Adding and Subtracting Fractions with Unequal Denominators

Adding and Subtracting Fractions with Unequal Denominators

When dealing with fractions, it's common to encounter situations where the denominators are not the same. In such cases, you need to find a common denominator before performing addition and subtraction. This process may seem involved at first, but with a clear understanding of the steps, you'll find that adding and subtracting fractions with unlike denominators is quite straightforward.

Finding a Common Denominator

The first step when dealing with fractions with unequal denominators is to find a common denominator. The common denominator is the smallest whole number that is a multiple of both denominators. For example, if we have (\frac{2}{5}) and (\frac{1}{3}), the common denominator could be 15 (since both 5 and 3 are factors of 15).

Here's a systematic way to find the common denominator for any pair of fractions:

1. List all the factors of the larger denominator.
2. Check if any of these factors are also factors of the smaller denominator.
3. Select the smallest factor that is common to both denominators.

For our example, the factors of 5 are 1, 5, and 5 × 5 = 25. The factors of 3 are 1, 3, and 3 × 3 = 9. The smallest factor that is common to 5 and 3 is 3, so the common denominator for (\frac{2}{5}) and (\frac{1}{3}) is 3 × 5 = 15.

Adding Fractions with Unequal Denominators

To add fractions with unlike denominators, we need to find their equivalent fractions with a common denominator. Here's the step-by-step process:

1. Find the common denominator.
2. Convert each fraction to an equivalent fraction with the common denominator.
3. Add the numerators of the equivalent fractions.
4. Write the answer as a fraction with the common denominator as the denominator.

For example, let's add (\frac{2}{5}) and (\frac{1}{3}) with the common denominator 15:

1. Common denominator: 15
2. Convert (\frac{2}{5}) to an equivalent fraction with a denominator of 15: (\frac{2 \times 3}{5 \times 3} = \frac{6}{15})
3. Convert (\frac{1}{3}) to an equivalent fraction with a denominator of 15: (\frac{1 \times 5}{3 \times 5} = \frac{5}{15})
4. Add the numerators: (\frac{6}{15} + \frac{5}{15} = \frac{6 + 5}{15} = \frac{11}{15})

The final answer is (\frac{11}{15}).

Subtracting Fractions with Unequal Denominators

Subtracting fractions with unlike denominators is done in the same manner as adding them, but with a negative sign in front of one of the fractions. Here's the step-by-step process:

1. Find the common denominator.
2. Convert each fraction to an equivalent fraction with the common denominator.
3. Subtract the numerators of the equivalent fractions.
4. Write the answer as a fraction with the common denominator as the denominator.

For example, let's subtract (\frac{2}{5}) from (\frac{3}{4}) with the common denominator 20:

1. Common denominator: 20
2. Convert (\frac{2}{5}) to an equivalent fraction with a denominator of 20: (\frac{2 \times 4}{5 \times 4} = \frac{8}{20})
3. Convert (\frac{3}{4}) to an equivalent fraction with a denominator of 20: (\frac{3 \times 5}{4 \times 5} = \frac{15}{20})
4. Subtract the numerators: (\frac{15}{20} - \frac{8}{20} = \frac{15 - 8}{20} = \frac{7}{20})

The final answer is (\frac{7}{20}).

To summarize, adding and subtracting fractions with unlike denominators requires finding a common denominator and converting each fraction to an equivalent fraction with that common denominator. This approach will ensure that your calculations are accurate and that your answers are presented in the simplest form possible.