Problem 28 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{7}{a}+\frac{5}{b}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{7b + 5a}{ab}\).

Step 1: Identify the denominators

Here one can observe two fractions, \(\frac{7}{a}\) and \(\frac{5}{b}\). The denominators are 'a' and 'b' respectively.

Step 2: Find common denominator

To add the fractions, they must have the same denominator. In this case, the simplest common denominator is the product of the denominators 'a' and 'b'.

Step 3: Convert the fractions

Each fraction is multiplied by the other fraction's denominator divided by its own denominator, which results in equivalent fractions: \(\frac{7}{a} * \frac{b}{b} = \frac{7b}{ab}\) and \(\frac{5}{b} * \frac{a}{a} = \frac{5a}{ab}\).

Step 4: Add the fractions

Now the fractions have the same denominator, they can be added: \(\frac{7b}{ab} + \frac{5a}{ab} = \frac{7b + 5a}{ab}\).

Key Concepts

Common Denominator

Understanding the concept of a common denominator is essential when working with fractions. It's the key to being able to add or subtract fractions effectively.

To add fractions, like \(\frac{7}{a}\) and \(\frac{5}{b}\), you need to combine them into one fraction. However, before you can do that, they need to have the same denominator. This shared denominator is known as the common denominator. Intuitively, it's like making sure you are measuring in the same units before you total up quantities. If the denominators are different, the fractions are like apples and oranges—similar, but not quite the same.

In our exercise, the denominators ‘a’ and ‘b’ are different. The easiest way to find a common denominator for them is to multiply them together, resulting in ‘ab’. Using the common denominator allows you to convert unlike fractions into like fractions, which can then be added or subtracted easily.

It's important to remember that finding the smallest common denominator will simplify your calculations, but any common denominator will work mathematically, as long as it is a multiple of both original denominators.

Simplifying Fractions

Simplifying fractions is a process that makes them easier to understand and work with. After finding a common denominator and adding the fractions, you might end up with a complex-looking result. However, this can often be simplified to a more manageable expression.

To simplify a fraction, you should look for factors common to both the numerator and the denominator. These can be divided out, or 'cancelled', to create a simpler fraction. For the sum \(\frac{7b + 5a}{ab}\) obtained in our exercise, we cannot simplify it further without knowing the values of ‘a’ and ‘b’. The simplification process would require these values to have common factors that can be cancelled.

However, simplifying also involves writing your answer in a way that's easy to understand. Even if you can't cancel out terms, ensuring the fraction looks neat and is presented clearly is part of the simplification process. Always present your final answer in its simplest form, whether that's a reduced fraction or a mixed number, if applicable.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. When solving for a common denominator, we create equivalent fractions that allow us to add or subtract fractions with different denominators.

For instance, \(\frac{7}{a}\) and \(\frac{7b}{ab}\) are equivalent fractions. To make them equivalent, you multiply the numerator and the denominator of the first fraction by ‘b’. Similarly, \(\frac{5}{b}\) and \(\frac{5a}{ab}\) are made equivalent by multiplying both the numerator and the denominator by ‘a’. This process doesn't change the fraction's value, rather it adjusts its form so it can be combined with another fraction.

Understanding that multiplying or dividing both the numerator and the denominator of a fraction by the same nonzero number gives an equivalent fraction is fundamental to working with fractions. This concept underlies many operations with fractions and is essential for simplifying them, solving equations, and rationalizing denominators.

Remember, equivalent fractions look different, but their positions on the number line are the same, illustrating that, fundamentally, they are one and the same quantity.