Problem 62 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \frac{5}{8}-\left(-\frac{2}{9}\right) $$

Step-by-Step Solution

Answer

\( \frac{5}{8} - \left( -\frac{2}{9} \right) = \frac{61}{72} \).

Step 1: Rewrite the problem

The given expression is \( \frac{5}{8} - \left( -\frac{2}{9} \right) \). Notice the subtraction of a negative is equivalent to adding its positive counterpart. So, the expression becomes \( \frac{5}{8} + \frac{2}{9} \).

Step 2: Find a common denominator

The fractions need a common denominator to perform the addition. The denominators are 8 and 9. The least common multiple (LCM) of 8 and 9 is 72. So, we need to convert each fraction to have 72 as the denominator.

Step 3: Convert fractions

Convert \( \frac{5}{8} \) to a fraction with denominator 72 by multiplying both the numerator and the denominator by 9: \( \frac{5}{8} \times \frac{9}{9} = \frac{45}{72} \).Convert \( \frac{2}{9} \) to a fraction with denominator 72 by multiplying both the numerator and the denominator by 8: \( \frac{2}{9} \times \frac{8}{8} = \frac{16}{72} \).

Step 4: Add the fractions

Now that both fractions have the same denominator, add them together: \( \frac{45}{72} + \frac{16}{72} = \frac{61}{72} \).

Step 5: Simplify the result

The fraction \( \frac{61}{72} \) cannot be simplified further because 61 is a prime number and does not divide evenly into 72. Hence, \( \frac{61}{72} \) is the final simplified form.

Key Concepts

Common Denominator

Before adding fractions, they must have a common denominator. The common denominator is simply a shared bottom number that will allow you to combine the fractions easily. When fractions have different denominators, it's like trying to mix apples and oranges. To make them compatible, we find a number they can both transform into equally.
In the given exercise, the original denominators were 8 and 9. Neither of these numbers can neatly divide into the other without leaving a remainder. Hence, we can't just add the fractions outright.

• Think of the common denominator as creating a common language between two fractions.
• It must be large enough to be a multiple of each original denominator.
• Through this common ground, you can accurately add, subtract, or otherwise compare fractions.

Least Common Multiple (LCM)

When searching for a common denominator, the least common multiple (LCM) plays a key role. The LCM of two numbers is the smallest number that can be divided by both of those numbers without leaving a remainder.
This is crucial as using the LCM keeps numbers manageable and calculations simpler. With numbers like 8 and 9, finding the LCM involves listing out the multiples of each number

• Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72...
• Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72...

The smallest matching number here is 72. This 72 becomes our common denominator, allowing us to transform and add the fractions smoothly. Remember, finding the LCM isn't just a step, it's a crucial strategy for managing fractions efficiently.

Adding Fractions

Let's get to the exciting part: adding fractions. Once both fractions have been expressed with a common denominator, you can easily perform addition. Here's how it works:
First, convert each fraction to its equivalent with the common denominator, which we’ve found to be 72.

• For \( \frac{5}{8} \), multiply both the numerator and the denominator by 9: \( \frac{5}{8} \times \frac{9}{9} = \frac{45}{72} \).
• For \( \frac{2}{9} \), do the same but multiply by 8: \( \frac{2}{9} \times \frac{8}{8} = \frac{16}{72} \).

Now, with \( \frac{45}{72} \) and \( \frac{16}{72} \), adding them is straightforward: simply add the numerators while keeping the common denominator: \( \frac{45 + 16}{72} = \frac{61}{72} \).
In this case, since 61 is a prime number, it doesn't divide evenly into 72, so the result is already in its simplest form. Thus, \( \frac{61}{72} \) is our final answer. Remember, when fractions share a common denominator, addition becomes a simple matter of addition in the numerators! This is the magic of getting a common denominator first.