Problem 32 - Prealgebra

Question

Subtract. $$5 \frac{1}{8}-2 \frac{5}{8}$$

Step-by-Step Solution

Answer

The result of subtracting is \(2 \frac{1}{2}\).

Step 1: Convert Mixed Numbers to Improper Fractions

The problem involves mixed numbers: \(5 \frac{1}{8}\) and \(2 \frac{5}{8}\). First, convert these mixed numbers to improper fractions. To convert \(5 \frac{1}{8}\), multiply the whole number 5 by the denominator 8, and add the numerator 1: \(5 \times 8 + 1 = 41\). Thus, \(5 \frac{1}{8} = \frac{41}{8}\). Now do the same for \(2 \frac{5}{8}\): multiply 2 by 8, and add 5: \(2 \times 8 + 5 = 21\). So, \(2 \frac{5}{8} = \frac{21}{8}\).

Step 2: Subtract the Improper Fractions

With both numbers converted to improper fractions, subtract \(\frac{21}{8}\) from \(\frac{41}{8}\). Since they have the same denominator, simply subtract the numerators: \(41 - 21 = 20\). Therefore, \(\frac{41}{8} - \frac{21}{8} = \frac{20}{8}\).

Step 3: Simplify the Fraction

The fraction \(\frac{20}{8}\) can be simplified. Find the greatest common divisor (GCD) of 20 and 8, which is 4. Divide both the numerator and the denominator by their GCD: \(\frac{20}{8} = \frac{20 \div 4}{8 \div 4} = \frac{5}{2}\).

Step 4: Convert Improper Fraction to Mixed Number

Finally, convert \(\frac{5}{2}\) back to a mixed number. Divide the numerator 5 by the denominator 2: 5 divided by 2 is 2 with a remainder of 1. Therefore, \(\frac{5}{2}\) is equivalent to \(2 \frac{1}{2}\).

Key Concepts

Understanding Mixed Numbers

Mixed numbers are a delightful mix of whole numbers and fractions. They appear commonly in everyday life, such as when recipes call for certain measurements. A mixed number consists of:

• A whole number part
• A fractional part

In the example given in the exercise, we have two mixed numbers: \(5 \frac{1}{8}\) and \(2 \frac{5}{8}\). The whole number is to the left, while the fraction (a simple fraction with a numerator and a denominator) follows. Mixed numbers are unique because they offer a more straightforward view, helping us visualize quantities better than improper fractions. But when subtracting, adding, or performing any arithmetic, it's helpful to convert them into improper fractions to simplify the operation.

Working with Improper Fractions

Improper fractions might seem intimidating at first, but they are pretty simple. An improper fraction is when the numerator (the top part of the fraction) is larger than or equal to the denominator (the bottom part). For example, \(\frac{41}{8}\) and \(\frac{21}{8}\) are improper fractions.
Why do we convert mixed numbers into improper fractions?

• It simplifies arithmetic operations like addition and subtraction.
• It provides consistency when handling fractions with unlike denominators.

To convert a mixed number to an improper fraction, multiply the whole number by the denominator, and add the numerator. This gives you the total number of eighths, for instance, which makes calculations easier. Once you've performed your operations, you often convert back to a mixed number to make the result more intuitive.

Finding the Greatest Common Divisor

The greatest common divisor (GCD) is a key player in simplifying fractions. Whenever you need to reduce a fraction to its simplest form, the GCD helps you out. The GCD of two numbers is the largest number that divides both without leaving a remainder.
In our exercise, we have the fraction \(\frac{20}{8}\). To simplify it, we find the GCD of 20 and 8. The divisors of 20 are 1, 2, 4, 5, 10, and 20, and for 8, they are 1, 2, 4, and 8. The greatest common divisor here is 4.

• Divide both the numerator and the denominator by the GCD.
• This gives you a simplified fraction: \(\frac{20}{8} = \frac{5}{2}\).

Simplifying fractions not only makes them more aesthetically pleasing but also often easier to interpret, especially in contexts like measurement and finance.