Ratios and proportions

Ratio: A ratio can be written in three different ways and all are read as “the ratio of a to b”

a t o

a/

Proportion: A proportion is an equation which says that two ratios are equivalent. it can be written in two ways:

two equal fractions, or using a colon, a:b = c:d

For example, if one package of cookie mix results in 20 cookies than that would be the same as to say that two packages will result in 40 cookies.

A proportion is read as “a is to b as c is to d”

a/b =c/d w h e r e b, d \neq 0

If one number in a proportion is unknown you can find that number by solving the proportion.

Example

You know that to make 20 cakes you have to use 2 eggs. How many eggs are needed to make 100 cakes?

	Eggs	cakes
Small amount	2	20
Large amount	x	100

e g g s/ c a k e s = e g g s/ c a k e s o r c a k es/ e g g s = c a k e s/ e g g s

If we write the unknown number in the nominator then we can solve this as any other equation

x/ 100 = 2/20

Multiply both sides by 100

100 \cdot x/ 100 = 100 \cdot 2/ 20

x = 200/ 20

x = 10

If the unknown number i.e. x is in the denominator, we can use another method that involves the cross product.

The cross product is the product of the numerator of one of the ratios and the denominator of the second ratio. The cross products of a proportion is always equal.

If we again use the example with the cookie mix used above

20/ 1 = 40/ 2

1 \cdot 40 = 2 \cdot 20 = 40

It is said that in a proportion if

a/ b =c/d w h e r e b, d \neq 0

a d = b c

If you look at a map it always tells you in one of the corners that 1 inch of the map correspond to a much bigger distance in reality. This is called a scaling. We often use scaling in order to depict various objects. Scaling involves recreating a model of the object and sharing its proportions, but where the size differs. One may scale up (enlarge) or scale down (reduce).

For example, the scale of 1:4 represents a fourth. Thus any measurement we see in the model would be 1/4 of the real measurement. If we wish to calculate the inverse, where we have a 20ft high wall and wish to reproduce it in the scale of 1:4, we simply calculate:

20 \cdot 1 : 4 = 20 \cdot 1/ 4 = 5