Question

Find each sum or difference. $$\frac{12}{15}-\frac{1}{15}-\frac{1}{15}$$

Short answer

\( \frac{2}{3} \)

Step-by-step solution

Identify the common denominator

All fractions have the same denominator: 15. This makes it simpler to combine them.

Combine the numerators

Since the denominators are the same, subtract the numerators. The given problem is: \[ \frac{12}{15} - \frac{1}{15} - \frac{1}{15} \]Combine the numerators: \[ 12 - 1 - 1 = 10 \]

Write the final fraction

Place the combined numerator over the common denominator:\[ \frac{10}{15} \]

Simplify the fraction

To simplify \( \frac{10}{15} \), find the greatest common divisor (GCD) of 10 and 15, which is 5. Divide both the numerator and the denominator by the GCD:\[ \frac{10 \text{ ÷ } 5}{15 \text{ ÷ } 5} = \frac{2}{3} \]

Key concepts

subtracting fractions

Subtracting fractions might seem tricky at first, but it's all about ensuring the denominators are the same. If the denominators are the same, you can directly subtract or add the numerators. For example, in the exercise: \[ \frac{12}{15} - \frac{1}{15} - \frac{1}{15} \] we see the same denominator (15) in all terms. So, we subtract the numerators: 12 - 1 - 1 to get 10. The fraction then becomes \[ \frac{10}{15}. \]

common denominators

Having common denominators is essential for adding or subtracting fractions. The denominator is the bottom number of the fraction and represents the total number of equal parts. If two fractions have the same denominator, you can easily combine them. In our exercise, the denominators were all 15, making the process straightforward. If you ever face different denominators, convert them to a common one by finding the least common multiple (LCM). Thus, \[ \frac{a}{b} + \frac{c}{d} \] can only be added if b=d or if both b and d are converted to the same value.

simplifying fractions

Once you've subtracted or added fractions, the last step is often to simplify the resulting fraction. Simplifying fractions makes them easier to understand and use. To simplify \[ \frac{10}{15}, \] find the greatest common divisor (GCD) of the numerator (10) and the denominator (15). The GCD of 10 and 15 is 5. Divide both the numerator and the denominator by their GCD: \[ \frac{10 \text{ ÷ } 5}{15 \text{ ÷ } 5} = \frac{2}{3}. \] Hence, simplified \[ \frac{10}{15} \] is \[ \frac{2}{3}. \]