Problem 101 - Elementary Algebra Within Reach

Question

In Exercises 97-102, evaluate the expression. $$ \frac{5}{16}-\frac{3}{10} $$

Step-by-Step Solution

Answer

The answer is \(\frac{1}{80}\).

Step 1: Find the Least Common Denominator (LCD)

In order to subtract these fractions, they need to share the same denominator. The LCD or Least Common Denominator of 16 and 10 is 80. This is the smallest number that both 10 and 16 divide evenly into.

Step 2: Convert the Fractions to have the LCD as the Denominator

For each fraction, multiply both the numerator and denominator by the factor needed to turn the original denominator into 80. For \(\frac{5}{16}\), multiply by 5. For \(\frac{3}{10}\), multiply by 8. This gives the new fractions \(\frac{25}{80}\) and \(\frac{24}{80}\).

Step 3: Subtract the Fractions

Now both fractions have the same denominator, so they can be subtracted. Subtraction is carried out on the numerators only: \(\frac{25}{80}-\frac{24}{80}=\frac{1}{80}\).

Key Concepts

Understanding Least Common Denominator

When subtracting or adding fractions, it is crucial for them to have the same denominator. The least common denominator (LCD) is the smallest multiple that is common to all denominators in the set. Finding the LCD simplifies the subtraction or addition process by aligning the fractions to a common base. For example, let's consider the fractions \( \frac{1}{3} \) and \( \frac{1}{4} \). The denominators here are 3 and 4.

The multiples of 3 are: 3, 6, 9, 12, 15, 18, and so forth. The multiples of 4 are: 4, 8, 12, 16, 20, etc. As you can see, the smallest number common to both lists is 12. Thus, 12 is the LCD for the fractions \( \frac{1}{3} \) and \( \frac{1}{4} \). Once you have found the LCD, you can proceed to convert the fractions so that they share this denominator.

Determining the LCD is an essential skill that helps streamline the fraction subtraction process.

Converting Fractions to Have the Same Denominator

Once the least common denominator is determined, the next step is converting the fractions so that they have this same denominator. This process involves adjusting the numerators according to the factor by which their original denominators are multiplied to reach the LCD. This ensures the value of the fraction remains unchanged.

For instance, take the fractions \( \frac{2}{5} \) and \( \frac{3}{7} \). If their LCD is 35, here's how you convert them:

• Multiply both the numerator and denominator of \( \frac{2}{5} \) by 7 (since \( 5 \times 7 = 35 \)), resulting in \( \frac{14}{35} \).
• Multiply both the numerator and denominator of \( \frac{3}{7} \) by 5 (because \( 7 \times 5 = 35 \)), resulting in \( \frac{15}{35} \).

By converting them in this manner, both fractions now have a denominator of 35, making them ready for subtraction or addition.

Steps for Subtracting Fractions

Subtracting fractions with different denominators involves a few clear steps, ensuring each fraction has the same base to guarantee accuracy in subtraction.

Here’s a quick rundown of the process:

• Identify the LCD: Determine the smallest common multiple of the two denominators.
• Convert the Fractions: Adjust the numerators and denominators so both fractions have the LCD as the new denominator.
• Subtract the Numerators: With both fractions now sharing a common denominator, you can simply subtract the numerators and keep the denominator the same.
• Simplify the Result: If possible, reduce the resulting fraction to its simplest form.

Let's break it down with an example: For \( \frac{7}{12} - \frac{5}{18} \), with an LCD of 36, the fractions convert to \( \frac{21}{36} - \frac{10}{36} \), resulting in \( \frac{11}{36} \). This calculated difference remains in its simplest form, concluding the subtraction.