Problem 72 - Elementary and Intermediate Algebra

Question

Perform the indicated operation and, if possible, simplify. If there are no variables, check using a calculator. $$ \frac{5}{7}-\frac{5}{21} $$

Step-by-Step Solution

Answer

\(\frac{10}{21}\)

Step 1: Find a Common Denominator

To subtract fractions, first make sure they have the same denominator. Identify the least common denominator (LCD) for the fractions \(\frac{5}{7}\) and \(\frac{5}{21}\). The LCD of 7 and 21 is 21.

Step 2: Adjust the Fractions

Convert the fraction \(\frac{5}{7}\) to have a denominator of 21. Multiply the numerator and the denominator by 3: \ \frac{5}{7} \times \frac{3}{3} = \frac{15}{21}. Now the fractions are \(\frac{15}{21}\) and \(\frac{5}{21}\).

Step 3: Perform the Subtraction

Subtract the fractions now that they have the same denominator: \(\frac{15}{21} - \frac{5}{21} = \frac{15-5}{21} = \frac{10}{21}\).

Step 4: Simplify if Possible

Check if the resulting fraction can be simplified. The greatest common divisor of 10 and 21 is 1, so \(\frac{10}{21}\) is already in its simplest form.

Step 5: Verify Using a Calculator

Confirm the result using a calculator. \(\frac{5}{7} = 0.714285714\) and \(\frac{5}{21} = 0.238095238\). Subtracting these gives: \(0.714285714 - 0.238095238 = 0.476190476\). Converting \(\frac{10}{21}\) to decimal form also gives \(0.476190476\), confirming the answer.

Key Concepts

least common denominator

When subtracting fractions, the first step is to find a common base or denominator. The least common denominator (LCD) is the smallest number that is a multiple of the denominators of both fractions involved. For the fractions \(\frac{5}{7}\) and \(\frac{5}{21}\), we need to find the smallest number that both 7 and 21 can evenly divide into. The prime factorization method or LCM (Least Common Multiple) method can be helpful, but here it's clear that 21 is the smallest common multiple. This means, we'll convert both fractions to have a denominator of 21, allowing us to combine them easily.

fraction subtraction

Once fractions share the same denominator, the subtracting process becomes straightforward. With fractions like \(\frac{15}{21}\) and \(\frac{5}{21}\), keep the denominator the same and subtract the numerators: \(\frac{15-5}{21} = \frac{10}{21}\). The shared denominator simplifies the process because you focus solely on the top parts (numerators). Ensuring both fractions have the same denominator before subtracting makes everything less complicated and more systematic.

fraction simplification

Simplifying fractions means reducing them to their most basic form. Check if the numerator and denominator share any common factors. For \(\frac{10}{21}\), the greatest common divisor (GCD) of 10 and 21 is 1, meaning they don’t share any prime factors other than 1. Therefore, \(\frac{10}{21}\) is already simplified. Simplification helps in presenting the fraction in its cleanest, most understandable form.

numerator and denominator adjustment

Adjusting numerators and denominators is key when working with different denominators. For our example, \(\frac{5}{7}\) needed to be converted to a fraction with a denominator of 21. To do this, multiply both the numerator and the denominator by the same number (in this case, 3). This gives us: \(\frac{5}{7} \times \frac{3}{3} = \frac{15}{21}\). This step ensures that both fractions have a common base, permitting easy subtraction. Proper adjustment maintains the value of the fractions while aligning their denominators.