coffeeWARNING: this is a long lesson! Maybe get yourself a cup of coffee and get comfortable before you begin.approve

Are you starting to get the hang of working with opposites to solve equations with addition and subtraction? If so, let's try more working with opposites to solve multiplication and division problems.

Do you remember that when a variable and a number are placed together with no sign between them, it means they should be multiplied? So when you see ...

4y = 20

... it means that 4 times y equals 20. You can probably guess that y = 5. But how did you do this? You divided 20 by 4! Division is the opposite of multiplication.

Mathematically speaking, you divide both sides of the equation by 4:

4y ÷ 4 = 20 ÷ 4

Can you remember why we have to do it this way? Think back to the 2 main rules of solving algebra questions:
opposites and balancing. The variable 'y' is being multiplied by 4, so we have to do the opposite (dividing) to get the variable alone. And because we divide by 4 on one side of the equation, we must divide by 4 on the other side of the equation to obey the rule of balancing. This shows what's happening:


The 4's on the left side of the equation cancel each other out (because 4 ÷ 4 = 1) and you are left with 20 ÷ 4 = 5.

Remember that when you are doing algebra, you can recognize a division problem because the numbers and variable look like a fraction. So...
a ÷ 8 is usually written like this:


To solve a division problem in algebra, you just multiply both sides of the equation by the same number that it is being divided by. This is the number in the denominator (the number on the bottom of the fraction), like this example:


Remember: because the internet may make it hard to display fractions properly, you will sometimes see them written like this:

a/8 = 6

The denominator (the number on the bottom of the fraction) is 8. So we must multiply both sides by 8:

$$\LARGE \frac{a}{8}$$ x 8 = 6 x 8

The 8's cancel each other out (the mathematical reason is because $$\LARGE \frac{8}{8} = 1$$) and you are left with:

a = 48

Here are a few more examples of multiplying and dividing, worked out:

Example 1:

12b = 36

This is a multiplication problem: 12b means "12 times b". So we must divide by the same number (12) to get the answer.

12b ÷ 12 = 36 ÷ 12

And now for the answer:

b = 3

If you substitute the number 3 back into the original equation, you can convince yourself that this is true!

12b = 36
12 x 3 = 36 .... TRUE!

Example 2:

$$\LARGE \frac{p}{3} = 25$$

This is a division problem:
$$\LARGE \frac{p}{3}$$ means p ÷ 3. So we must multiply to get the answer:

$$\LARGE \frac{p}{3}$$ x 3 = 25 x 3

p = 75

Again, if you substitute the number 75 back into the original equation, you can check to see that you're right:

$$\LARGE \frac{75}{3}$$ = 25 ... YES! We're right again!

Are you ready to try a few on your own? Go to the next activity, the multiplication and division quiz.

Last modified: Tuesday, 26 July 2011, 5:04 PM