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Adding and Subtracting Fractions with Different Denominators

Adding and Subtracting Fractions with Different Denominators

Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators.

Adding and Subtracting Fractions with Different Denominators

1. Find the LCD.
2. Convert each fraction to an equivalent form with the LCD as the denominator.
3. Add or subtract the fractions.
4. Write the result in simplified form.

Example

Add: $$\frac{1}{2}+\frac{1}{3}.$$

Solution

 $$\frac{1}{2}+\frac{1}{3}$$ Find the LCD of 2, 3. Change into equivalent fractions with the LCD 6. Simplify the numerators and denominators. $$\frac{3}{6}+\frac{2}{6}$$ Add. $$\frac{5}{6}$$

Remember, always check to see if the answer can be simplified. Since $$5$$ and $$6$$ have no common factors, the fraction $$\frac{5}{6}$$ cannot be reduced.

Example

Subtract: $$\frac{1}{2}-\left(-\frac{1}{4}\right).$$

Solution

 $$\frac{1}{2}-\left(-\frac{1}{4}\right)$$ Find the LCD of 2 and 4. Rewrite as equivalent fractions using the LCD 4. Simplify the first fraction. $$\frac{2}{4}-\left(-\frac{1}{4}\right)$$ Subtract. $$\frac{2\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\left(-1\right)}{4}$$ Simplify. $$\frac{3}{4}$$

One of the fractions already had the least common denominator, so we only had to convert the other fraction.

Example

Add: $$\frac{7}{12}+\frac{5}{18}.$$

Solution

 $$\frac{7}{12}+\frac{5}{18}$$ Find the LCD of 12 and 18. Rewrite as equivalent fractions with the LCD. Simplify the numerators and denominators. $$\frac{21}{36}+\frac{10}{36}$$ Add. $$\frac{31}{36}$$

Because $$31$$ is a prime number, it has no factors in common with $$36.$$ The answer is simplified.

When we use the Equivalent Fractions Property, there is a quick way to find the number you need to multiply by to get the LCD. Write the factors of the denominators and the LCD just as you did to find the LCD. The “missing” factors of each denominator are the numbers you need.

The LCD, $$36,$$ has $$2$$ factors of $$2$$ and $$2$$ factors of $$3.$$

Twelve has two factors of $$2,$$ but only one of $$3$$—so it is ‘missing‘ one $$3.$$ We multiplied the numerator and denominator of $$\frac{7}{12}$$ by $$3$$ to get an equivalent fraction with denominator $$36.$$

Eighteen is missing one factor of $$2$$—so you multiply the numerator and denominator $$\frac{5}{18}$$ by $$2$$ to get an equivalent fraction with denominator $$36.$$ We will apply this method as we subtract the fractions in the next example.

Example

Subtract: $$\frac{7}{15}-\frac{19}{24}.$$

Solution

 $$\frac{7}{15}-\frac{19}{24}$$ Find the LCD.  15 is ‘missing’ three factors of 2 24 is ‘missing’ a factor of 5 Rewrite as equivalent fractions with the LCD. Simplify each numerator and denominator. $$\frac{56}{120}-\frac{95}{120}$$ Subtract. $$-\frac{39}{120}$$ Rewrite showing the common factor of 3. $$-\frac{13·3}{40·3}$$ Remove the common factor to simplify. $$-\frac{13}{40}$$

Example

Add: $$-\phantom{\rule{0.2em}{0ex}}\frac{11}{30}+\frac{23}{42}.$$

Solution

 $$-\frac{11}{30}+\frac{23}{42}$$ Find the LCD. Rewrite as equivalent fractions with the LCD. Simplify each numerator and denominator. $$-\frac{77}{210}+\frac{115}{210}$$ Add. $$\frac{38}{210}$$ Rewrite showing the common factor of 2. $$\frac{19·2}{105·2}$$ Remove the common factor to simplify. $$\frac{19}{105}$$

In the next example, one of the fractions has a variable in its numerator. We follow the same steps as when both numerators are numbers.

Example

Add: $$\frac{3}{5}+\frac{x}{8}.$$

Solution

The fractions have different denominators.

 $$\frac{3}{5}+\frac{x}{8}$$ Find the LCD. Rewrite as equivalent fractions with the LCD. Simplify the numerators and denominators. $$\frac{24}{40}+\frac{5x}{8}$$ Add. $$\frac{24\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}5x}{40}$$

We cannot add $$24$$ and $$5x$$ since they are not like terms, so we cannot simplify the expression any further.

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