Simplify: $\frac{1}{9}+\frac{2}{9}+\frac{4}{9}$
The objective is to simplify the sum of three rational numbers: \(\frac{1}{9} + \frac{2}{9} + \frac{4}{9}\). All three fractions share a common denominator of \(9\). Adding fractions with common denominators is straightforward; we simply add the numerators while keeping the common denominator unchanged.
To add these fractions, we follow the rule \(\frac{a}{c} + \frac{b}{c} + \frac{d}{c} = \frac{a + b + d}{c}\). Applying this to our expression:
\[\frac{1}{9} + \frac{2}{9} + \frac{4}{9} = \frac{1 + 2 + 4}{9}\]
Summing the values in the numerator, we get \(1 + 2 + 4 = 7\). Thus, the sum is \(\frac{7}{9}\).
The resulting fraction is \(\frac{7}{9}\). We check if this can be simplified further by looking for common factors between the numerator (7) and the denominator (9). Since 7 is a prime number and is not a factor of 9 (which is \(3^2\)), the fraction is already in its simplest form. Therefore, the final result is \(\frac{7}{9}\).
Imagine you are baking a recipe that calls for different portions of a cup of flour, and you have measuring spoons representing different fractions. If you need \(\frac{1}{9}\) of a cup, \(\frac{2}{9}\) of a cup, and \(\frac{4}{9}\) of a cup for different steps in your recipe, you can simply add them together to find that you need a total of \(\frac{7}{9}\) of a cup of flour. This demonstrates how adding fractions helps in precise measurements for daily tasks like cooking, carpentry, or crafting.