Question

Subtract the following fractions and mixed numbers. Reduce to lowest terms. \(2 \frac{3}{4}-\frac{1}{4}=\) ______

Short answer

\(\frac{5}{2}\)

Step-by-step solution

Convert Mixed Number to Improper Fraction

The first step is to convert the mixed number into an improper fraction. In this problem, we need to convert \(2 \frac{3}{4}\) into an improper fraction. The whole number is 2, and the fraction is \(\frac{3}{4}\). Multiply the whole number by the denominator: \(2 \times 4 = 8\). Add the numerator: \(8 + 3 = 11\). Therefore, \(2 \frac{3}{4}\) becomes \(\frac{11}{4}\).

Align Denominators for Subtraction

Both fractions, \(\frac{11}{4}\) and \(\frac{1}{4}\), share the same denominator. This means they are already aligned for subtraction, so we can proceed with subtracting them directly.

Subtract the Numerators

Subtract the numerators of the two fractions. This involves subtracting \(\frac{1}{4}\) from \(\frac{11}{4}\), so we calculate \(11 - 1 = 10\). Therefore, the result of the subtraction is \(\frac{10}{4}\).

Reduce the Fraction to Lowest Terms

The last step is to reduce the fraction \(\frac{10}{4}\) to its simplest form. Find the greatest common divisor (GCD) of 10 and 4, which is 2. Divide both the numerator and the denominator by the GCD: \(\frac{10 \div 2}{4 \div 2} = \frac{5}{2}\). So, the reduced form is \(\frac{5}{2}\).

Key concepts

Mixed Numbers

Mixed numbers consist of two parts: a whole number and a fraction. For example, in the mixed number \(2 \frac{3}{4}\), 2 is the whole number, and \(\frac{3}{4}\) is the fraction. To perform operations like addition or subtraction, mixed numbers are often converted into improper fractions.

Why do this? Because it simplifies the calculation process. Converting a mixed number into an improper fraction involves the following steps:

• Multiply the whole number by the denominator of the fraction. For \(2 \frac{3}{4}\), you multiply 2 by 4 to get 8.
• Add the result to the numerator of the fraction. Here, adding 8 and 3 gives you 11.
• The improper fraction is \(\frac{11}{4}\).

When the fractions of a mixed number are aligned, operations become straightforward, even if improper fractions look a bit odd at first. This is especially useful when adding, subtracting, or even comparing fractions.

Improper Fractions

An improper fraction is one where the numerator is larger than the denominator, like \(\frac{11}{4}\). This contrasts with proper fractions like \(\frac{1}{4}\), where the numerator is smaller than the denominator. Improper fractions can be easier to work with when it comes to arithmetic operations.

To illustrate, let's see why they're useful in subtraction. With both \(\frac{11}{4}\) and \(\frac{1}{4}\) having the same denominator, subtraction becomes straightforward. Here, you simply subtract the numerators: \(11 - 1 = 10\). So, the result is \(\frac{10}{4}\).

Improper fractions can often appear in intermediate steps when dealing with mixed numbers. After conversion, you can do your math before converting the result back to a mixed number, if needed.

Lowest Terms

Finding the lowest terms for a fraction means simplifying it to its most basic form. This is crucial for clarity and ease of understanding across mathematical operations. Each fraction should be presented in its simplest form, where the numerator and denominator have no common factors other than 1.

To simplify \(\frac{10}{4}\), follow these steps:

• Identify the greatest common divisor (GCD) of the numerator and the denominator. For \(10\) and \(4\), the GCD is 2.
• Divide both the numerator and the denominator by the GCD. Here, \(10 \div 2 = 5\) and \(4 \div 2 = 2\).
• The fraction in lowest terms becomes \(\frac{5}{2}\).

Reducing to lowest terms conveys a fraction in its simplest and most understandable state. It makes life easier, whether you're comparing sizes, performing calculations, or simply communicating mathematical ideas clearly.