Problem 185 - Intermediate Algebra

Question

In the following exercises, add or subtract. (a) \(\frac{2}{3}+\frac{1}{6}\) (b) \(\frac{2}{3} \div \frac{1}{6}\)

Step-by-Step Solution

Answer

(a) \(\frac{5}{6}\) (b) 4

Step 1 - Find a common denominator (Part a)

To add the fractions \(\frac{2}{3} + \frac{1}{6}\), first find the least common denominator (LCD) of 3 and 6. The LCD of 3 and 6 is 6.

Step 2 - Convert fractions to a common denominator (Part a)

Convert \(\frac{2}{3}\) to a fraction with denominator 6 by multiplying both the numerator and the denominator by 2: \(\frac{2 \times 2}{3 \times 2} = \frac{4}{6}\). Now the fractions are \(\frac{4}{6}\) and \(\frac{1}{6}\).

Step 3 - Add the fractions (Part a)

Add the fractions: \(\frac{4}{6} + \frac{1}{6} = \frac{4 + 1}{6} = \frac{5}{6}\).

Step 4 - Write the equation in simplified form (Part b)

For the division \(\frac{2}{3} \div \frac{1}{6}\), rewrite the expression as multiplication by the reciprocal: \(\frac{2}{3} \times \frac{6}{1}\).

Step 5 - Multiply the fractions (Part b)

Multiply the numerators together and the denominators together: \(\frac{2 \times 6}{3 \times 1} = \frac{12}{3}\).

Step 6 - Simplify the result (Part b)

Simplify \(\frac{12}{3}\) by dividing the numerator by the denominator: \(\frac{12}{3} = 4\).

Key Concepts

least common denominator

When dealing with the addition or subtraction of fractions, the least common denominator (LCD) is crucial. The LCD is the smallest number that both denominators can divide into evenly. For instance, in the problem \(\frac{2}{3}+\frac{1}{6}\), you need to find the LCD of the denominators 3 and 6. The least common denominator for 3 and 6 is 6.

• Determine the multiples of each denominator.
• Identify the smallest multiple common to both denominators.

Now, you build equivalent fractions that share this LCD. For \(\frac{2}{3}\), multiply both the numerator and the denominator by 2 (because 6 divided by 3 equals 2) to get \(\frac{4}{6}\).
This way, you convert the fractions, sharing the same denominator, making them easier to add or subtract.

fraction multiplication

Fraction multiplication is straightforward compared to addition or subtraction of fractions. You simply multiply across the numerators and the denominators. For example, in the division problem \(\frac{2}{3} \div \frac{1}{6}\), you need to convert the division into multiplication by using the reciprocal. So, \(\frac{2}{3} \div \frac{1}{6} \) becomes \(\frac{2}{3} \times \frac{6}{1} \).

• Flip the second fraction (find the reciprocal) when dividing.
• Multiply the numerators together.
• Multiply the denominators together.

This results in \(\frac{2 \times 6}{3 \times 1} = \frac{12}{3}\) which you then proceed to simplify. Simple as that.

fraction simplification

Simplifying fractions is the process of reducing them to their simplest form, which means the numerator and denominator are the smallest possible integers. In the example \(\frac{12}{3} \), you divide both the numerator and the denominator by their greatest common divisor (GCD).

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

For \(\frac{12}{3}\), the GCD is 3. Dividing both numbers by 3 gives \(\frac{12}{3} = 4\). This reduces the fraction to its simplest form, making it easier to understand and work with in further calculations.