4.1 Rule of Three
The expression rule of three comes from the Latin Regula de tribus. The rule is used when you have four numbers which are proportional, like for example the similarity:
3
6 
= 
5
10

This can be read as 3 is to 6 as 5 is to 10.
If three of these numbers are given, then you can determine the fourth number by using the rule of three. For example, if we have:
x
6 
= 
5
10 
We are looking for the number x, and calculate it like this:
x

= 
5 · 6
10 
= 3 
This method for finding x in one calculation is called cross multiplication.
In order to use cross multiplication, we need to have proportionality. Read page 1.1 in this section to review what proportionality is.
In order to clarity how this is used for more practical calculations, here are a few examples:
Example 1
Driving a car between Stockholm and Gävle takes 2 h and the distance is 150 km. How much time does it take to drive between Stockholm and Luleå if the speed is the same? The distance between Stockholm and Luleå is 900 km.
Driving 150 km takes 2 h.
Driving 900 km takes x h.
150
900 
= 
2
x 
The correlation above is a proportionality because 150 km is to 900 km as 2 h is to x h.
The equation can be solved in two ways, either by cross multiplication or by solving for x in the usual manner by multiplying both sides by x. We will show both methods.
We cross multiply:
The arrows show which numbers we multiply with each other. 

We multiply both sides by x:
150 · x
900 
= 
2 · x
x 
150 · x
900 
= 
2 
2. We multiply both sides by 900.
150 · x · 900
900 
= 
2 · 900 

Cross multiplication gives:



150 · x
150 
= 
1,800
150 
x = 12


150 · x
150 
= 
1,800
150 
x = 12

Answer: It takes 12 h to drive between Stockholm and Luleå.
Which method is easier to use? Use the method you like best.
Example 2
3 kg bananas cost 13.50 kr. How much does 7 kg bananas cost if the price per kilo is the same no matter how much you buy?
We solve the exercise with cross multiplication.
3 kg costs 13.50 kr.
7 kg costs x kr.
3
7 
= 
13.50
x 
Cross multiplication gives:
3 · x = 13.50 · 7
3x = 94.50
3 · x
3 
= 
94.50
3 
x = 31.50 kr
Answer: 7 kg bananas cost 31.50 kr.
Example 3
The triangles below are similar. If two triangles are similar, this means that their corresponding sides are proportional. When corresponding sides are proportional, the relationship between the sides is the same.
How long is side x?
From similarity, we get:
x is to 7, as
6 is to 8.
This gives:
x
7 
= 
6
8 
Cross multiplication gives:
8 · x = 6 · 7
8x = 42
8 · x
8 
= 
42
8 
x = 5.25 cm
Answer: The side marked with the x is 5.25 cm.
