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מדריכי לימוד > Prealgebra

Subtracting Fractions With Common Denominators

Learning Outcomes

  • Use fraction circles to find the difference between two fractions with like denominators
  • Subtract fractions with a like denominator without fraction circles
  • Subtract fractions with like denominators that contain variables

Model Fraction Subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into 1212 slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or 712\frac{7}{12} of the pizza) left in the box. If Leonardo eats 22 of these remaining pieces (or 212\frac{2}{12} of the pizza), how much is left? There would be 55 pieces left (or 512\frac{5}{12} of the pizza). 712−212=512\frac{7}{12}-\frac{2}{12}=\frac{5}{12} Let’s use fraction circles to model the same example, 712−212\frac{7}{12}-\frac{2}{12}. Start with seven 112\frac{1}{12} pieces. Take away two 112\frac{1}{12} pieces. How many twelfths are left? The bottom reads 7 twelfths minus 2 twelfths equals 5 twelfths. Above 7 twelfths, there is a circle divided into 12 equal pieces, with 7 pieces shaded in orange. Above 2 twelfths, the same circle is shown, but 2 of the 7 pieces are shaded in grey. Above 5 twelfths, the 2 grey pieces are no longer shaded, so there is a circle divided into 12 pieces with 5 of the pieces shaded in orange. Again, we have five twelfths, 512\frac{5}{12}. Doing the Manipulative Mathematics activity "Model Fraction Subtraction" will help you develop a better understanding of subtracting fractions.

Example

Use fraction circles to find the difference: 45−15\frac{4}{5}-\frac{1}{5} Solution: Start with four 15\frac{1}{5} pieces. Take away one 15\frac{1}{5} piece. Count how many fifths are left. There are three 15\frac{1}{5} pieces left. The bottom reads 4 fifths minus 1 fifth equals 3 fifths. Above 4 fifths, there is a circle divided into 5 equal pieces, with 4 pieces shaded in orange. Above 1 fifth, the same circle is shown, but 1 of the 4 shaded pieces is shaded in grey. Above 3 fifths, the 1 grey piece is no longer shaded, so there is a circle divided into 5 pieces with 3 of the pieces shaded in orange.

Try It

#146190 [ohm_question height="270"]146190[/ohm_question]

Subtract Fractions with a Common Denominator

We subtract fractions with a common denominator in much the same way as we add fractions with a common denominator.

Fraction Subtraction

If a,b, and ca,b,\text{ and }c are numbers where c≠0c\ne 0, then ac−bc=a−bc\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c} To subtract fractions with a common denominators, we subtract the numerators and place the difference over the common denominator.

Example

Find the difference: 2324−1424\frac{23}{24}-\frac{14}{24}

Answer: Solution:

2324−1424\frac{23}{24}-\frac{14}{24}
Subtract the numerators and place the difference over the common denominator. 23−1424\frac{23 - 14}{24}
Simplify the numerator. 924\frac{9}{24}
Simplify the fraction by removing common factors. 38\frac{3}{8}

Try It

#146191 [ohm_question height="270"]146191[/ohm_question]
Watch the following video for more examples of subtracting fractions with like denominators. https://youtu.be/7CeAQcpOJw0

Example

Find the difference: y6−16\frac{y}{6}-\frac{1}{6}

Answer: Solution:

y6−16\frac{y}{6}-\frac{1}{6}
Subtract the numerators and place the difference over the common denominator. y−16\frac{y - 1}{6}
The fraction is simplified because we cannot combine the terms in the numerator.

Try it

#146192 [ohm_question height="270"]146192[/ohm_question]

Example

Find the difference: −10x−4x-\frac{10}{x}-\frac{4}{x}

Answer: Solution: Remember, the fraction −10x-\frac{10}{x} can be written as −10x\frac{-10}{x}.

−10x−4x-\frac{10}{x}-\frac{4}{x}
Subtract the numerators. −10−4x\frac{-10 - 4}{x}
Simplify. −14x\frac{-14}{x}
Rewrite with the negative sign in front of the fraction. −14x-\frac{14}{x}

Try It

#146249 [ohm_question height="270"]146249[/ohm_question]
Now lets do an example that involves both addition and subtraction.

Example

Simplify: 38+(−58)−18\frac{3}{8}+\left(-\frac{5}{8}\right)-\frac{1}{8}

Answer: Solution:

38+(−58)−18\frac{3}{8}+\left(-\frac{5}{8}\right)-\frac{1}{8}
Combine the numerators over the common denominator. 3+(−5)−18\frac{3+\left(-5\right)-1}{8}
Simplify the numerator, working left to right. −2−18\frac{-2 - 1}{8}
Subtract the terms in the numerator. −38\frac{-3}{8}
Rewrite with the negative sign in front of the fraction. −38-\frac{3}{8}

Try It

#146250 [ohm_question height="270"]146250[/ohm_question]
In the next video we show more examples of subtracting fractions with a common denominator.  Make note of the second example, it addresses a common mistake made by students when simplifying fractions with variables. https://youtu.be/-mLFZT2KgWI

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