🔪 Solving Fractions

#math #fractions

We are going to solve these!!!!

$$\frac{a}{30}=3$$

Why? Because I never had them in high school. WHAT!? yea 😦. ||Dutch school system sucks|| This is a reference sheet for me to use when I have to use these fractions again.

Naming

English isn't my native langue so here are some early clarifications.

$$\frac{\text{3 <- Numerator}}{\text{4 <- Denominator}}$$

Fraction multiplication rules

If you wish to multiply 2 fractions you can just multiply the top and bottom so easy

$$\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}$$

Multiply whole numbers

If you want to multiply whole numbers you can write 3 as $\frac{3}{1}$ which means you can apply the multiplication rule again.

$$3\times \frac{a}{b}=\frac{3}{1}\times \frac{a}{b}=\frac{3a}{1b}=\frac{3a}{b}$$

Write whole numbers like this

Again you can write whole numbers like this. Important!

$$3=\frac{3}{1}$$

Fraction addition rules

The denominator's need to be the same.

$$\frac{a}{b}+\frac{c}{b}$$

Fractions don't have the same denominator

You need to find the Least Common Denominator LCD. This means you need to try to get the bottom parts to be the same.

You are free to multiply the numerator and denominator of the fraction with what ever you want. YAY. You need to make sure that what happens for the numerator also happens for the denominator. Example:

$$\frac{1}{3}+\frac{1}{6}$$

First step is to multiply the numerator with a number we want $\frac{1\ast 2}{3\ast 2}$ I choose 2. (Warning this is not the same as fraction multiplication!! that would result in $\frac{2}{1}$) Now the result is $\frac{2}{6}$ and yay we have the same denominator so we can add them up.

Help I can't find the least common denominator.

If you can't find the LCD you can always use this trick. multiply the 2 Denominator's with each other. Then you will always get a LCD

$$\frac{1}{6}+\frac{7}{15}$$$$\frac{1\ast 15}{6\ast 15}+\frac{7\ast 6}{15\ast 6}$$$$\frac{15}{90}+\frac{42}{90}$$

You will get huge numbers but its a valid answer.

Adding whole numbers

This is pretty easy, following [[#Write whole numbers like this]]. We can write our whole number in that from and then convert the denominator into the same number.

$$6+\frac{5}{8}=\frac{6}{1}+\frac{5}{8}$$$$\frac{48}{8}+\frac{5}{8}=\frac{53}{8}$$

Canceling fractions - Trick 1

When you have 2 variables at the top and bottom they become $\frac{1}{1}$ and cancel out.

$$\frac{a}{a}=1=\frac{1}{1}$$

Remember [[#Write whole numbers like this]] 1 is the same as $\frac{1}{1}$. Is this true? Lets check.

$$\frac{3\times 4}{3}=\frac{12}{3}=\frac{4}{1}=4$$

First we calculate the top part $3\times 4=12$ then we end up with $\frac{12}{3}$. Then to make the denominator $1$ we can divide by $3$ on numerator and denominator which results in $\frac{4}{1}$ which you can also write as $4$. So its true! we can just remove $\frac{3}{3}$. Warning $\frac{3}{3}$ becomes $\frac{1}{1}$!

So in conclusion this is the formula.

$$\frac{ab}{a}=\frac{\cancel{a}b}{\cancel{a}}=b=\frac{b}{1}$$

Gotcha 1

But what if the $b$ is a denominator not a numerator? Do we get the same result?

$$\frac{a}{ab}\ne b$$

How do we solve this? well remember $\frac{a}{a}$ results in $\frac{1}{1}$ so it becomes.

$$\frac{a}{ab}=\frac{1}{1b}=\frac{1}{b}$$

So the result is $\frac{1}{b}$

The magic triangle - Trick 2

Lets say we have a fraction and you want to reorient the formula to get some result example: $a=\frac{b}{c}$. An easy way to think about it is using this fraction.

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

You can see that if you want 3 you do $\frac{6}{2}$. If you want the result 6 you can do $3\times 2$ if you want 2 you can do $\frac{6}{3}$. Its easy to do these in your head. Cross reference these with the $a=\frac{b}{c}$ to convert the forumla

The logic behind it

Moving by division

When you have a equation (equality) you can move variables around. Here we want to solve for $a$.

$$3=10a$$

What we can do is divide both sides with $10$. Why? because I said so. (sadly this is how it works).

$$\frac{3}{10}=\frac{10a}{10}$$

Now we can apply the [[#Canceling fractions - Trick 1]] Trick.

$$\frac{3}{10}=\frac{\cancel{10}a}{\cancel{10}}$$$$\frac{3}{10}=a$$

Another example:

$$x=ba$$$$\frac{x}{a}=\frac{ba}{a}$$$$\frac{x}{a}=\frac{b\cancel{a}}{\cancel{a}}$$$$\frac{x}{a}=b$$

Moving by multiply

To go the other way we can multiply

$$b=\frac{x}{a}$$

Here we multiply both sides with a Again look how we converted $a$ to $\frac{a}{1}$

$$b\times a=\frac{x}{a}\frac{a}{1}$$$$ba=\frac{xa}{a}$$

We can use the [[#Canceling fractions - Trick 1]]

$$ba=\frac{x\cancel{a}}{\cancel{a}}$$$$ba=x$$

gotcha 2 Solve for $a$

Again when looking at [[#gotcha 1]] we are now going to solve for $a$

$$\frac{x}{a}=b$$

Now we can't use any of our previous tricks.

• Multiplying with $a$ only make it into $x=ab$
• Diving with $a$ does a $\frac{1}{a}$ which isn't what we want. So what to do?

2 steps

1. We multiply with a which gives us $x=ab$
2. Then we divide with b to remove it form the right which gives us. $\frac{x}{b}=a$

Conclusion

These are the tricks I would highly recommend to practice a lot as you should be able to do this automaticlly sadly. Remember [[#The magic triangle - Trick 2]] and Good luck!!!

To practice: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:solve-equations-inequalities

Sources

• https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-ratio-proportion
• https://www.mathsisfun.com/least-common-denominator.html