Addition and subtraction: General case:
Notes
1. This formula works for ordinary fractions as well, and also when the two expressions have the same denominator (although cancellation is necessary to simplify the answer in that case).
2. When adding or subtracting expressions with different denominators, it helps to factor and/or cancel before starting as you would with products or quotients. This makes it easier to simplify the final answer.
3. If the denominators are the same, it is better to use the rule for addition and subtraction with common denominator; otherwise you will need to do additional work to simplify the answer.
| $\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}} + \dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}} = \dfrac{\color{#026fc1}{P}\color{#c1026f}{S} + \color{#026fc1}{Q}\color{#c1026f}{R}}{\color{#026fc1}{Q}\color{#c1026f}{S}} \qquad \quad \ \ $ | Cross multiply to get the numerator: |
$\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}$  $\dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}}$
|
| Multiply across to get the denominator: |
$\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}$  $\dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}}$
| |
| $\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}} - \dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}} = \dfrac{\color{#026fc1}{P}\color{#c1026f}{S} - \color{#026fc1}{Q}\color{#c1026f}{R}}{\color{#026fc1}{Q}\color{#c1026f}{S}} \qquad \quad \ \ $ | Cross multiply to get the numerator: |
$\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}$  $\dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}}$
|
| Multiply across to get the denominator: |
$\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}$  $\dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}}$
|
1. This formula works for ordinary fractions as well, and also when the two expressions have the same denominator (although cancellation is necessary to simplify the answer in that case).
2. When adding or subtracting expressions with different denominators, it helps to factor and/or cancel before starting as you would with products or quotients. This makes it easier to simplify the final answer.
3. If the denominators are the same, it is better to use the rule for addition and subtraction with common denominator; otherwise you will need to do additional work to simplify the answer.
#[Examples][Ejemplos]#
#[Some for you][Algunas para ti]#
$\dfrac{\color{#026fc1}{3}}{\color{#026fc1}{2x+1}} + \dfrac{\color{#c1026f}{4}}{\color{#c1026f}{x-5}} $ \t ${}= \dfrac{\color{#026fc1}{3}\color{#c1026f}{(x-5)} + \color{#026fc1}{(2x+1)}\color{#c1026f}{4}}{\color{#026fc1}{(2x+1)}\color{#c1026f}{(x-5)}}$ \t
Cross multiply for the numerator and multiply across for the denominator.
\\ \t ${}= \dfrac{11x - 9}{(2x+1)(x-5)}$
\t
Calculate the numerator.
\\
\\ $\dfrac{\color{#026fc1}{2x}}{\color{#026fc1}{y-1}} - \dfrac{\color{#c1026f}{y+1}}{\color{#c1026f}{x}} $ \t ${}= \dfrac{\color{#026fc1}{2x}\color{#c1026f}{(x)} - \color{#026fc1}{(y-1)}\color{#c1026f}{(y+1)}}{\color{#026fc1}{(y-1)}\color{#c1026f}{(x)}}$ \t
Cross multiply for the numerator and multiply across for the denominator.
\\ \t ${}= \dfrac{2x^2-y^2+1}{(2x+1)(x-5)}$
\t
Calculate the numerator.
\\
\\ $\dfrac{\color{#026fc1}{5}}{\color{#026fc1}{2x}} - \dfrac{\color{#c1026f}{3}}{\color{#c1026f}{2(x+5)}} $ \t ${}= \dfrac{\color{#026fc1}{5}\cdot \color{#c1026f}{2(x+5)} - \color{#026fc1}{(2x)}\color{#c1026f}{3}}{\color{#026fc1}{2x}\cdot \color{#c1026f}{2(x+5)}}$ \t
Cross multiply for the numerator and multiply across for the denominator.
\\ \t ${}= \dfrac{4x + 50}{4x(x+5)}$
\t
Calculate the numerator.
\\ \t ${}= \dfrac{2(2x + 25)}{4x(x+5)}$
\t
Factor the numerator.
\\ \t ${}= \dfrac{2x + 25}{2x(x+5)}$
\t
Simplify: Cancel the 2.
Calculate the following. In your answer, the numerator and denominator should not have a factor in common.
$%0 \qquad = \qquad$
\t | BOX* |
| BOX* |
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$%1 \qquad = \qquad$
\t | BOX* |
| BOX* |
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\\ \t ACTIVEMATH
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$%2 \qquad = \qquad$
\t | BOX* |
| BOX* |
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\\ \t ACTIVEMATH
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$%3 \qquad = \qquad$
\t | BOX* |
| BOX* |
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\\ \t ACTIVEMATH
MESSAGE