Prospect Prep

In a fraction, the numerator is above the line, and the denominator is below the line. One mneumonic to remember this is "the **d**enominator is **d**ownstairs."

So, in the problem ^{2}⁄_{3} + ^{1}⁄_{5} , 2 and 1 are the numerators, and 3 and 5 are the denominators.

## Adding and Subtraction Fractions

To add and subtract fractions, they must have a **common denominator**. ^{2}⁄_{3} + ^{1}⁄_{5} does not equal (2 + 1) / (3 + 5) = ^{3}⁄_{8} .

### Finding a common denominator

To get the common denominator to add and subtract fractions, you must find the least common multiple of the original denominators.

^{2}⁄_{3} + ^{1}⁄_{5}

Find the least common multiple:

- Multiples of 3: 3, 6, 9, 12,
**15** - Multiples of 5: 5, 10,
**15**

****Convert each fraction to its new form.

^{2}⁄_{3}is the same as^{4}⁄_{6},^{6}⁄_{9},^{8}⁄_{12}, and^{10}⁄_{15}^{1}⁄_{5}is the same as^{2}⁄_{10}and^{3}⁄_{15}

To properly convert each fraction, determine what number multiplied by the original denominator equals the least common multiple. Then, multiply that number by the original numerator of each fraction to get the new numerator.

To get 15 in the denominator of ^{2}⁄_{3}, you need to multiply 3 by 5 because 3(5) = 15. You can't do something to the denominator without also doing the same thing to the numerator. So, multiply 2 by 5 to get 10 in the numerator. Here, you are essentially multiplying ^{2}⁄_{3 } by ^{5}⁄_{5} (which also equals 1) to get a fraction that you can use.

**So ^{2}⁄_{3} + ^{1}⁄_{5 }is the same as ^{10}⁄_{15 }+ ^{3}⁄_{15 }= ^{13}⁄_{15 }**

### The Butterfly Trick Shortcut

When adding or subtracting two fractions, try this shortcut:

- Multiply up from the lower left to the upper right and write the product.

- Multiply up from the lower right to the upper left and write the product.

- Multiply across the bottom to get a common denominator.

- Add or subtract the numerators.

- Reduce fraction if needed.

### Adding Fractions with Variables

The easiest way to add two fractions with variables is to use the Butterfly Trick above.

However, to add or subtract three or more fractions, you need to find the common denominator for all of them. Often, the least common multiple is just the product of all of the denominators.

*Example 1: *

The least common multiple of 3c, a, and b is **3abc**.

- Convert each fraction:

- Add them all to get the solution:

*Example 2:*

^{a}⁄_{2} + ^{b}⁄_{3} + ^{c}⁄_{4}

The least common multiple of 2, 3, and 4 is **12**.

- Convert each fraction:

^{3}⁄_{2a} (^{2b}⁄_{2b}) = ^{6b}⁄_{4ab}

^{3}⁄_{4b} (^{a}⁄_{a}) = ^{3a}⁄_{4ab}

- Subtract to find solution:

^{(6b - 3a)}⁄_{4ab}

### Practice Problems

^{3}⁄_{4}-^{2}⁄_{5}^{1}⁄_{2}-^{1}⁄_{10}^{6}⁄_{7}+^{1}⁄_{8}^{2}⁄_{3}+^{2}⁄_{7}^{3a}⁄_{5}+^{a}⁄_{4}^{a}⁄_{2b}-^{a}⁄_{3b}^{4x}⁄_{y}+^{2y}⁄_{x}^{mn}⁄_{7}-^{2n}⁄_{21}+^{m}⁄_{3}^{6g}⁄_{5}-^{7g}⁄_{10}-^{2g}⁄_{15}^{3r}⁄_{5s}+^{2s}⁄_{r}+^{6s}⁄_{5r}

*Answers*

^{7}⁄_{20}^{2}⁄_{5}^{55}⁄_{56}^{20}⁄_{21}^{17a}⁄_{20}^{a}⁄_{6b}^{(4x^2 + 2y^2)}⁄_{xy}^{(3mn - 2n + 7m)}⁄_{21}^{11g}⁄_{30}^{(3r^2 + 16s^2)}⁄_{5rs}

_{}*David Travis is the founder and CEO of Prospect Prep, a New York City-based tutoring agency dedicated to helping students earn better grades, higher scores, and acceptance letters from the colleges of their dreams.*

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