working with both the numerator and the denominator.
When you divide both by their GCD, you make the numbers smaller while keeping the ratio the same. For instance, dividing both 28 and 36 by the GCD (which is 4) gives us: \(\frac{28 \/ 4}{36 \/ 4} = \frac{7}{9}\).
Thus, 7 is the new numerator, and 9 is the new denominator.
Fraction Simplification
Simplifying fractions is the process of reducing the numbers in the fraction to their simplest form. After identifying the GCD of the numerator and denominator, you divide both by that GCD.
Using the example \(\frac{28}{36}\), the GCD of 28 and 36 is 4.
Now, you simplify the fraction by dividing both the numerator and the denominator by 4:
\(\frac{28 \/ 4}{36 \/ 4} = \frac{7}{9}\).
Now \(\frac{7}{9}\) is in its simplest form because 7 and 9 have no common divisors other than 1.
This method ensures you maintain the original ratio's value but in a smaller and easier-to-use form.

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