Find the following sum or difference, and write in lowest terms.
$\frac{5}{12}-\frac{5}{18}$
To compute the difference between two fractions, \(\frac{5}{12}\) and \(\frac{5}{18}\), we first observe that they have different denominators. To subtract them, we must find a common denominator. The least common multiple (LCM) of 12 and 18 will serve as the smallest common denominator. We can find the prime factorization of each denominator: \(12 = 2^2 \times 3\) and \(18 = 2 \times 3^2\). The least common multiple is formed by taking the highest power of each prime present: \(2^2 \times 3^2 = 4 \times 9 = 36\).
Now, we convert each fraction to an equivalent one with a denominator of 36. For the first fraction: \(\frac{5}{12} = \frac{5 \times 3}{12 \times 3} = \frac{15}{36}\). For the second fraction: \(\frac{5}{18} = \frac{5 \times 2}{18 \times 2} = \frac{10}{36}\). Now that the denominators are identical, we subtract the numerators: \(\frac{15}{36} - \frac{10}{36} = \frac{15-10}{36} = \frac{5}{36}\).
To ensure the result is in its lowest terms, we check if the numerator and the denominator share any common factors other than 1. The prime factors of 5 are simply 5. Since 36 is not divisible by 5, the fraction \(\frac{5}{36}\) is already in its simplest form. Thus, \(\frac{5}{12} - \frac{5}{18} = \frac{5}{36}\).
In daily life, working with fractions is common when measuring ingredients or splitting portions. For example, if you have \(\frac{5}{12}\) of a liter of juice in a bottle and you pour out \(\frac{5}{18}\) of a liter into a glass, calculating how much is left requires subtracting these fractions. Using the method of a common denominator ensures you correctly determine that \(\frac{5}{36}\) of a liter remains in the bottle. This skill is essential for precise cooking, measuring, and resource management.