Ratios

A ratioÂ comparesÂ values.

A ratio says how much of one thing there is compared to another thing.

There are 3 blue squares to 1 yellow square

Ratios can be shown in different ways:

Using the “:” to separate the values: 3 : 1
Instead of the “:” we can use the word “to”: 3 to 1
Or write it like a fraction:
 3 1

A ratio can be scaled up:

Here the ratio is alsoÂ 3 blue squares to 1 yellow square,
even though there are more squares.

Using Ratios

The trick with ratios is to always multiply or divide the numbersÂ by the same value.

Example:

 4 : 5Â is the same asÂ 4Ã—2Â : 5Ã—2Â = 8 : 10

Example: A Recipe for pancakes uses 3 cups of flour and 2 cups of milk.

So the ratio of flour to milk isÂ 3 : 2

To make pancakes for a LOT of people we might need 4 times the quantity, so we multiply the numbers by 4:

3Ã—4Â : 2Ã—4Â = 12 : 8

In other words, 12 cups of flour and 8 cups of milk.

The ratio is still the same, so the pancakes should be just as yummy.

“Part-to-Part” and “Part-to-Whole” Ratios

The examples so far have been “part-to-part” (comparing one part to another part).

But a ratio can also show a part compared to theÂ whole lot.

Example: There are 5 pups, 2 are boys, and 3 are girls

 Part-to-Part:The ratio of boys to girls isÂ 2:3Â orÂ 2/3 The ratio of girls to boys isÂ 3:2Â orÂ 3/2 Part-to-Whole: The ratio of boys toÂ allÂ pups isÂ 2:5Â orÂ 2/5 The ratio of girls toÂ allÂ pups isÂ 3:5Â orÂ 3/5

Proportions

ProportionÂ says that twoÂ ratiosÂ (or fractions) are equal.

Example:

SoÂ 1-out-of-3Â is equal toÂ 2-out-of-6

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

Example: Rope

A ropesÂ lengthÂ andÂ weightÂ are in proportion.

WhenÂ 20mÂ of rope weighsÂ 1kg,Â then:

• 40mÂ of that rope weighsÂ 2kg
• 200mÂ of that rope weighsÂ 10kg
• etc.

So:

201Â =Â 402

Sizes

When shapes are “in proportion” their relative sizes are the same.

 Here we see that the ratios of head length to body length are the same in both drawings.So they areÂ proportional. Making the head too long or short would look bad!

Working With Proportions

NOW, how do we use this?

Example: you want to draw the dog’s head, and would like to know how long it should be:

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

 ? Â =Â 10 42 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 Ã— 10) / 20 = 420 / 20 =Â 21

So you should draw the headÂ 21Â long.

Using Proportions to Solve Percents

A percent is actually a ratio! Saying “25%” is actually saying “25 per 100”:

 Â 25% =Â 25 100

We can use proportions to solve questions involving percents.

First, put what we know into this form:

 Part Â =Â Percent Whole 100

Example: what is 25% of 160 ?

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

 Part Â =Â 25 160 100

Find the Part:

Example: what is 25% of 160 (continued) ?

 Part Â =Â 25 160 100

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

Part = (160 Ã— 25) / 100 = 4000 / 100 =Â 40

Answer: 25% of 160 is 40.

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

Part = 160 Ã— (25 / 100) = 160 Ã— 0.25 =Â 40

Either method works fine.

We can also find a Percent:

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

Fill in what we know:

 \$12 Â =Â Percent \$80 100

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

Percent = (\$12 Ã— 100) / \$80 = 1200 / 80 =Â 15%

Answer: \$12 isÂ 15%Â of \$80

Or find the Whole:

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:

 \$150 Â =Â 80 Whole 100

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

Whole = (\$150 Ã— 100) / 80 = 15000 / 80 =Â 187.50

Answer: the phone’s normal price wasÂ \$187.50

Using Proportions to Solve Triangles

We can use proportions to solve similar triangles.

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:

h Â =Â  2.4 m
2.9 m 1.3 m

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

h = (2.9 Ã— 2.4) / 1.3 = 6.96 / 1.3 =Â 5.4 mÂ (to nearest 0.1)

Answer: the tree is 5.4 m tall.

And he didn’t even need a ladder!

The “Height” could have been at the bottom, so long as it was on the bottom for BOTH ratios, like this:

Let us try the ratio of “Shadow Length to Height”:

2.9 m Â =Â  1.3 m
h 2.4 m

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

h = (2.9 Ã— 2.4) / 1.3 = 6.96 / 1.3 =Â 5.4 mÂ (to nearest 0.1)

It is the same calculation as before.

A “Concrete” Example

Ratios can haveÂ more than two numbers!

For example concrete is made by mixing cement, sand, stones and water.

A typical mix of cement, sand and stones is written as a ratio, such asÂ 1:2:6.

We can multiply all values by the same amount and still have the same ratio.

10:20:60 is the same as 1:2:6

So when we use 10 buckets of cement, we should use 20 of sand and 60 of stones.

Example: you have just put 12 buckets of stones into a mixer, how much cement and how much sand should you add to make aÂ 1:2:6Â 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 ofÂ everythingÂ 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 sameÂ relativeÂ sizes)

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

Why are they the same ratio?Â Well, theÂ 1:2:6Â ratio says to have:

• twice as much Sand as Cement (1:2:6)
• 6 times as much Stones as Cement (1:2:6)

In our mix we have:

• twice as much Sand as Cement (2:4:12)
• 6 times as much Stones as Cement (2:4:12)

So it should be just right!

That is the good thing about ratios. You can make the amounts bigger or smaller and so long as theÂ relativeÂ sizes are the same then the ratio is the same.