Problem 88 - Elementary Algebra

Question

For exercises 81-96, evaluate. $$ \frac{9}{10}-\frac{1}{16} $$

Step-by-Step Solution

Answer

\( \frac{67}{80} \)

Step 1: Identify the fractions

The given problem involves subtracting two fractions, \(\frac{9}{10} - \frac{1}{16}\).

Step 2: Find a common denominator

To subtract these fractions, they need a common denominator. The denominators are 10 and 16. The least common multiple (LCM) of 10 and 16 is 80.

Step 3: Convert fractions to have the common denominator

Rewrite the fractions with the common denominator of 80. For \(\frac{9}{10}\), multiply the numerator and denominator by 8 to get \(\frac{72}{80}\). For \(\frac{1}{16}\), multiply the numerator and denominator by 5 to get \(\frac{5}{80}\).

Step 4: Subtract the fractions

Subtract the new fractions: \(\frac{72}{80} - \frac{5}{80} = \frac{67}{80}\).

Step 5: Simplify, if possible

The fraction \(\frac{67}{80}\) is already in simplest form.

Key Concepts

least common multiple

When subtracting fractions, it’s important to have a common denominator. The reason we need a common denominator is that fractions are parts of a whole. To properly subtract or add them, they must represent parts of the same sized slice. The least common multiple (LCM) is essential to this process. The LCM of two numbers is the smallest number that both original numbers can divide into without leaving a remainder. For our problem involving \(\frac{9}{10} - \frac{1}{16}\), we find the LCM of the denominators 10 and 16. We do this by listing the multiples of each number and finding the smallest common one.

Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80...
Multiples of 16: 16, 32, 48, 64, 80...
The smallest common multiple is 80. Therefore, the LCM of 10 and 16 is 80.
This LCM becomes our new common denominator for both fractions. By using the LCM, we ensure that we are accurately combining the fractions in terms of the same sized pieces.

common denominator

Now that we have our least common multiple, we need to convert each fraction to have this common denominator. The common denominator makes it easier to subtract one fraction from another. For \(\frac{9}{10}\), we see that 10 * 8 = 80. Therefore, we multiply both the numerator and the denominator by 8: \(\frac{9 \times 8}{10 \times 8} = \frac{72}{80}\).

For \(\frac{1}{16}\), we see that 16 * 5 = 80. Therefore, we multiply both the numerator and the denominator by 5: \(\frac{1 \times 5}{16 \times 5} = \frac{5}{80}\).
Using a common denominator allows us to directly subtract the numerators while keeping the denominator the same. So our problem now looks like this: \(\frac{72}{80} - \frac{5}{80} = \frac{67}{80}\). We simply subtract 5 from 72 to get 67.
This method ensures that we are dealing with like terms, making the process of subtraction straightforward.

simplifying fractions

After finding the difference between our fractions, we look at \(\frac{67}{80}\). Simplifying a fraction means reducing it to its simplest form where the numerator and the denominator have no common factors other than 1. In our case, \(\frac{67}{80}\) is already in simplest form. There is no number (other than 1) that divides both 67 and 80 evenly.
Notice: It's often a good practice to check whether the resulting fraction can be simplified. Here are a few tips for simplifying:

• Look for the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.
• If the GCD is 1, the fraction is already in simplest form.

For instance, if our result had been \(\frac{70}{80}\), we would recognize that both 70 and 80 can be divided by their GCD, which is 10, simplifying to \(\frac{7}{8}\).
This process ensures the fraction is as straightforward and easy to understand as possible.