Question

Evaluate each expression. $$ \frac{2}{3}+\frac{4}{5} \cdot \frac{1}{6} $$

Short answer

\frac{4}{5}

Step-by-step solution

Identify the operations

The expression \(\frac{2}{3} + \frac{4}{5} \times \frac{1}{6}\) includes addition and multiplication. According to the order of operations (PEMDAS/BODMAS), handle the multiplication first.

Multiply the fractions

Calculate \(\frac{4}{5} \times \frac{1}{6}\). Multiply the numerators together and the denominators together: \(\frac{4 \times 1}{5 \times 6} = \frac{4}{30}\). Simplify the fraction by finding the greatest common divisor (GCD) of 4 and 30, which is 2. Therefore, \(\frac{4}{30} = \frac{2}{15}\).

Add the fractions

Now add \(\frac{2}{3}\) and \(\frac{2}{15}\). Find a common denominator, which is 15. Convert \(\frac{2}{3}\) to \(\frac{2 \times 5}{3 \times 5} = \frac{10}{15}\). Now add \(\frac{10}{15} + \frac{2}{15} = \frac{12}{15}\).

Simplify the result

Simplify \(\frac{12}{15}\) by dividing the numerator and the denominator by their GCD, which is 3. So, \(\frac{12}{15} = \frac{4}{5}\).

Key concepts

Fraction Multiplication

When multiplying fractions, it is important to follow a simple process. You multiply the numerators (the top numbers) together and then multiply the denominators (the bottom numbers) together.
For example, if you have \(\frac{4}{5} \times \frac{1}{6}\), you multiply 4 by 1, giving you 4, and 5 by 6, giving you 30. The result is \(\frac{4}{30}\).
This method works regardless of the size of the fractions, making it a straightforward way to find the product of any two fractions.

Simplifying Fractions

Simplifying fractions means making them easier to work with by reducing them to their smallest form.
To do this, find the greatest common divisor (GCD) of the numerator and the denominator. For \(\frac{4}{30}\), the GCD is 2.
Divide both the numerator and denominator by 2 to get \(\frac{4 ÷ 2}{30 ÷ 2} = \frac{2}{15}\). Simplifying makes fractions easier to understand and use in further calculations.

Common Denominator

Adding fractions requires them to have a common denominator. This means the bottom numbers must be the same.
To find a common denominator, identify the least common multiple (LCM) of the denominators. For \(\frac{2}{3}\) and \(\frac{2}{15}\), the LCM of 3 and 15 is 15.
Convert \(\frac{2}{3}\) to an equivalent fraction with 15 as the denominator by multiplying both the numerator and the denominator by 5, giving \(\frac{2 \times 5}{3 \times 5} = \frac{10}{15}\).
Now you can easily add the fractions: \(\frac{10}{15} + \frac{2}{15} = \frac{12}{15}\). Simplify \(\frac{12}{15}\) by dividing by their GCD, which is 3, to get \(\frac{4}{5}\).