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Chapter 4: Problem 286

In the following exercises, find the difference. $$\frac{4}{21}-\frac{19}{21}$$

Short Answer

Expert verified
\frac{-5}{7}

Step by step solution

01

- Identify the Common Denominator

Both fractions \(\frac{4}{21}\) and \(\frac{19}{21}\) have the same denominator, which is 21.
02

- Subtract the Numerators

Since the denominators are the same, subtract the numerators. This gives us: \(\frac{4 - 19}{21} = \frac{-15}{21}\).
03

- Simplify the Fraction

Simplify \(\frac{-15}{21}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3. So, \(\frac{-15 \div 3}{21 \div 3} = \frac{-5}{7}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

common denominator
When subtracting fractions, it鈥檚 essential to work with a common denominator. The denominator is the bottom part of the fraction. Here, both fractions \(\frac{4}{21}\) and \(\frac{19}{21}\) already share the denominator 21.
This common denominator means you can subtract the numerators directly. The numerator is the top part of a fraction.
Think of the denominator like slices of a pie. If both fractions have the same number of slices in total, it鈥檚 easy to see how many slices they differ by.
greatest common divisor
To simplify fractions, you often need to find the greatest common divisor (GCD). The GCD is the largest number that can evenly divide both the numerator and the denominator.
In our exercise, the fraction after subtraction is \(\frac{-15}{21}\).
First, identify the GCD of -15 and 21. Start by listing out the factors:
  • Factors of 15: 1, 3, 5, 15
  • Factors of 21: 1, 3, 7, 21

The common factors are 1 and 3. The greatest one is 3. This means the GCD is 3.
simplifying fractions
Once you have the greatest common divisor, you can simplify the fraction. Simplifying means reducing it to its lowest terms.
To simplify \(\frac{-15}{21}\), divide both the numerator and the denominator by their GCD, which is 3: \(\frac{-15 \div 3}{21 \div 3} = \frac{-5}{7}\).
Now, \(\frac{-5}{7}\) is the simplified form. This process helps make fractions easier to understand and work with.

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Most popular questions from this chapter

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