Question

Evaluate each expression. $$ \frac{3}{20}-\frac{2}{15} $$

Short answer

\(\frac{1}{60}\)

Step-by-step solution

Identify the Least Common Denominator (LCD)

For the fractions \(\frac{3}{20}\) and \(\frac{2}{15}\), find the least common denominator. The denominators are 20 and 15. The LCD of 20 and 15 is the smallest number that both denominators can divide into, which is 60.

Convert Fractions to Equivalent Fractions with the LCD

Convert each fraction to an equivalent fraction with the denominator of 60.For \(\frac{3}{20}\): Multiply the numerator and the denominator by 3. \[\frac{3 \times 3}{20 \times 3} = \frac{9}{60}\]For \(\frac{2}{15}\): Multiply the numerator and the denominator by 4. \[\frac{2 \times 4}{15 \times 4} = \frac{8}{60}\]

Subtract the Fractions

Subtract the fractions with the common denominator:\[\frac{9}{60} - \frac{8}{60} = \frac{9 - 8}{60} = \frac{1}{60}\]

Simplify the Result

In this case, \(\frac{1}{60}\) is already in its simplest form.

Key concepts

Least Common Denominator

When subtracting fractions, it's important to have a common denominator. This denominator must be common to both fractions. This makes subtraction easier. The least common denominator (LCD) is the smallest number that both denominators can divide into evenly. For \(\frac{3}{20}\) and \(\frac{2}{15}\), the denominators are 20 and 15. You start by finding the prime factors of 20 (which are 2, 2, and 5) and 15 (which are 3 and 5). The LCD is found by taking the highest power of each prime number present in any of the denominators. So, for 20 and 15:
\[ \text{Prime factors of 20: } 2^2 \times 5 \]
\[ \text{Prime factors of 15: } 3 \times 5 \]
Hence, the LCD is 60 because it includes both prime factors at their highest powers (2^2, 3, and 5). This means you can evenly divide 60 by both 20 and 15.

Equivalent Fractions

Once you have the LCD, the next step is to convert each fraction to an equivalent fraction with this common denominator. This is done by multiplying both the numerator and the denominator of each fraction by the same number. For \(\frac{3}{20}\), we need to convert it to an equivalent fraction with a denominator of 60. Since 20 times 3 equals 60, multiply both the numerator and the denominator of \(\frac{3}{20}\) by 3:
\[ \frac{3 \times 3}{20 \times 3} = \frac{9}{60} \]
Similarly, for \(\frac{2}{15}\), multiply both the numerator and the denominator by 4 because 15 times 4 equals 60:
\[ \frac{2 \times 4}{15 \times 4} = \frac{8}{60} \]
Now, both fractions \(\frac{9}{60}\) and \(\frac{8}{60}\) have a common denominator of 60, making them equivalent fractions with a common base.

Fraction Simplification

With both fractions having a common denominator, you can now easily subtract them. Subtract the numerators while keeping the common denominator:
\[ \frac{9}{60} - \frac{8}{60} = \frac{9 - 8}{60} = \frac{1}{60} \]
In this case, \(\frac{1}{60}\) is already in its simplest form. Fraction simplification involves dividing the numerator and the denominator by their greatest common divisor (GCD). For instance, if the result was \(\frac{4}{8}\), you could simplify it by dividing both the numerator and the denominator by 4, the GCD of both 4 and 8.
This will give you \(\frac{1}{2}\). Simplifying fractions makes them easier to understand and compare. However, since \(\frac{1}{60}\) cannot be simplified any further, it is already as simple as it can be.