Problem 86 - Prealgebra

Question

Painting Recently you purchased \(\frac{1}{2}\) gallon of paint to paint your dorm room. Once the job was finished you realized that you only used \(\frac{1}{3}\) of the gallon. What fractional amount of the paint is left in the can?

Step-by-Step Solution

Answer

\(\frac{1}{6}\) of a gallon of paint is left.

Step 1: Understanding the Problem

You initially purchased \(\frac{1}{2}\) gallon of paint. After painting, you've used \(\frac{1}{3}\) of a gallon. Now, we need to calculate the remaining paint after subtracting the used amount from the initial amount.

Step 2: Setting Up the Subtraction Problem

The problem can be set up as a subtraction of fractions: \(\frac{1}{2} - \frac{1}{3}\). To perform this subtraction, both fractions need to have the same denominator.

Step 3: Finding a Common Denominator

The least common denominator (LCD) for the fractions \(\frac{1}{2}\) and \(\frac{1}{3}\) is 6. We'll convert both fractions to have this denominator.

Step 4: Converting the Fractions

Convert \(\frac{1}{2}\) to \(\frac{3}{6}\) and \(\frac{1}{3}\) to \(\frac{2}{6}\). This is done by multiplying the numerator and the denominator by the necessary values: \(\frac{1}{2} \times \frac{3}{3} = \frac{3}{6}\) and \(\frac{1}{3} \times \frac{2}{2} = \frac{2}{6}\).

Step 5: Subtracting the Fractions

With a common denominator, you can now subtract: \(\frac{3}{6} - \frac{2}{6} = \frac{1}{6}\).

Step 6: Conclusion

The amount of paint left in the can is \(\frac{1}{6}\) of a gallon.

Key Concepts

Understanding Least Common Denominator

When subtracting fractions, it's essential for both fractions to share the same denominator. A denominator is simply the number at the bottom of a fraction. It's the total number that represents the whole to be divided. However, fractions might often have different denominators. To make subtraction easier, we find what is called the "Least Common Denominator" (LCD). The LCD is the smallest number that both denominators can divide into evenly.

Let's consider the example from the exercise: you have fractions \(\frac{1}{2}\) and \(\frac{1}{3}\). The denominators here are 2 and 3. The least common denominator for these fractions is 6, because it is the smallest number that both 2 and 3 can divide without leaving any remainder. So, keep in mind that finding the LCD is all about identifying a common ground for your denominators to help bring the fractions together in one unified view.

Steps for Subtracting Fractions

Once you've found the least common denominator, like the 6 in our example, it's time to rewrite both fractions so they can "play nice" together. This means converting each fraction to an equivalent fraction that has the LCD as its new denominator.

Here's how it works: take the fraction \(\frac{1}{2}\) and you know it needs to become a fraction out of 6. So you multiply both the numerator (top number) and the denominator (bottom number) by 3, giving you \(\frac{3}{6}\). Similarly, take \(\frac{1}{3}\), and convert it to sixths by multiplying both the numerator and the denominator by 2 to get \(\frac{2}{6}\).

Now they can be easily subtracted: subtract the numerators and keep the denominator the same. So, \(\frac{3}{6} - \frac{2}{6} = \frac{1}{6}\). And there you have it! The subtraction is complete and the difference gives you the amount of paint left.

Converting Fractions to Same Denominator

Converting fractions involves making both fractions comparable by having the same denominator so they can be subtracted or added easily. This is mostly required when neither fraction initially shares a common denominator. Using multiplication in conversion allows you to create equivalent fractions – different-looking fractions that represent the same amount.

Here's a recap with our example for converting:

• For \(\frac{1}{2}\), multiply the numerator and the denominator by 3 to get \(\frac{3}{6}\). You're scaling it up to match the common denominator.
• For \(\frac{1}{3}\), multiply the numerator and the denominator by 2 to get \(\frac{2}{6}\). The idea is similar: make it match the denominator we've chosen.

Once both fractions have the same denominator, operations become straightforward. Just work on the numerators while the denominator stays the same, creating a smooth path to solving addition or subtraction tasks with fractions.