Problem 102 - Prealgebra

Question

Add. $$\frac{5}{6}+\frac{2}{3}$$

Step-by-Step Solution

Answer

The sum is \(\frac{3}{2}\).

Step 1: Find a Common Denominator

To add fractions, they must have the same denominator. The denominators here are 6 and 3. The least common denominator (LCD) is the smallest multiple both denominators share. The LCD of 6 and 3 is 6.

Step 2: Convert Fractions to Equivalent Fractions

Convert the fractions so they both have the common denominator of 6. The fraction \(\frac{5}{6}\) already has the denominator 6, so it remains unchanged. For \(\frac{2}{3}\), convert it by multiplying both the numerator and denominator by 2 to get \(\frac{4}{6}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, add them by adding their numerators: \(\frac{5}{6} + \frac{4}{6} = \frac{5+4}{6} = \frac{9}{6}\).

Step 4: Simplify the Fraction

Simplify \(\frac{9}{6}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3. Thus, \(\frac{9}{6} = \frac{9\div3}{6\div3} = \frac{3}{2}\).

Key Concepts

Common Denominator

When adding fractions, a common denominator is essential. The denominators indicate how many equal parts make up a whole. To add fractions, they must refer to the same size parts, which only happens when their denominators are the same.

The least common denominator (LCD) is the smallest number that both denominators can divide into without leaving a remainder. Think of it as the smallest multiple both numbers share. In our example, the fractions are \( \frac{5}{6} \) and \( \frac{2}{3} \).

The denominators are 6 and 3 respectively. Here:

• The multiples of 6 are: 6, 12, 18, 24, ...
• The multiples of 3 are: 3, 6, 9, 12, 15, ...

The smallest common multiple of both is 6, which is the LCD. Once the LCD is determined, every fraction involved in the equation can be expressed with this same denominator, facilitating their addition.

Equivalent Fractions

Equivalent fractions represent the same part of a whole, even though they may seem different at a glance. Converting fractions to have a common denominator often involves rewriting them as equivalent fractions.

To create an equivalent fraction, multiply both the numerator and the denominator by the same number. This transformation does not change the value of the fraction. For instance, \( \frac{2}{3} \) needs to have the same denominator as \( \frac{5}{6} \), which is 6.

To do this:

• Multiply the numerator (2) and the denominator (3) by 2, resulting in \( \frac{4}{6} \).

Doing so makes it possible to add \( \frac{4}{6} \) to \( \frac{5}{6} \), as both fractions now have the same parts of the whole, thanks to the common denominator they share.

Simplifying Fractions

After performing operations with fractions, like addition, simplifying the result is important. Simplifying involves making a fraction as straightforward as possible, using the smallest numbers available that still have the same value.

To simplify \( \frac{9}{6} \), find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that can divide both the numerator and the denominator without leaving any remainder. For 9 and 6, the GCD is 3.

By dividing the numerator (9) and the denominator (6) by 3, we get:

• Numerator: \( 9 \div 3 = 3 \)
• Denominator: \( 6 \div 3 = 2 \)

Thus, \( \frac{9}{6} \) simplifies to \( \frac{3}{2} \). Simplifying fractions not only makes them neater but also easier to understand and compare.