Proportions

Example:

We see that 1-out-of-3 is equal to 2-out-of-6

The ratios are the same, so they are in proportion.

Example: Rope

A rope's length and weight are in proportion.

When 20m of rope weighs 1kg, then:

• 40m of that rope weighs 2kg
• 200m of that rope weighs 10kg
• etc.

So:

20 1 = 40 2

Example: International paper sizes (like A3, A4, A5, etc) all have the same proportions:

So any artwork or document can be resized to fit on any sheet. Very neat.

Example: you want to draw the dog's head ... how long should it be?

Let us write the proportion with the help of the 10/20 ratio from above:

? 42 = 10 20

Now we solve it using a special method:

Multiply across the known corners,
then divide by the third number

And we get this:

? = (42 × 10) / 20
= 420 / 20
= 21

So you should draw the head 21 long.

Example: what is 25% of 160 ?

The percent is 25, the whole is 160, and we want to find the "part":

Part160 = 25100

Multiply across the known corners, then divide by the third number:

Part = (160 × 25) / 100
= 4000 / 100
= 40

Answer: 25% of 160 is 40.

 

Note: we could have also solved this by doing the divide first, like this:

Part = 160 × (25 / 100)
= 160 × 0.25
= 40

Either method works fine.

Example: what is $12 as a percent of $80 ?

Fill in what we know:

$12$80 = Percent100

Multiply across the known corners, then divide by the third number. This time the known corners are top left and bottom right:

Percent = ($12 × 100) / $80
= 1200 / 80
= 15%

Answer: $12 is 15% of $80

Example: The sale price of a phone was $150, which was only 80% of normal price. What was the normal price?

Fill in what we know:

$150Whole = 80100

Multiply across the known corners, then divide by the third number:

Whole = ($150 × 100) / 80
= 15000 / 80
= 187.50

Answer: the phone's normal price was $187.50

Example: How tall is the Tree?

Sam tried using a ladder, tape measure, ropes and various other things, but still couldn't work out how tall the tree was.

But then Sam has a clever idea ... similar triangles!

Sam measures a stick and its shadow (in meters), and also the shadow of the tree, and this is what he gets:

Now Sam makes a sketch of the triangles, and writes down the "Height to Length" ratio for both triangles:

Height: Shadow Length:     h 2.9 m = 2.4 m 1.3 m

Multiply across the known corners, then divide by the third number:

h = (2.9 × 2.4) / 1.3
= 6.96 / 1.3
= 5.4 m (to nearest 0.1)

Answer: the tree is 5.4 m tall.

And he didn't even need a ladder!

Let us try the ratio of "Shadow Length to Height":

Shadow Length: Height:     2.9 m h = 1.3 m 2.4 m

Multiply across the known corners, then divide by the third number:

h = (2.9 × 2.4) / 1.3
= 6.96 / 1.3
= 5.4 m (to nearest 0.1)

It is the same calculation as before.

Example: you have just put 12 buckets of stones into a mixer, how much cement and how much sand should you add to make a 1:2:6 mix?

Let us lay it out in a table to make it clearer:

	Cement	Sand	Stones
Ratio Needed:	1	2	6
You Have:			12

You have 12 buckets of stones but the ratio says 6.

That is OK, you simply have twice as many stones as the number in the ratio ... so you need twice as much of everything to keep the ratio.

Here is the solution:

	Cement	Sand	Stones
Ratio Needed:	1	2	6
You Have:	2	4	12

And the ratio 2:4:12 is the same as 1:2:6 (because they show the same relative sizes)

So the answer is: add 2 buckets of Cement and 4 buckets of Sand. (You will also need water and a lot of stirring....)