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Chapter 5: Problem 409

In the following exercises, write each ratio as a fraction. $$84 \text { to } 36$$

Short Answer

Expert verified
The ratio 84 to 36 as a fraction is \( \frac{7}{3} \).

Step by step solution

01

Identify the terms in the ratio

The given ratio is 84 to 36. Identify the two numbers in the ratio: 84 and 36.
02

Write the ratio as a fraction

Write the ratio as a fraction, with the first term as the numerator and the second term as the denominator: \ \[ \frac{84}{36} \]
03

Simplify the fraction

Simplify the fraction by finding the greatest common divisor (GCD) of 84 and 36. The GCD of 84 and 36 is 12. Divide both the numerator and the denominator by their GCD: \ \[ \frac{84 \div 12}{36 \div 12} = \frac{7}{3} \]
04

Write the simplified fraction

The simplified form of the ratio is \( \frac{7}{3} \).

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Key Concepts

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

simplifying fractions
Simplifying fractions means to make the fraction as simple as possible. This involves reducing the fraction to its smallest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).
It's essential because it makes fractions easier to understand and compare.

For example, to simplify the fraction \[ \frac{84}{36} \], we need to find their GCD, which is 12. By dividing both the numerator (84) and the denominator (36) by 12, we get \[ \frac{7}{3} \].
This process ensures the fraction is in its simplest form.
greatest common divisor
The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that can evenly divide two given numbers. Finding the GCD is crucial for simplifying fractions.
Here is how you can calculate the GCD:
  • List the factors of both numbers.
  • Identify the highest common factor they share.

For instance, to find the GCD of 84 and 36, the factors are:
- Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

The largest common factor is 12, making the GCD of 84 and 36 equal to 12.
numerator and denominator
In a fraction, the numerator is the top number, and the denominator is the bottom number.
Both parts play essential roles in the fraction's value.
  • The numerator indicates how many parts are taken.
  • The denominator shows how many parts make up a whole.

For example, in the fraction \[ \frac{84}{36} \], 84 is the numerator and 36 is the denominator. After simplifying, we get the fraction \[ \frac{7}{3} \], where 7 is the numerator, and 3 is the denominator.

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