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Chapter 3: Problem 13

Subtract and simplify. $$ \frac{3}{4}-\frac{3}{20} $$

Short Answer

Expert verified
\[ \frac{3}{4} - \frac{3}{20} = \frac{3}{5} \]

Step by step solution

01

Find a common denominator

To subtract two fractions, the denominators need to be the same. The least common multiple (LCM) of 4 and 20 is 20.
02

Convert fractions to have the same denominator

Rewrite \frac{3}{4} to have the denominator 20. Multiply both the numerator and the denominator by 5:\[ \frac{3}{4} \times \frac{5}{5} = \frac{15}{20} \]Now, \frac{3}{4} is \frac{15}{20}.
03

Perform the subtraction

Subtract the second fraction \frac{3}{20} from the converted first fraction \frac{15}{20}:\[ \frac{15}{20} - \frac{3}{20} = \frac{15 - 3}{20} = \frac{12}{20} \]
04

Simplify the result

Find the greatest common divisor (GCD) of 12 and 20, which is 4. Then divide both the numerator and the denominator by 4:\[ \frac{12}{20} = \frac{12 \div 4}{20 \div 4} = \frac{3}{5} \]

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Key Concepts

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

Common Denominator
When dealing with fractions, a common denominator is essential for addition or subtraction. It ensures that the fractions are part of the same whole. For example, to subtract \(\frac{3}{4} - \frac{3}{20}\), we need both fractions to have the same denominator. This common denominator allows us to compare and subtract the fractions directly.
Least Common Multiple
To find a common denominator, we often use the least common multiple (LCM). The LCM of two numbers is the smallest multiple that they both share. In our example, to subtract \(\frac{3}{4} - \frac{3}{20}\), we look at the denominators 4 and 20. The LCM of 4 and 20 is 20, as 20 is the smallest number both can divide into without a remainder. Once we have the LCM, we can rewrite the fractions so they share this common denominator.
Simplifying Fractions
Simplifying fractions makes them easier to work with. After performing operations like subtraction, we often find that the resulting fraction can be reduced. This is where simplifying comes in. For our example, after subtracting and obtaining \(\frac{12}{20}\), we can simplify it by finding common factors. By dividing both the numerator and the denominator by their greatest common divisor (GCD), we simplify \(\frac{12}{20}\) to \(\frac{3}{5}\).
Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that can evenly divide both the numerator and the denominator of a fraction. To simplify \(\frac{12}{20}\), we find that the GCD of 12 and 20 is 4. By dividing both the numerator and denominator by 4, we get \(\frac{3}{5}\). Using the GCD helps reduce fractions to their simplest form, making calculations more manageable and results clearer.

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