Question

Perform the indicated operation. Write all results in lowest terms. $$\frac{1}{4}+\frac{4}{9}-\frac{1}{9}$$

Short answer

Question: Simplify the following expression: $$\frac{1}{4} + \frac{4}{9} - \frac{1}{9}$$ Answer: $$\frac{7}{12}$$

Step-by-step solution

Find the least common denominator (LCD) for the fractions

To find the LCD of the three fractions, we need to find the least common multiple (LCM) of their denominators. In this case, the denominators are 4 and 9. Both numbers are divisible by 1, so their LCM is 4 x 9 = 36.

Convert the fractions to equivalent fractions with the LCD

Now, we'll convert each fraction to an equivalent fraction with a denominator of 36: $$\frac{1}{4} = \frac{1 \times 9}{4 \times 9} = \frac{9}{36}$$ $$\frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36}$$ $$\frac{1}{9} = \frac{1 \times 4}{9 \times 4} = \frac{4}{36}$$

Perform the arithmetic operations

Now, we'll perform the addition and subtraction on the equivalent fractions: $$\frac{9}{36} + \frac{16}{36} - \frac{4}{36} = \frac{9 + 16 - 4}{36} = \frac{21}{36}$$

Simplify the resulting fraction

To simplify the fraction, we find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 21 and 36 is 3. So, we can divide both the numerator and the denominator by 3: $$\frac{21}{36} = \frac{21 \div 3}{36 \div 3} = \frac{7}{12}$$ The result of the expression is $$\frac{7}{12}$$.