Problem 49 - Prealgebra

Question

Add and subtract as indicated. $$\frac{5}{6}-\frac{1}{3}+\frac{4}{3}$$

Step-by-Step Solution

Answer

The answer is \( \frac{11}{6} \).

Step 1: Find a Common Denominator

To add or subtract fractions, they must have the same denominator. The fractions given are \( \frac{5}{6} \), \( \frac{1}{3} \), and \( \frac{4}{3} \). The least common denominator (LCD) for the denominators 6 and 3 is 6.

Step 2: Convert Fractions

Convert \( \frac{1}{3} \) and \( \frac{4}{3} \) so they have the denominator of 6. - Multiply the numerator and denominator of \( \frac{1}{3} \) by 2 to get \( \frac{2}{6} \).- Multiply the numerator and denominator of \( \frac{4}{3} \) by 2 to get \( \frac{8}{6} \).

Step 3: Perform Subtraction and Addition

Subtract \( \frac{2}{6} \) from \( \frac{5}{6} \): \[ \frac{5}{6} - \frac{2}{6} = \frac{3}{6} \].Add \( \frac{8}{6} \) to the result: \[ \frac{3}{6} + \frac{8}{6} = \frac{11}{6} \].

Step 4: Simplify the Expression if Necessary

In this case, the fraction \( \frac{11}{6} \) is already in its simplest form as an improper fraction.

Key Concepts

Adding Fractions

To add fractions, a common denominator is essential. This ensures that the pieces we are adding are of the same size. Think of it like this: you can't directly add apples and oranges unless you convert them to the same unit, say, pieces of fruit. Similarly, in fractions, the denominator tells you the size of the fraction's pieces.

Once fractions share a common denominator, simply add the numerators. For example, if we have \( \frac{3}{6} \) and \( \frac{8}{6} \), their denominators are already the same. Therefore, adding them involves adding 3 and 8 to get \( \frac{11}{6} \).

• Ensure the denominators are the same.
• Add the numerators.
• Write the sum over the common denominator.

Improper fractions, like \( \frac{11}{6} \), sometimes need to be simplified further depending on the problem's requirements. However, \( \frac{11}{6} \) here is its simplest improper form.

Subtracting Fractions

Subtracting fractions follows a similar principle to adding fractions: they must have the same denominator to be subtracted directly. Once you have a common denominator, subtraction involves taking the difference of the numerators.

Let's look at \( \frac{5}{6} - \frac{2}{6} \). With a shared denominator, subtract the numerators 5 and 2, resulting in \( \frac{3}{6} \). This is like taking away 2 slices of a 6-slice pizza from a pizza with 5 slices left.

• Ensure denominators are the same.
• Subtract the numerators.
• Write the difference over the common denominator.

Remember to check if further simplification is possible. For instance, \( \frac{3}{6} \) can be reduced to \( \frac{1}{2} \) by dividing both numerator and denominator by their greatest common divisor, which is 3.

Common Denominator

A common denominator is the foundation for adding or subtracting fractions. Think of it as finding a common language that the fractions can communicate in. The least common denominator (LCD) is the smallest number that the denominators can all divide into evenly.

In fractions like \( \frac{5}{6} \), \( \frac{1}{3} \), and \( \frac{4}{3} \), the LCD is 6 because both 3 and 6 can divide into 6 without leaving a remainder.

• Identify all denominators.
• Find the LCD.
• Convert other fractions to this common denominator.

This conversion might involve adjusting fractions by multiplying their numerator and denominator by the same number to maintain their size. For \( \frac{1}{3} \), multiply by 2, resulting in the equivalent \( \frac{2}{6} \). Similarly, \( \frac{4}{3} \) also converts to \( \frac{8}{6} \). By achieving a common denominator, fractions are ready for direct addition or subtraction.