Problem 94 - Prealgebra

Question

Give the difference between \(5 \frac{1}{3}\) and \(1 \frac{5}{6}\).

Step-by-Step Solution

Answer

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

Step 1: Convert mixed numbers to improper fractions

First, we need to convert the mixed numbers to improper fractions. For the first number, multiply the whole number by the denominator and add the numerator: \(5 \times 3 + 1 = 16\). So, \(5 \frac{1}{3} = \frac{16}{3}\). For the second number, \(1 \times 6 + 5 = 11\), so \(1 \frac{5}{6} = \frac{11}{6}\).

Step 2: Find a common denominator

To subtract these fractions, we need a common denominator. The denominators are 3 and 6. The least common multiple of 3 and 6 is 6. Thus, \(\frac{16}{3}\) can be converted to \(\frac{32}{6}\) to have the same denominator as \(\frac{11}{6}\).

Step 3: Subtract the fractions

Now, subtract \(\frac{11}{6}\) from \(\frac{32}{6}\). This gives us: \(\frac{32}{6} - \frac{11}{6} = \frac{21}{6}\).

Step 4: Simplify the result

Simplify the fraction \(\frac{21}{6}\). Divide both the numerator and the denominator by their greatest common divisor, which is 3: \(\frac{21}{6} \div \frac{3}{3} = \frac{7}{2}\). Finally, you can convert \(\frac{7}{2}\) back to a mixed number, which is \(3 \frac{1}{2}\).

Key Concepts

Improper Fractions

When you come across mixed numbers in mathematics, an important step is converting them to improper fractions before performing operations like addition or subtraction. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This is different from a proper fraction, where the numerator is less than the denominator.

To convert a mixed number such as \( 5 \frac{1}{3} \) to an improper fraction, you multiply the whole number by the denominator and add the numerator from the fractional part. In this example, you multiply 5 (the whole number) by 3 (the denominator), resulting in 15. You then add the numerator of the fraction, which is 1, so 15 + 1 equals 16. Thus, \( 5 \frac{1}{3} \) becomes \( \frac{16}{3} \).

By understanding and using improper fractions, it becomes easier to handle mixed numbers in calculations. It standardizes the operation, making it straightforward to find common denominators and simplify results later.

Common Denominator

When subtracting fractions, it's critical that they have a common denominator, which ensures that the pieces we are subtracting from are the same size. This concept is analogous to being sure to measure everything with the same unit when solving real-world problems.

To find a common denominator, you first need the least common multiple (LCM) of the denominators. For example, when working with \( \frac{16}{3} \) and \( \frac{11}{6} \), the denominators are 3 and 6. The LCM of 3 and 6 is 6, because 6 is the smallest number that both 3 and 6 can divide into completely.

After finding the LCM, convert the fractions so they have this common denominator. For \( \frac{16}{3} \), you multiply both the numerator and the denominator by 2 to get \( \frac{32}{6} \) since 6 divided by 3 equals 2. Now, both fractions, \( \frac{32}{6} \) and \( \frac{11}{6} \), share a denominator of 6 and can easily be subtracted from each other.

Simplifying Fractions

After performing operations like subtraction with fractions, simplification is usually the final step. Simplifying fractions means adjusting the fraction to its smallest, simplest form. You do this by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Consider the fraction \( \frac{21}{6} \). To simplify it, calculate the GCD of 21 and 6, which is 3. You divide 21 by 3 obtaining 7, and divide 6 by 3 obtaining 2, thus simplifying \( \frac{21}{6} \) to \( \frac{7}{2} \).

Sometimes, simplifying might result in an improper fraction, such as \( \frac{7}{2} \). You can leave it as is, or if needed, convert it back to a mixed number. Dividing 7 by 2 gives 3 with a remainder of 1, so \( \frac{7}{2} \) becomes the mixed number \( 3 \frac{1}{2} \). Learning and practicing simplification helps in making fraction operations cleaner and results more interpretable.