Problem 60 - Prealgebra

Question

Add or subtract the following fractions, as indicated. $$\frac{3}{4}+\frac{5}{6}$$

Step-by-Step Solution

Answer

The sum is \( \frac{19}{12} \) or \( 1 \frac{7}{12} \).

Step 1: Find the Least Common Denominator (LCD)

To add fractions, they must have the same denominator. The denominators here are 4 and 6. The least common multiple of 4 and 6 is 12. So, the LCD is 12.

Step 2: Adjust the First Fraction

Convert the first fraction to an equivalent fraction with the denominator 12. \( \frac{3}{4} \) is converted by multiplying both the numerator and the denominator by 3, resulting in \( \frac{9}{12} \).

Step 3: Adjust the Second Fraction

Convert the second fraction to an equivalent fraction with the denominator 12. \( \frac{5}{6} \) is converted by multiplying both the numerator and the denominator by 2, resulting in \( \frac{10}{12} \).

Step 4: Add the Fractions

Now that the fractions have the same denominator, add the numerators together while keeping the denominator: \( \frac{9}{12} + \frac{10}{12} = \frac{19}{12} \).

Step 5: Simplify if Needed

The fraction \( \frac{19}{12} \) is an improper fraction. It can also be expressed as a mixed number: \( 1 \frac{7}{12} \).

Key Concepts

Understanding the Least Common Denominator

When adding or subtracting fractions, it is crucial that they share the same denominator. This is because the denominator, which is the bottom number of a fraction, indicates into how many equal parts the whole is divided. If the parts are not the same size, we cannot directly add or subtract the quantities represented by the fractions.
To achieve this, we find the least common denominator (LCD). The LCD is the smallest multiple that both denominators share. For instance, when working with the fractions \( \frac{3}{4} \) and \( \frac{5}{6} \), our denominators are 4 and 6. We look for the smallest number into which both 4 and 6 can divide evenly. Listing multiples might help:

• Multiples of 4: 4, 8, 12, 16, ...
• Multiples of 6: 6, 12, 18, 24, ...

The smallest common multiple here is 12. Thus, 12 is our least common denominator, allowing the fractions to be combined seamlessly.
Understanding and finding the LCD streamlines the process of fraction addition, ensuring accuracy.

Creating Equivalent Fractions

Once the least common denominator is identified, the next step is creating equivalent fractions. Equivalent fractions are different fractions that name the same amount. To make two fractions have the same denominator, we adjust them so although their appearance changes, their value does not.
For \( \frac{3}{4} \), we convert it to an equivalent fraction with a denominator of 12. Since 4 times 3 equals 12, we multiply both the numerator and the denominator by 3, giving us \( \frac{9}{12} \). Similarly, for \( \frac{5}{6} \), we multiply both the numerator and denominator by 2, turning it into \( \frac{10}{12} \). Now both fractions look different, but they are equivalent to their original forms and have matching denominators, allowing us to add them together.
Creating equivalent fractions is akin to changing the size of pieces to ensure a match, making the operation easy and logical.

Exploring Improper Fractions

Once you have added two fractions with a common denominator, the result might be an improper fraction. An improper fraction is when the numerator, or top number, is larger than the denominator.
In our example, summing \( \frac{9}{12} \) and \( \frac{10}{12} \) results in \( \frac{19}{12} \). Since 19 is greater than 12, this is an improper fraction. In many situations, you may prefer to convert this into a mixed number, which combines a whole number with a fraction.

• To convert \( \frac{19}{12} \) into a mixed number, divide 19 by 12, which equals 1 with a remainder of 7.
• This means \( \frac{19}{12} \) can also be expressed as \( 1 \frac{7}{12} \).

Understanding improper fractions and converting them when necessary helps to see the full picture of the quantity described, making mathematical communication clearer and more intuitive.