Problem 82 - Elementary Algebra

Question

For exercises 81-96, evaluate. $$ \frac{21}{16}-\frac{5}{24} $$

Step-by-Step Solution

Answer

\frac{53}{48}

Step 1: Identify the Problem

The problem requires finding the difference between the fractions \(\frac{21}{16}\) and \(\frac{5}{24}\).

Step 2: Find a Common Denominator

To subtract two fractions, they must have a common denominator. The denominators here are 16 and 24. The least common multiple (LCM) of 16 and 24 is 48.

Step 3: Convert Fractions to Common Denominator

Convert each fraction to an equivalent fraction with a denominator of 48. Multiply the numerator and denominator of \(\frac{21}{16}\) by 3: \[ \frac{21}{16} = \frac{21 \times 3}{16 \times 3} = \frac{63}{48} \] Multiply the numerator and denominator of \(\frac{5}{24}\) by 2: \[ \frac{5}{24} = \frac{5 \times 2}{24 \times 2} = \frac{10}{48} \]

Step 4: Subtract the Numerators

Now that the fractions have the same denominator, subtract the numerators: \[ \frac{63}{48} - \frac{10}{48} = \frac{63 - 10}{48} = \frac{53}{48} \]

Step 5: Simplify the Result (if needed)

The fraction \(\frac{53}{48}\) is in its simplest form as 53 is a prime number and does not share any common factors with 48 besides 1.

Key Concepts

Common Denominator

When subtracting fractions, we need a common denominator. A common denominator is a shared multiple of the denominators of the fractions involved. For the fractions \(\frac{21}{16} \) and \(\frac{5}{24} \), their denominators are 16 and 24. To find a common denominator, we look for the smallest number that both denominators can divide evenly into. This is crucial because making the denominators the same allows us to combine the fractions easily. Let's move on to finding that number!

Least Common Multiple

The Least Common Multiple (LCM) is the smallest common multiple of two or more numbers. For our problem, we need the LCM of 16 and 24. The multiples of 16 are: 16, 32, 48, 64, and so on. The multiples of 24 are: 24, 48, 72, and so on. The smallest common multiple between these lists is 48. So, 48 is our LCM. This LCM will serve as our common denominator to convert both fractions to equivalent fractions sharing the same denominator.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same amount. To subtract \(\frac{21}{16} \) and \(\frac{5}{24} \), we convert them into equivalent fractions with a common denominator. We found that 48 was the LCM of 16 and 24. To convert \(\frac{21}{16} \) into a fraction with 48 as the denominator, multiply the numerator and denominator by 3: \(\frac{21}{16} = \frac{21 \times 3}{16 \times 3} = \frac{63}{48} \). Similarly, convert \(\frac{5}{24} \) by multiplying the numerator and denominator by 2: \(\frac{5}{24} = \frac{5 \times 2}{24 \times 2} = \frac{10}{48} \). Now, we have \(\frac{63}{48} \) and \(\frac{10}{48} \).

Simplifying Fractions

Once the fractions have the same denominator, you can subtract them by subtracting their numerators: \(\frac{63}{48} - \frac{10}{48} = \frac{53}{48} \). To simplify a fraction means to make it as simple as possible by ensuring the numerator and the denominator share no common factors besides 1. In this case, since 53 is a prime number and does not have any common factors with 48 other than 1, \(\frac{53}{48} \) is already in its simplest form. And that's our final answer!