Fractions

Fractions are difficult for many to understand, but any knowledge gaps here will create big problems down the road. A fraction represents a number of parts written as one number over another with a line in between. Fractions are the same as division, \( \frac{3}{4} \) is equal to 3 divided by 4, which is also equal to three fourths of a whole. The top number in any fraction is called the numerator, while the bottom number is called the denominator. The numerator gives the parts out of the denominator of a whole that we have. \( \frac{4}{5} \) has a numerator of 4 and a denominator of 5, it represents four fifths of a whole (or 80%).

Fractions can also be greater than 1 or negative.

The fraction \( \frac{7}{3} \) is equal to \( 2\frac{1}{3} \), or two and one third. When you break fractions out into whole number (2) and fractional parts \( (\frac{1}{3}) \) you get a mixed number \( (2\frac{1}{3}) \). While mixed numbers can be easier to interpret they make computations like multiplication more difficult so it is better to work with the pure fractions (like \( \frac{7}{3} \)) instead. Here are a few examples of conversions between mixed numbers and pure fractions:

• \( 1\frac{1}{4}=\frac{5}{4} \)
• \( 4\frac{1}{2}=\frac{9}{2} \)
• \( 2\frac{2}{5}=\frac{12}{5} \)

Some examples of negative fractions are:

$$-\frac{2}{7},\;\frac{-22}{3},\;\frac{6}{-13}$$

The negative sign can come before the fraction, in the numerator, or in the denominator, but the first two cases are the most conventional.

Multiplying fractions

This is the easiest computation to do with fractions. To multiply two fractions you simply multiply the numerators to get a new numerator, then multiply the denominators to get a new denominator; combine these for your new fraction. Example:

$$\frac{3}{5} \cdot \frac{4}{7} = \frac{3 \cdot 4}{5 \cdot 7} = \frac{12}{35}$$

Sometimes you’ll be able simplify the resulting fraction by dividing out numbers that factor into both the numerator and denominator. For example:

$$\frac{3}{6} = \frac{3 \cdot 1}{3 \cdot 2} = \frac{1}{2}$$

The last example was simplified by dividing three out of both the numerator and denominator. Here’s another example of multiplication and simplification:

$$\frac{5}{8}\cdot\frac{6}{9} = \frac{5\cdot6}{8\cdot9} = \frac{30}{72} = \frac{6\cdot5}{6\cdot12} = \frac{5}{12}$$

Negative and larger fractions follow the same rules.

$$\frac{-3}{7}\cdot\frac{15}{4}=\frac{-3\cdot15}{7\cdot4}=-\frac{45}{28}$$

Dividing fractions

To divide one fraction by another, follow this rule: flip the fraction you’re dividing by (switch the numerator and denominator) then multiply the flipped fraction by the other. In the example below notice how we flip \( \frac{1}{3} \) into \( \frac{3}{1} \) and change the division to multiplication:

$$\frac{1}{2}/\frac{1}{3} = \frac{1}{2}\cdot\frac{3}{1} = \frac{3}{2}$$

If that didn’t demonstrate the technique well enough, here is another example:

$$\frac{11}{16}/\frac{3}{4}=\frac{11}{16}\cdot\frac{4}{3}=\frac{44}{48}=\frac{4\cdot11}{4\cdot12}=\frac{11}{12}$$

Again, don’t let negative numbers throw you off:

$$\frac{6}{5}/\frac{-1}{3}=\frac{6}{5}\cdot\frac{-3}{1}=\frac{-18}{5}$$

When multiplying or dividing two fractions, your answer isn’t always a fraction:

$$\frac{4}{9}/\frac{-2}{9} = \frac{4}{9}\cdot\frac{-9}{2} = -2$$

Can you see all the cancellations that led to the previous answer?

Adding and subtracting fractions

To add or subtract two fractions they must have matching denominators. If two fractions have common denominators, you can add or subtract them by simply adding of subtracting the numerators to create a new fraction. You won’t always begin with this luxury of equal denominators so you’ll often need to rescale one or both of your fractions. Observe that:

$$\frac{1}{3} = \frac{2}{6} = \frac{3}{9} = \frac{4}{12} = \frac{11}{33}$$

Whenever you multiply (or divide) both the numerator and denominator of a fraction by the same number, you create an equal fraction written with different numbers. To better understand this, think of a pizza cut into 8 equal slices. If you eat 4 of the 8 slices you’ve eaten \( \frac{4}{8} \) of the pizza, which is equal to \( \frac{4\cdot1}{4\cdot2} = \frac{1}{2} \) of the pizza. Think of \( \frac{4}{8} \) as a rescaled, but equivalent fraction to \( \frac{1}{2} \). You will usually need to rescale both fractions to find common denominators before you can add or subtract them. Example:

$$\frac{1}{2}+\frac{1}{3}=\frac{3}{6}+\frac{2}{6}=\frac{5}{6}$$

In the above example we found the common denominator of 6 for our two fractions. We converted the first fraction by multiplying its numerator and denominator by 3. We converted the second fraction by multiplication both the top and bottom by 2. This gave us two rescaled fractions with equal denominators. At this point the addition becomes easy: add the numerators and leave the denominator alone.

Like we did above, you can always find common denominators by multiplying the top and bottom of each fraction by the denominator of the opposing fraction. Example:

$$\frac{11}{4}-\frac{6}{5}=\frac{11\cdot5}{4\cdot5}-\frac{6\cdot4}{5\cdot4}=\frac{55}{20}-\frac{24}{20}=\frac{31}{20}$$

However, this method will sometimes lead you to work with much bigger numbers than you need to:

$$\frac{9}{16}-\frac{5}{8}=\frac{9\cdot8}{16\cdot8}-\frac{5\cdot16}{8\cdot16}=\frac{72}{128}-\frac{80}{128}=\frac{-8}{128}=\frac{8\cdot-1}{8\cdot16}=-\frac{1}{16}$$

The following could have been solved more easily if we found a smaller common denominator:

$$\frac{9}{16}-\frac{5}{8}=\frac{9}{16}-\frac{5\cdot2}{8\cdot2}=\frac{9}{16}-\frac{10}{16}=-\frac{1}{16}$$

If you can see a simpler way to reach common denominators you can save time by avoiding the first approach, but when in doubt, just go that route.

Practice questions

Solve and reduce to lowest terms

1. \( \frac{3}{4}\cdot\frac{1}{4} \)
2. \( \frac{4}{3}\cdot\frac{3}{16} \)
3. \( -\frac{2}{7}\cdot-\frac{4}{5} \)
4. \( \frac{-3}{4}\cdot\frac{4}{5} \)
5. \( \frac{3}{4}/\frac{1}{4} \)
6. \( \frac{8}{3}/\frac{7}{6} \)
7. \( \frac{-3}{8}/\frac{-2}{3} \)
8. \( \frac{-2}{5}/\frac{1}{6} \)
9. \( \frac{3}{4}+\frac{1}{4} \)
10. \( \frac{3}{4}+\frac{2}{5} \)
11. \( \frac{6}{7}+\frac{5}{9} \)
12. \( \frac{-3}{5}+\frac{7}{10} \)
13. \( \frac{7}{10}-\frac{3}{5} \)
14. \( \frac{3}{4}-\frac{5}{4} \)
15. \( \frac{11}{5}-\frac{5}{3} \)
16. \( \frac{-2}{7}-\frac{-4}{3} \)

Solutions

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1. \( \frac{3}{4}\cdot\frac{1}{4}=\frac{3\cdot1}{4\cdot4}=\frac{3}{16} \)
2. \( \frac{4}{3}\cdot\frac{3}{16}=\frac{12}{48}=\frac{1}{4} \)
3. \( -\frac{2}{7}\cdot-\frac{4}{5}=\frac{8}{35} \)
4. \( \frac{-3}{4}\cdot\frac{4}{5}=\frac{-12}{20}=-\frac{3}{5} \)
5. \( \frac{3}{4}/\frac{1}{4}=\frac{3}{4}\cdot\frac{4}{1}=\frac{12}{4}=3 \)
6. \( \frac{8}{3}/\frac{7}{6}=\frac{8}{3}\cdot\frac{6}{7}=\frac{48}{21}=\frac{16}{7}\;\text{or}\;2\frac{2}{7} \)
7. \( \frac{-3}{8}/\frac{-2}{3}=\frac{-3}{8}\cdot\frac{-3}{2}=\frac{9}{16} \)
8. \( \frac{-2}{5}/\frac{1}{6}=\frac{-2}{5}\cdot6=-\frac{12}{5}\;\text{or}\;-2\frac{2}{5} \)
9. \( \frac{3}{4}+\frac{1}{4}=\frac{3+1}{4}=\frac{4}{4}=1 \)
10. \( \frac{3}{4}+\frac{2}{5} = \frac{3\cdot5}{4\cdot5}+\frac{2\cdot4}{5\cdot4} = \frac{15}{20}+\frac{8}{20} = \frac{23}{20}\;\text{or}\;1\frac{3}{20} \)
11. \( \frac{6}{7}+\frac{5}{9} = \frac{6\cdot9}{7\cdot9}+\frac{5\cdot7}{9\cdot7} = \frac{54}{63}+\frac{35}{63} = \frac{89}{63}\;\text{or}\;1\frac{26}{63} \)
12. \( \frac{-3}{5}+\frac{7}{10}=\frac{-3\cdot2}{5\cdot2}+\frac{7}{10}=\frac{-6}{10}+\frac{7}{10}=\frac{1}{10} \)
13. \( \frac{7}{10}-\frac{3}{5}=\frac{7}{10}-\frac{6}{10}=\frac{1}{10} \)
14. \( \frac{3}{4}-\frac{5}{4}=\frac{-2}{4}=-\frac{1}{2} \)
15. \( \frac{11}{5}-\frac{5}{3}=\frac{33}{15}-\frac{25}{15}=\frac{8}{15} \)
16. \( \frac{-2}{7}-\frac{-4}{3}=\frac{-6}{21}+\frac{28}{21}=\frac{22}{21}\;\text{or}\;1\frac{1}{21} \)

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