## Key Questions

• To write fractions with a common denominator, you will most likely need to scale some numbers up! I will explain how.

Let's try it with the fractions $\frac{2}{3}$ and $\frac{3}{12}$

12 is larger than 3, so we will have to multiply the 3 by some number to equal 12. (We are really finding the Least Common Multiple of the two denominators!) To do this, you have to multiply the 3 by 4, because 3x4=12. But now the numerator doesn't match the denominator. When you scale the denominator up, you have to scale the numerator up too! So the 2 must be multiplied by 4 also.

Now you have the following: $\frac{8}{12}$ and $\frac{3}{12}$

These fractions now have common denominators! Now they're all set for adding or subtracting fractions.

Try another: $\frac{2}{6}$ and $\frac{3}{5}$: The least common multiple of 6 and 5 is 30. (the product of the denominators)

Transform each fraction by multiplying by "1":
$\frac{2}{6} \cdot \frac{5}{5}$ = $\frac{10}{30}$ and $\frac{3}{5} \cdot \frac{6}{6}$ = $\frac{18}{30}$

One last problem: $\frac{4}{9}$ and $\frac{7}{6}$ What is the least common multiple of 9 and 6? Could you use 54? Absolutely, but it is not the LEAST number that you could use. How about 18? YES!

$\frac{4}{9} \cdot \frac{2}{2}$ = $\frac{8}{18}$ and $\frac{7}{6} \cdot \frac{3}{3}$ = $\frac{21}{18}$ Ready to go...

Hope this helped!

Explanation down below...

#### Explanation:

A mixed fraction is a fraction written with a whole and a fraction.

Eg $2$$\frac{1}{2}$

An improper fraction is a fraction with a numerator that is larger than the denominator.

Eg $\frac{22}{5}$

To write a mixed fraction into an improper one you have to take your whole number next to your fraction, multiply that number by your denominator and then take your original numerator and add it on to your answer. Finally put the number over the denominator.

Example:

$2$$\frac{1}{2}$

$2$x$2 = 4$

$4 + 1 = 5$

$\frac{5}{2}$

• I assume you know that if you multiply both numerator and denominator of a fraction by a same number, you get an equivalent fraction. Thus, for example, if you start from 2/3 and multiply both numerator and denominator by 3, you get 6/9, which is indeed equivalent to 2/3.

Now, if you want to add two fraction, you first of all transform both of them as just shown, obtaining two equivalent fractions with the same denominator. At this point, you have a sum of two fraction of the form $\setminus \frac{a}{b} + \setminus \frac{c}{b}$, which is easily $\setminus \frac{a + c}{b}$.

To do so, you look for the least common multiple of the two denominator. Let's say that we have to calculate $\setminus \frac{3}{5} + \setminus \frac{5}{8}$. The least common multiple of 5 and 8 is 40, so we have to transform $\setminus \frac{3}{5}$ into $\setminus \frac{24}{40}$ (multiplying numerator and denominator by 8), and then we transform $\setminus \frac{5}{8}$ into $\setminus \frac{25}{40}$ (multiplying numerator and denominator by 5).

These are equivalent fraction, so we can be sure that $\setminus \frac{3}{5} + \setminus \frac{5}{8}$ equals $\setminus \frac{24}{40} + \setminus \frac{25}{40}$. The advantage is, of course, that the second one is much easier to compute, since one immediately gets that $\setminus \frac{24}{40} + \setminus \frac{25}{40} = \setminus \frac{49}{40}$

If something isn't clear, don't hesitate to ask:)