Fraction Addition and Subtraction Basics

Adding Fractions

When working with fractions, it's essential to understand how to perform basic arithmetic operations like addition and subtraction. In this tutorial, we will explore different cases of fraction addition and subtraction, starting with unlike denominators and progressing to common denominators.

Adding Fractions With Unlike Denominators

Adding fractions with unlike denominators requires finding a lowest common denominator (LCD) before performing the actual operation. Here is a step-by-step guide to adding two fractions with unlike denominators:

1. Identify the LCD. This is usually done by multiplying the denominators together. If the original denominators have any prime factors in common, these must also be included in the product.

2. Convert each fraction into an equivalent fraction with the new LCD as the denominator. To do this, you first identify what multiple of each numerator equals the LCD, and divide both the numerator and the denominator by that amount. For example, if the LCD were (6), (\frac{3}{8}) would become (\frac{3\cdot3}{8\cdot3} = \frac{9}{24}=\frac{3}{8}).

For instance, let's say we want to add [\frac{7}{8}+\frac{3}{4}.] We start by identifying the LCD, which can be found using the Least Common Multiple (LCM): [8\times4=32.] Then, we convert our fractions into their simplest form within this LCD: [\frac{7\cdot2}{8\cdot2}=7] and [\frac{3\cdot8}{4\cdot8}=\frac{24}{32},] so we get [7+\frac{24}{32}=\boxed{\frac{112}{32}}=\frac{31}{8}.]

This process ensures consistency when combining unlike fractions and ensures they are combined correctly in terms of their values when expressed in a single fraction.

Subtracting Fractions With Unlike Denominators

Subtracting fractions with unlike denominators follows a similar approach to adding them. Instead of converting each fraction to a common denominator, we need to find a negative reciprocal of one of the fractions:

1. Identify the fraction whose denominator cancels out with the other. This means the fraction becomes an integer.

2. Negate all parts of this chosen fraction, while leaving the remaining fraction unchanged. Note that negating a positive fraction results in a negative fraction (e.g., -(x)), whereas negating a negative fraction gives rise to a positive fraction (e.g., +(y)).

Let's see an example: Consider [\frac{5}{8}-\frac{3}{4}.] First, we look for a fraction whose denominator divides evenly into the denominator of the other fraction: [\frac{5}{8}\Rightarrow 8,\quad \text{and} \quad \frac{3}{4}\Rightarrow 4.] So, (8\div4=2), and this divisor works for both fractions. Now, we subtract the second fraction from the first via cancellation:[\frac{5-3\cdot2}{8}=\frac{-5}{8}.]Finally, since we started with (-3) in the second fraction, we reverse its sign: [-(-5)=+5,]so the result is [\boxed{\frac{5}{8}}]which does indeed equal [\frac{5}{8}-\frac{3}{4}.]

Note that when calculating fractions with small numbers, there may appear to be differences between calculators, but this is due to rounding errors rather than true discrepancies.

Subtracting Fractions With Common Denominators

If we encounter fractions with the same denominator, such as [\frac{3}{8}-\frac{5}{8},]we immediately know that they share a common denominator. Therefore, the steps for subtracting fractions with unlike denominators apply here as well:

1. Simply subtract the second fraction from the first.

Regarding the above problem, [\frac{3}{8}-\frac{5}{8}=\boxed{\frac{-2}{8}}=\frac{-1}{4}=-\frac{1}{4}.]Again, note the difference in signs. When we started, (-5) was in the second fraction; reversing its sign gave us "(+5)" in the final answer.

Adding Fractions With Common Denominators

The principle behind adding fractions with common denominators remains the same as for unlike denominators except now we don't need to worry about changing the respective (common) denominator to find a common factor:

1. Add the two fractions directly, keeping the common denominator intact.

Taking an example of [\frac{3}{8}+\frac{5}{8},]the sum is simply [\boxed{\frac{3+5}{8}}=\frac{8}{8}=1.]We did not change the common denominator ((8)), nor did we complicate things by introducing new factors. As far as algebraic expressions go, this is generally considered 'clean'.

In conclusion, understanding the principles involved in manipulating fractions allows us to tackle increasingly complex problems involving multiplication and division of fractions, or mixed numbers. Mastery of these skills opens up many doors mathematically, enabling us to delve deeper into more advanced mathematical concepts.