Problem 230 - Intermediate Algebra

Question

In the following exercises, evaluate. \(\frac{5}{12}-w\) when (a) \(w=\frac{1}{4}\) (b) \(w=-\frac{1}{4}\)

Step-by-Step Solution

Answer

(a) \(\frac{1}{6}\), (b) \(\frac{2}{3}\)

Step 1 - Substitute the value of w for part (a)

In the given expression \(\frac{5}{12} - w\), substitute \(w\) with \(\frac{1}{4}\): \(\frac{5}{12} - \frac{1}{4}\)

Step 2 - Find a common denominator for part (a)

The common denominator for \(\frac{5}{12}\) and \(\frac{1}{4}\) is 12. Convert \(\frac{1}{4}\) to have a denominator of 12: \(\frac{1}{4} = \frac{3}{12}\)

Step 3 - Perform the subtraction for part (a)

Now subtract \(\frac{3}{12}\) from \(\frac{5}{12}\): \(\frac{5}{12} - \frac{3}{12} = \frac{2}{12} = \frac{1}{6}\)

Step 4 - Substitute the value of w for part (b)

In the given expression \(\frac{5}{12} - w\), substitute \(w\) with \(-\frac{1}{4}\): \(\frac{5}{12} - (-\frac{1}{4})\)

Step 5 - Simplify the expression for part (b)

Subtracting a negative number is the same as adding the positive equivalent: \(\frac{5}{12} + \frac{1}{4}\)

Step 6 - Find a common denominator for part (b)

The common denominator for \(\frac{5}{12}\) and \(\frac{1}{4}\) is 12. Convert \(\frac{1}{4}\) to have a denominator of 12: \(\frac{1}{4} = \frac{3}{12}\)

Step 7 - Perform the addition for part (b)

Now add \(\frac{3}{12}\) to \(\frac{5}{12}\): \(\frac{5}{12} + \frac{3}{12} = \frac{8}{12} = \frac{2}{3}\)

Key Concepts

Subtraction of Fractions

Subtraction of fractions can seem tricky at first, but once you understand the core principles, it becomes straightforward. The most important thing to remember is that you cannot directly subtract fractions unless their denominators (bottom numbers) are the same. Consider the fractions \(\frac{5}{12}\) and \(\frac{1}{4}\). To subtract \(\frac{1}{4}\) from \(\frac{5}{12}\), we need to express both fractions with a common denominator. After that’s done, all you need to do is subtract the numerators (top numbers), leaving the denominator the same.
For instance, when we changed \(\frac{1}{4}\) to \(\frac{3}{12}\), this allowed us to directly perform the subtraction: \(\frac{5}{12} - \frac{3}{12} = \frac{2}{12}\), which simplifies to \(\frac{1}{6}\).

Common Denominators

A common denominator is essential when performing operations like addition and subtraction of fractions. It means both fractions have the same bottom number, which allows for easy calculation between them.
To find a common denominator, look for the least common multiple (LCM) of the denominators of your fractions. In our example, we needed a common denominator for 12 and 4. The LCM of these two numbers is 12. Here’s what we did:

• Converted \(\frac{1}{4}\) to an equivalent fraction with a denominator of 12 by multiplying both its numerator and denominator by 3, resulting in \(\frac{3}{12}\).

With both fractions now having the same denominators, they are ready for straightforward subtraction.

Negative Numbers

Understanding how to handle negative numbers is crucial for correctly solving many math problems, including those involving fractions. When we subtract a negative number, it’s the same as addition.
For example, in part (b), we were asked to evaluate \(\frac{5}{12} - (-\frac{1}{4})\). Here’s how we tackled this:

• First, recognize that subtracting \(-\frac{1}{4}\) is the same as adding \(\frac{1}{4}\).
• Next, convert \(\frac{1}{4}\) to \(\frac{3}{12}\) just like we did in part (a).

Now we simply add: \(\frac{5}{12} + \frac{3}{12} = \frac{8}{12}\), which simplifies to \(\frac{2}{3}\).
Remember, flipping a negative to a positive can simplify your calculations immensely!