Problem 93 - Prealgebra

Question

Find the difference between \(6 \frac{1}{5}\) and \(2 \frac{7}{10}\).

Step-by-Step Solution

Answer

The difference is \(3 \frac{1}{2}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. The mixed number \(6 \frac{1}{5}\) can be expressed as \(\frac{31}{5}\) and the mixed number \(2 \frac{7}{10}\) can be expressed as \(\frac{27}{10}\).

Step 2: Find a Common Denominator

To subtract fractions, they need the same denominator. The denominators here are 5 and 10. The least common denominator is 10. So, convert \(\frac{31}{5}\) by finding an equivalent fraction with the denominator 10. Multiply the numerator and the denominator of \(\frac{31}{5}\) by 2 to get \(\frac{62}{10}\).

Step 3: Subtract the Fractions

Now, subtract the second fraction from the first. Compute \(\frac{62}{10} - \frac{27}{10}\). Subtract the numerators: 62 - 27 = 35. The denominator remains 10, so the result is \(\frac{35}{10}\).

Step 4: Simplify the Resulting Fraction

Simplify \(\frac{35}{10}\) by finding the greatest common divisor of 35 and 10, which is 5. Divide both the numerator and denominator by 5 to get \(\frac{7}{2}\).

Step 5: Convert Back to a Mixed Number (if needed)

If necessary, convert the improper fraction \(\frac{7}{2}\) back to a mixed number by dividing 7 by 2. This gives 3 with a remainder of 1: \(3 \frac{1}{2}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are usually easier to understand at a glance, as they show both parts of a number clearly. For example, in the exercise, we had the mixed numbers \(6 \frac{1}{5}\) and \(2 \frac{7}{10}\).
A mixed number combines the simplicity of a whole number with the precision of a fraction. However, when performing math operations like subtraction, mixed numbers can complicate the process.
Thus, we first convert mixed numbers to improper fractions, which are more straightforward to calculate with.

Improper Fractions

Improper fractions have a numerator that is larger than or equal to the denominator. This is the opposite of a proper fraction, where the numerator is smaller than the denominator.
For instance, converting \(6 \frac{1}{5}\) to an improper fraction means calculating:\[6 \times 5 + 1 = 31\]
So, we get \(\frac{31}{5}\). Similarly, \(2 \frac{7}{10}\) converts to \(\frac{27}{10}\). Improper fractions make it easier to perform arithmetic operations such as addition or subtraction, as there's no confusing overlap with whole numbers. This directness is why mathematicians often prefer them in calculations.

Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions. It's like finding a common language so both fractions can "talk" to each other.
The original exercise had denominators 5 and 10. With 10 being the larger number, it can accommodate both 5 and 10 (since 10 is a multiple of 5).
Converting \(\frac{31}{5}\) to \(\frac{62}{10}\) aligns the denominators, allowing straightforward subtraction. Once the fractions have a common denominator, you only need to calculate with the numerators, keeping the denominator the same.

Fraction Simplification

Simplifying fractions means making them as simple as possible. This involves dividing both the numerator and denominator by their greatest common divisor (GCD).
In our exercise, we simplify \(\frac{35}{10}\) by recognizing the GCD of 35 and 10 is 5.
Therefore, when dividing both by 5, you arrive at \(\frac{7}{2}\). This fraction is simpler and easier to understand.
Often, the simplest form of a fraction is preferred, especially in answers, as it is universally accepted and typically more concise.