- Convert \(\frac{1}{6}\) to \(\frac{5}{30}\) to match denominators.
- Set up the equation \(\frac{5}{30} + U = \frac{21}{30}\).
- Subtract \(\frac{5}{30}\) from both sides to find \(U = \frac{16}{30}\).
- Simplify the final result to \(\boxed{\frac{8}{15}}\).
Explanation
- Problem Analysis
We are given the equation \(\frac{1}{6} + U = \frac{21}{30}\), where \(U\) is the unknown number we need to determine. Our goal is to isolate \(U\) by using algebraic manipulation and common denominators.
- Finding a Common Denominator
To add or subtract fractions, it is helpful to express them with a common denominator. The denominators are 6 and 30. Since 30 is a multiple of 6, we can use 30 as the common denominator. We convert the fraction \(\frac{1}{6}\) by multiplying both the numerator and denominator by 5:
\[\frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30}\]
- Isolating and Solving for U
Now, we substitute \(\frac{5}{30}\) back into our original equation:
\[\frac{5}{30} + U = \frac{21}{30}\]
To solve for \(U\), we subtract \(\frac{5}{30}\) from both sides of the equation:
\[U = \frac{21}{30} - \frac{5}{30}\]
\[U = \frac{21 - 5}{30} = \frac{16}{30}\]
- Simplifying the Result
Finally, we simplify the fraction \(\frac{16}{30}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
\[U = \frac{16 \div 2}{30 \div 2} = \frac{8}{15}\]
The unknown number \(U\) is \(\frac{8}{15}\).
Examples
In everyday life, we often need to balance equations when splitting resources. For example, if you have a recipe that requires \(\frac{1}{6}\) of a cup of sugar and you want to know how much more sugar you need to reach a total of \(\frac{21}{30}\) of a cup for a larger batch, solving this type of linear equation helps ensure you get the proportions exactly right. Mastering fraction arithmetic ensures accuracy in cooking, construction, and budgeting.