Problem 37 - Beginning and Intermediate Algebra

Question

Add or subcract as indicared. Write the answer in lower terms. See Erample 5 $$ \frac{17}{21}-\frac{10}{21} $$

Step-by-Step Solution

Answer

\( \frac{1}{3} \)

Step 1: Identify Like Denominators

Both fractions in the expression \( \frac{17}{21} - \frac{10}{21} \) have the same denominator, which is 21. This allows us to proceed directly with the subtraction.

Step 2: Subtract the Numerators

Since the denominators are the same, subtract the numerators: \( 17 - 10 = 7 \). The resulting fraction is \( \frac{7}{21} \).

Step 3: Simplify the Fraction

To simplify \( \frac{7}{21} \), find the greatest common divisor (GCD) of 7 and 21, which is 7. Divide both the numerator and the denominator by their GCD: \( \frac{7 \div 7}{21 \div 7} = \frac{1}{3} \).

Step 4: Final Step: Write the Solution

The simplified result of the subtraction \( \frac{17}{21} - \frac{10}{21} \) is \( \frac{1}{3} \).

Key Concepts

Like Denominators

When working with fractions, like denominators make the arithmetic a little simpler. **Like denominators** mean that the bottom numbers of the fractions you are working with are the same. For instance, in \[ \frac{17}{21} - \frac{10}{21} \] the denominator for both fractions is 21.

• This uniformity allows us to focus only on the numerators (or top numbers) during operations like addition or subtraction.
• You don't have to worry about converting the fractions, which saves you time.

Tip: When the denominators are like this, you can directly add or subtract the numerators.
This is why in our example, we can go straight to subtracting the numerators.

Subtracting Fractions

Subtracting fractions with like denominators can be a breeze if you follow the right steps. First, ensure that the denominators are indeed the same. Once confirmed, focus solely on the numerators. For \[ \frac{17}{21} - \frac{10}{21} \] we begin by simply subtracting the top numbers:

• Subtract 10 from 17, which gives us 7.
• Keep the denominator the same, which is 21.

This operation results in the fraction \[ \frac{7}{21} \] after subtraction. Remember, the whole operation hinges on having like denominators, which lets you only work with the numerators.
Keep in mind: If the denominators differ, you'd first need to find a common denominator.

Simplifying Fractions

Once you have completed the subtraction, what you often end up with is an expression that might not be in its simplest form. Simplifying fractions takes your result down to its most reduced version. To simplify \[ \frac{7}{21} \] you need to find the greatest common divisor (GCD) of both the numerator and the denominator. In this case:

• The GCD of 7 and 21 is 7.
• Divide both the numerator (7) and the denominator (21) by their GCD.

This division results in \[ \frac{7 \div 7}{21 \div 7} = \frac{1}{3} \].
Simplifying fractions:

• Helps make fractions easier to understand and compare.
• Is a crucial final step that ensures your answer is presented in the most clear and concise way.