Problem 51 - Algebra A Combined Function

Question

Add or subtract as indicated. See Examples 15 through \(20,22,\) and \(23 .\) $$ \frac{10}{3}-\frac{5}{21} $$

Step-by-Step Solution

Answer

The result is \( \frac{65}{21} \).

Step 1: Find a Common Denominator

To subtract fractions, we need a common denominator. The denominators are 3 and 21. The least common multiple (LCM) of 3 and 21 is 21. So, we'll use 21 as the common denominator.

Step 2: Convert Fractions

Convert each fraction to an equivalent fraction with the common denominator of 21.For the first fraction \( \frac{10}{3} \), multiply both the numerator and the denominator by 7 to get \( \frac{70}{21} \).For the second fraction \( \frac{5}{21} \), it already has the denominator 21, so it remains \( \frac{5}{21} \).

Step 3: Subtract Fractions

Now that both fractions have a common denominator, subtract the second fraction from the first:\[\frac{70}{21} - \frac{5}{21} = \frac{70 - 5}{21} = \frac{65}{21}.\]

Step 4: Simplify the Result

The last step is to simplify the fraction if possible. Divide both the numerator and the denominator by their greatest common divisor (GCD). For \( \frac{65}{21} \), the GCD is 1, so the fraction is already in its simplest form.

Key Concepts

Least Common Multiple

When working with the subtraction of fractions, finding the Least Common Multiple (LCM) is key. The LCM of two numbers is the smallest number that is a multiple of both. For example, when subtracting fractions such as \( \frac{10}{3} \) and \( \frac{5}{21} \), the denominators are 3 and 21.
The LCM of 3 and 21 is the smallest number divisible by both. To determine if a number is a common multiple, you can list the multiples of each number and find the smallest one they share:

• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, ...
• Multiples of 21: 21, 42, 63, ...

Here, 21 is the LCM because it is the first number that appears on both lists. Using the LCM as the common denominator simplifies adding or subtracting fractions.

Common Denominator

Obtaining a common denominator is crucial when adding or subtracting fractions. Once you identify the least common multiple, you can convert each fraction to an equivalent fraction sharing this common denominator. This simplifies the computation.
For the fractions \( \frac{10}{3} \) and \( \frac{5}{21} \), we use 21 as the common denominator. The steps include:

• Multiply the numerator and the denominator of \( \frac{10}{3} \) by 7 (since \(3 \times 7 = 21\)), to get \( \frac{70}{21} \).
• Since \( \frac{5}{21} \) is already over 21, it remains \( \frac{5}{21} \).

These equivalent fractions allow us to easily perform subtraction as both fractions are now aligned over the same base.

Simplifying Fractions

After solving a fraction operation, the next step is often to simplify the result. Simplifying a fraction means reducing it to its lowest terms. This involves finding any common factors between the numerator and the denominator and dividing by these factors.
In our problem, after subtracting \( \frac{70}{21} - \frac{5}{21} \), we get \( \frac{65}{21} \). To simplify it, check for common factors. However, since the greatest common divisor (GCD) of 65 and 21 is 1, this fraction is already in its simplest form.
Always ensure to check if further reduction is possible, as this could make the work cleaner and more readable.

Greatest Common Divisor

The Greatest Common Divisor (GCD) of two numbers is the largest number that divides both without a remainder. It is a handy tool when simplifying fractions.
To simplify \( \frac{65}{21} \), find its GCD. Here, list the divisors:

• Divisors of 65: 1, 5, 13, 65
• Divisors of 21: 1, 3, 7, 21

Their GCD is 1, which means they have no larger common number other than 1. Consequently, \( \frac{65}{21} \) remains unchanged as it is already in its simplest form. Understanding GCD is essential for fraction operations, ensuring expressions are reduced correctly.