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Chapter 1: Problem 48

Simplify. $$ \frac{75}{80} $$

Short Answer

Expert verified
The simplified fraction is \(\frac{15}{16}\).

Step by step solution

01

Determine the Greatest Common Divisor (GCD)

To simplify the fraction \(\frac{75}{80}\), find the Greatest Common Divisor (GCD) of 75 and 80. The GCD is the largest number that divides both 75 and 80 without leaving a remainder. The GCD of 75 and 80 is 5.
02

Divide the Numerator and the Denominator by the GCD

Divide both the numerator (75) and the denominator (80) by their GCD (5): \(\frac{75 \div 5}{80 \div 5} = \frac{15}{16}\)
03

Write the Simplified Fraction

The fraction after simplification is \(\frac{15}{16}\).

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Key Concepts

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

greatest common divisor
When simplifying fractions, the first important step is to find the Greatest Common Divisor (GCD). The GCD is the largest number that can divide both the numerator and the denominator without any remainder. For example, in the fraction \(\frac{75}{80}\), we need to find the GCD of 75 and 80.
In this case, the GCD is 5. So, when looking at fractions, always start by determining the GCD.
numerator
The numerator is the top part of a fraction. It indicates how many parts of a whole are being considered. For the fraction \(\frac{75}{80}\), the numerator is 75.
To simplify fractions, you need to divide the numerator by the GCD. By dividing 75 by the GCD (which is 5), you get 15.
The simplified numerator helps you in reducing the fraction to its simplest form.
denominator
The denominator forms the bottom part of a fraction and tells you into how many equal parts the whole is divided. In the fraction \(\frac{75}{80}\), the denominator is 80.
Just like the numerator, you divide the denominator by the GCD to simplify the fraction. Dividing 80 by the GCD (which is 5), you get 16.
Simplifying the denominator is crucial to arriving at the fraction's simplest form.
fraction
A fraction represents a part of a whole and is written in the form of \(\frac{numerator}{denominator}\). It shows how many parts of a certain size are there.
For example, \(\frac{75}{80}\) begins as our initial fraction. Using the steps outlined:
  • Finding the GCD of 75 and 80 (which is 5).
  • Dividing both the numerator (75) by 5 to get 15.
  • Dividing the denominator (80) by 5 to get 16.
This means that the simplified fraction of \(\frac{75}{80}\) end up as \(\frac{15}{16}\), illustrating fewer but larger parts of the same whole.

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