Question

Subtract the following fractions and mixed numbers. Reduce to lowest terms. \(4 \frac{1}{6}-1 \frac{1}{3}=\) ______

Short answer

The result is \(2 \frac{5}{6}\).

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. For the number \(4 \frac{1}{6}\), multiply the whole number (4) by the denominator (6) and add the numerator (1): \((4 \times 6) + 1 = 25\). So, \(4 \frac{1}{6}\) becomes \(\frac{25}{6}\).For the number \(1 \frac{1}{3}\), multiply the whole number (1) by the denominator (3) and add the numerator (1): \((1 \times 3) + 1 = 4\). So, \(1 \frac{1}{3}\) becomes \(\frac{4}{3}\).

Find a Common Denominator

To subtract these fractions, they must have a common denominator. The denominators are 6 and 3. The least common denominator (LCD) of 6 and 3 is 6. \(\frac{25}{6}\) already has this denominator. Convert \(\frac{4}{3}\) to a denominator of 6 by multiplying both the numerator and denominator by 2, resulting in \(\frac{8}{6}\).

Subtract the Fractions

Now that both fractions have the same denominator, subtract the numerators: \(\frac{25}{6} - \frac{8}{6}\). Perform the subtraction: \(25 - 8 = 17\). The result is \(\frac{17}{6}\).

Simplify the Fraction

\(\frac{17}{6}\) is already in its simplest form because 17 is a prime number and does not share any common factors with 6. However, it can be expressed as a mixed number because 17 divided by 6 is 2 with a remainder of 5.So, \(\frac{17}{6}\) can be written as \(2 \frac{5}{6}\).

Key concepts

Mixed Numbers

Mixed numbers contain both a whole number and a fractional part. They are commonly used to express numbers that lie between the integers. For instance, in our exercise, we have the mixed numbers:

• 4 \( \frac{1}{6} \), which means 4 whole units plus an additional \( \frac{1}{6} \) of a unit.
• 1 \( \frac{1}{3} \), depicting 1 whole unit plus \( \frac{1}{3} \) of a unit.

Understanding mixed numbers is key in the arithmetic of fractions, as often we need to convert them to improper fractions to perform operations like addition or subtraction efficiently. This conversion allows for a seamless calculation of fractions as you just focus on the numerators and denominators.

Improper Fractions

An improper fraction is one where the numerator is larger than the denominator. This means the fraction is greater than or equal to 1. Converting mixed numbers into improper fractions is a crucial step before performing operations such as addition or subtraction. For example,

• The mixed number \(4 \frac{1}{6}\) is converted to the improper fraction \(\frac{25}{6}\). This is done by multiplying 4 (the whole number) by 6 (the denominator), adding 1 (the numerator), resulting in 25, forming \(\frac{25}{6}\).
• Likewise, \(1 \frac{1}{3}\) becomes \(\frac{4}{3}\), derived by multiplying 1 by 3, then adding 1.

This conversion is essential to handle calculations involving fractions neatly and accurately.

Common Denominator

To subtract or add fractions efficiently, they need to share the same denominator. This is because fractions are parts of a whole, and only by having common denominators can we directly compare or combine these parts. In our example:

• The original fractions were \(\frac{25}{6}\) and \(\frac{4}{3}\). The least common denominator (LCD) of 6 and 3 is 6.
• \(\frac{25}{6}\) already has this common denominator, while \(\frac{4}{3}\) is converted to \(\frac{8}{6}\) by multiplying both the numerator and denominator by 2.

Now, the fractions can be subtracted as they are expressed in terms of the same whole, making the operation straightforward.

Simplifying Fractions

Simplifying fractions means reducing them to their lowest terms, so they cannot be divided any further while still being whole numbers. This is important for clarity and ease of understanding and is often the final step in fraction operations.

• In our exercise result, \(\frac{17}{6}\), this fraction is already simplified since the numerator (17) and the denominator (6) have no common factors other than 1.
• However, \(\frac{17}{6}\) can be expressed as a mixed number: 2 whole parts and \(\frac{5}{6}\) remaining, becoming \(2 \frac{5}{6}\).

Expressing the result in the simplest form makes it easier to understand and compare with other numbers.