Problem 75 - Prealgebra

Question

Add or subtract as indicated. $$\frac{3}{2}+\frac{5}{10}$$

Step-by-Step Solution

Answer

The result of the addition is 2.

Step 1: Identify the Operation

The given expression is \( \frac{3}{2} + \frac{5}{10} \). This means we need to add the two fractions.

Step 2: Find a Common Denominator

Determine a common denominator for the fractions. The denominators are 2 and 10. The least common multiple of 2 and 10 is 10.

Step 3: Convert the Fractions

Convert \( \frac{3}{2} \) so that its denominator is 10. The equivalent fraction is \( \frac{3 \times 5}{2 \times 5} = \frac{15}{10} \). \( \frac{5}{10} \) already has a denominator of 10.

Step 4: Add the Fractions

Add the numerators of the converted fractions: \( \frac{15}{10} + \frac{5}{10} = \frac{15 + 5}{10} = \frac{20}{10} \).

Step 5: Simplify the Result

Simplify \( \frac{20}{10} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 10: \( \frac{20}{10} = 2 \).

Key Concepts

Understanding the Least Common Denominator

When adding fractions, it's incredibly important to have the same denominator for both fractions. This makes the arithmetic much simpler. The least common denominator (LCD) is the smallest number that both denominators can divide into without leaving a remainder. In our example, we are working with the fractions \( \frac{3}{2} \) and \( \frac{5}{10} \). Here, the denominators are 2 and 10.

To find the LCD, we need to determine the smallest common multiple of these numbers. Factors of 2 include 2, 4, 6, 8, 10, etc., while factors of 10 are 10, 20, 30, etc. The smallest common factor in both lists is 10, which makes it the least common denominator. With the LCD, you can rewrite fractions so they have the same denominator, simplifying the addition process.

Simplifying Fractions

After performing operations on fractions, you often end up with fractions that can be reduced or simplified. Simplifying a fraction makes it as simple as possible, where the numerator and the denominator share no common factors other than 1.

For example, once we've added our two fractions \( \frac{15}{10} + \frac{5}{10} \) and reached \( \frac{20}{10} \), we notice that this fraction can be simplified. To do so, divide both the numerator and the denominator by their greatest common divisor (GCD).

In \( \frac{20}{10} \), both 20 and 10 share 10 as a common factor, which is their GCD. When we divide both the numerator and the denominator by 10, we get \( 2 \). Thus, \( \frac{20}{10} \) simplifies to 2, giving us the final answer in its simplest form.

Fraction Addition Steps Explained

Adding fractions might seem tricky at first, but following a consistent set of steps can make it much easier.

First, identify the operation you need to perform. In this case, it's addition: \( \frac{3}{2} + \frac{5}{10} \). Determine if you need a common denominator. Their denominators are different, specifically 2 and 10, so you need to find the least common denominator, which is 10.

Next, convert the fractions if necessary. Convert \( \frac{3}{2} \) to have the same denominator as \( \frac{5}{10} \). This becomes \( \frac{15}{10} \). Once both fractions share a denominator, simply add their numerators: \( \frac{15}{10} + \frac{5}{10} = \frac{20}{10} \).

Finally, remember to simplify the result where possible. Simplifying \( \frac{20}{10} \) by dividing both terms by 10 gives \( 2 \). This step ensures your answer is in the simplest form, making it clear and concise.