Problem 24 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$\frac{5}{8}-\frac{1}{3}$$

Step-by-Step Solution

Answer

The difference is \(\frac{7}{24}\).

Step 1: Identify the Denominator

To subtract fractions, they must have the same denominator. Notice that each fraction has a different denominator: 8 and 3.

Step 2: Find the Least Common Denominator (LCD)

Identify the smallest number that both 8 and 3 can divide into evenly. The smallest common multiple of 8 and 3 is 24.

Step 3: Convert Each Fraction to Have the LCD

Convert \(\frac{5}{8}\) and \(\frac{1}{3}\) to equivalent fractions with a denominator of 24.For \(\frac{5}{8}\), multiply the numerator and the denominator by \(3\) to get \(\frac{15}{24}\).For \(\frac{1}{3}\), multiply the numerator and the denominator by \(8\) to get \(\frac{8}{24}\).

Step 4: Subtract the Fractions

With equivalent fractions, both having the denominator of 24, you can now subtract: \(\frac{15}{24} - \frac{8}{24} = \frac{7}{24}\).

Step 5: Simplify if Necessary

Examine the fraction \(\frac{7}{24}\). Since 7 and 24 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Least Common Denominator

When subtracting fractions like \( \frac{5}{8} - \frac{1}{3} \), the key is to work with a common denominator. Different denominators can make the problem tricky because you can't directly subtract the numerators. So, how do we overcome this? By finding the Least Common Denominator (LCD).

The least common denominator is the smallest number that both denominators (in this case 8 and 3) can be evenly divided into. To find the LCD, list the multiples of each denominator until you find the smallest common one:

• Multiples of 8: 8, 16, 24, 32...
• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24...

By checking these multiples, we see that 24 is the first number that appears in both lists, making it the LCD. This common denominator allows the fractions to speak the same 'language,' making it possible to subtract them.

Simplifying Fractions

Once you have subtracted two fractions and ended up with a result, the next step is to simplify the fraction if possible. Simplification makes the fraction easier to understand and smaller in terms of its numerator and denominator, but it must retain the same value.

To simplify \( \frac{7}{24} \), you need to find the greatest common factor (GCF) of the numerator and the denominator. The GCF is the largest number that divides evenly into both numbers. In this case:

• Factors of 7 are just 1 and 7.
• Factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

The only common factor is 1, indicating that \( \frac{7}{24} \) is already in its simplest form. Simplifying fractions involves dividing the numerator and the denominator by their GCF, but sometimes as here, no further reduction is possible.

Equivalent Fractions

The concept of equivalent fractions is crucial in operations like addition and subtraction where the denominators differ. An equivalent fraction is a different fraction that has the same value as the original one. For example, multiplying the numerator and the denominator of a fraction by the same non-zero number creates an equivalent fraction.

Let's apply this with \( \frac{5}{8} \) and \( \frac{1}{3} \) to align with the LCD of 24.

• For \( \frac{5}{8} \), multiply both the numerator (5) and the denominator (8) by 3 to get \( \frac{15}{24} \).
• For \( \frac{1}{3} \), multiply both the numerator (1) and the denominator (3) by 8 to get \( \frac{8}{24} \).

These new fractions, \( \frac{15}{24} \) and \( \frac{8}{24} \), are equivalent to the original fractions but have the same denominators, allowing you to subtract them easily. Equivalent fractions are like using different units to measure the same quantity, such as expressing the same length in inches or centimeters.