Question

Find each sum or difference. $$\frac{5}{7}-\frac{2}{7}-\frac{3}{7}$$

Short answer

0

Step-by-step solution

- Identify the common denominators

Notice that all fractions share the same denominator, which is 7. This allows us to combine them directly by focusing on the numerators.

- Combine the numerators

Subtract and add the numerators: \(5 - 2 - 3\)

- Simplify the numerators

Calculate the result of the numerators: \(5 - 2 - 3 = 0\)

- Combine the result with the common denominator

The result of the numerators is 0, so place this over the common denominator: \( \frac{0}{7} \)

- Simplify the fraction

Any number divided by any non-zero number is 0, so \( \frac{0}{7} = 0 \).

Key concepts

Fractions

A fraction represents a part of a whole. It consists of two parts: the numerator (top number) and the denominator (bottom number).
The numerator tells how many parts we have, and the denominator tells into how many parts the whole is divided. For example, in \(\frac{5}{7}\), 5 is the numerator, and 7 is the denominator.
Fractions are essential in various math operations like addition, subtraction, multiplication, and division. They help in expressing numbers that are not whole numbers.

Subtraction

Subtraction is the process of taking one quantity away from another. With fractions, this involves subtracting the numerators while keeping the denominator the same if they have the same denominator.
For instance, with \(\frac{5}{7} - \frac{2}{7}\):

• Both fractions have the same denominator, so subtract the numerators.
• \(5 - 2 = 3\)
• Keep the common denominator: \: \(\frac{3}{7}\)

In the given example, there is a further subtraction step: \(\frac{3}{7} - \frac{3}{7} = 0\).

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. It allows fractions to be added, subtracted, or compared.
In our example, all fractions have the same denominator, 7. This makes our task easier because we can directly subtract the numerators without changing the denominators.

• A common denominator helps avoid complex calculations.
• It simplifies combining or comparing fractions.
• In case denominators are different, we need to find the least common multiple (LCM) of the denominators and convert the fractions accordingly.

Having a common denominator streamlines the subtraction process.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form where the numerator and denominator have no common factors other than 1.
In our example: \(\frac{0}{7} = 0\)

• Any fraction with a numerator of 0 is zero, so it’s fully simplified.
• For non-zero numerators, you’d divide the numerator and denominator by their greatest common divisor (GCD).
• Example: \(\frac{6}{8}\) simplifies to \(\frac{3}{4}\) by dividing both by 2.

Simplifying fractions makes them easier to understand and work with in further calculations.