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Chapter 3: Problem 32

Subtract and simplify. $$ \frac{5}{12}-\frac{5}{16} $$

Short Answer

Expert verified
\( \frac{5}{48} \)

Step by step solution

01

Find a common denominator

To subtract fractions, they must have a common denominator. The denominators are 12 and 16. The least common multiple (LCM) of 12 and 16 is 48.
02

Convert the fractions

Convert each fraction to have the common denominator of 48. \[ \frac{5}{12} = \frac{5 \times 4}{12 \times 4} = \frac{20}{48} \] \[ \frac{5}{16} = \frac{5 \times 3}{16 \times 3} = \frac{15}{48} \]
03

Subtract the fractions

Subtract the numerators of the converted fractions while keeping the denominator the same: \[ \frac{20}{48} - \frac{15}{48} = \frac{20 - 15}{48} = \frac{5}{48} \]
04

Simplify the fraction

Check if the fraction can be simplified further. The fraction \( \frac{5}{48} \) is already in its simplest form since 5 and 48 have no common factors other than 1.

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Key Concepts

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

Least Common Multiple
Before subtracting fractions, we need them to have the same denominator. This is where the Least Common Multiple (LCM) comes in. The LCM of two numbers is the smallest number that both can divide evenly into. For example, to find the LCM of 12 and 16, list out the multiples of each number:
  • Multiples of 12: 12, 24, 36, 48, 60...
  • Multiples of 16: 16, 32, 48, 64...
We can see that the smallest common multiple is 48. Thus, 48 is the LCM of 12 and 16. Using the LCM helps to convert fractions to a common denominator, making them easy to subtract.
Converting Fractions
Once we have the LCM, the next step is to convert each fraction to have this common denominator. This involves converting both fractions to equivalent fractions with the denominator of 48. Here's how it's done:
  • For \(\frac{5}{12}\), multiply both the numerator and denominator by 4:\br> \[ \frac{5}{12} = \frac{5 \times 4}{12 \times 4} = \frac{20}{48} \]
  • For \(\frac{5}{16}\), multiply both the numerator and denominator by 3:\br> \[ \frac{5}{16} = \frac{5 \times 3}{16 \times 3} = \frac{15}{48} \]
By converting the fractions, we have \(\frac{20}{48}\) and \(\frac{15}{48}\), which are easier to work with because they share the same denominator.
Simplifying Fractions
After converting and subtracting the fractions, we may need to simplify the result. Simplifying fractions means making them as simple as possible by reducing them to their smallest form. Let's check if \(\frac{5}{48}\) can be simplified:
  • Look for the greatest common divisor (GCD) of 5 and 48. In this case, the GCD is 1 since 5 is a prime number and does not divide 48.
  • Since the GCD is 1, \(\frac{5}{48}\) is already in its simplest form and cannot be simplified further.
By completing these steps, we ensure the fractions are represented in their cleanest and most understandable form.

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