Question

Add the following fractions and mixed numbers. Reduce to lowest terms. \(101 \frac{3}{4}+33 \frac{1}{4}+5 \frac{1}{4}=\) ______

Short answer

140 \( \frac{1}{4} \)

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

The first step is to convert each mixed number into an improper fraction.For \(101 \frac{3}{4}\):\[101 \frac{3}{4} = \frac{101 \times 4 + 3}{4} = \frac{407}{4}\]For \(33 \frac{1}{4}\):\[33 \frac{1}{4} = \frac{33 \times 4 + 1}{4} = \frac{133}{4}\]For \(5 \frac{1}{4}\):\[5 \frac{1}{4} = \frac{5 \times 4 + 1}{4} = \frac{21}{4}\]

Add the Improper Fractions

Add the fractions obtained in the first step. Since they all have the same denominator, we can directly add their numerators:\[\frac{407}{4} + \frac{133}{4} + \frac{21}{4} = \frac{407 + 133 + 21}{4} = \frac{561}{4}\]

Convert Back to Mixed Number and Simplify

Now, convert the improper fraction back to a mixed number.Divide 561 by 4:561 ÷ 4 = 140 remainder 1Thus, \(\frac{561}{4} = 140 \frac{1}{4}\).

Check for Simplification

The fraction \(\frac{1}{4}\) is already in its simplest form. Therefore, the final answer is already reduced to the lowest terms.

Key concepts

Mixed Numbers

A mixed number is a special type of fraction that combines a whole number with a proper fraction. This is very common in everyday life since it's easier to represent quantities that are more than a whole but not quite two wholes this way. For example, if you have one whole cake and a quarter of another cake, instead of saying you have 1.25 cakes, you describe them as "one and a quarter cakes" or as the mixed number, \(1\frac{1}{4}\). To work with mixed numbers in mathematical operations, it is often easier to first convert them into improper fractions.

Improper Fractions

Improper fractions are a type of fraction where the numerator is greater than or equal to the denominator. This makes them handy when performing addition or subtraction of fractions since it avoids dealing with whole numbers separately. For example, the mixed number \(3\frac{1}{2}\) can be converted to the improper fraction \(\frac{7}{2}\). This is achieved by multiplying the whole number by the denominator and adding the numerator, all over the same denominator. Improper fractions allow you to see the whole quantity as a single number, simplifying arithmetic operations.

Simplification

Simplification is the process of reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. This makes the fraction easier to understand and work with. For example, \(\frac{8}{12}\) simplifies to \(\frac{2}{3}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 4 in this case. It's important to always check if a fraction can be simplified after performing any arithmetic operations like addition, subtraction, multiplication, or division, to ensure clarity and accuracy of the result.

Addition of Fractions

Adding fractions involves combining the numerators of fractions with the same denominator, as these have comparable parts. This is easiest when the denominators are the same, like adding \(\frac{3}{4}\) and \(\frac{1}{4}\), which becomes \(\frac{4}{4}\) or \(1\). However, with different denominators, you must first find a common denominator. In this exercise, we conveniently add \(\frac{407}{4}\), \(\frac{133}{4}\), and \(\frac{21}{4}\) because they already share the same denominator, making addition straightforward by summing up the numerators. The final outcome is often needed in the form of simplest fractions or mixed numbers, requiring further steps of conversion or simplification.