Problem 61 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ \frac{3}{10}+\frac{5}{10} $$

Step-by-Step Solution

Answer

\frac{4}{5}

Step 1: Identify Common Denominator

Both fractions \(\frac{3}{10}\) and \(\frac{5}{10}\) have the same denominator which is 10.

Step 2: Add the Numerators

Since the denominators are the same, add the numerators directly: \(\frac{3+5}{10}\)

Step 3: Simplify the Fraction

Perform the addition in the numerator: \(\frac{8}{10}\). This can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \(\frac{8÷2}{10÷2}=\frac{4}{5}\)

Key Concepts

Common Denominator

When dealing with the addition of fractions, the first step is to ensure they have a common denominator. The denominator is the number at the bottom of the fraction. It represents how many total parts the whole is divided into.
If fractions have the same denominator, they can be easily added or subtracted. For example, in our exercise \( \frac{3}{10} \) and \( \frac{5}{10} \) both have the denominator 10.
This means we can add them directly. Always look for a common denominator when working with fractions for seamless addition or subtraction.

Numerator

The numerator is the number at the top of the fraction. It shows how many parts we have out of the whole represented by the denominator. In our case, we have two numerators: 3 and 5 from the fractions \( \frac{3}{10} \) and \( \frac{5}{10} \) respectively.
When the denominators are the same, you simply add the numerators together. Here we get: \[ \frac{3 + 5}{10} = \frac{8}{10} \].
Therefore, understanding numerators helps in correctly adding the parts represented by each individual fraction.

Simplifying Fractions

After adding fractions, the next step is often simplifying the result. Simplifying a fraction means making it as simple as possible without changing its value. This can involve dividing the numerator and the denominator by the same number.
As in our problem, we obtained \( \frac{8}{10} \). To simplify, we look for the greatest common divisor (GCD) of 8 and 10, which is 2. Then, we divide both the numerator and the denominator by their GCD: \[ \frac{8 \div 2}{10 \div 2} = \frac{4}{5} \]
This results in the simplest form of the fraction.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. This is crucial when simplifying fractions.
Finding the GCD starts with identifying all the common divisors of both numbers. For 8 and 10, the divisors are:

• 8: 1, 2, 4, 8
• 10: 1, 2, 5, 10

Here, the largest common divisor is 2. We then use it to simplify the fraction by dividing both the numerator and the denominator by this number: \[ \frac{8\div 2}{10\div 2} = \frac{4}{5} \].
Knowing how to find the GCD makes working with fractions much easier and conversion cleaner.