Solving Proportions

If   a / b   =

c / d   then, a × d = b × c

We will illustrate this with a couple of examples.

Example #1:

Solve for x if   5 / x

=   10 / 16

Since these two fractions or ratios are in proportions, we know that the cross product must be equal.

Using the cross product, we get:

5 × 16 = x × 10

80 = 10x

If you know your multiplication table you can quickly get the answer.

If 10 × x = 80, then x should be 8 because 10 × 8 is 80.

x = 8

The proportion becomes   5 / 8

=   10 / 16

Notice that 5 × 16 = 8 × 10 = 80

You can also break the problem down into more steps if you like as shown below:

The first cross product also called product of the extremes is:
5 × 16 = 80

The second cross product also called product of the means is:
10 × x

Setting the cross products equal, we get:

10 × x = 80

There is a faster way to get the answer when solving proportions. Look at the proportion again:

5 / x

=    10 / 16

Notice that to get 10, 5 was multiplied by 2. By the same token, to get 16, something or a number must be multiplied by 2. What number multiplied by 2 will give you 16? No doubt it is 8!

Example #2:

Solve for n if   8 / 10

=   n / 25

Using the cross product, we get:

8 × 25 = 10 × n

200 = 10n

Instead of asking yourself " 10 times what equals 200? " we will this time solve the equation in order to show you another way to get n.

Divide both sides by 10

200 / 10

=    10n / 10

200 divided by 10 is 20 and 10 divided by 10 is 1

20 = 1n

20 = n

Useful equivalent proportions you can use when solving proportions.

Principle #1:

If   a / b

=    c / d   then,

a + b / b

=    c + d / d

Proof:

Add 1 to both sides of the equation and do the math as demonstrated:



The above can be useful if you are solving the proportion below:

x - 8 / 8

=    6 / 4

The above equation becomes

x - 8 + 8 / 8

=    6 + 4 / 4

Or

x / 8

=    10 / 4

The above is of course a lot easier to solve

Principle #2:

If  x / y

=    x / 4   then, y = 4

For instance if  50 / y

=    50 / 100   then, y = 100

If  18 / y

=    x / y   then, x = 18

Principle #3:

If   a / b

=    c / d   then,

a + c / b + d

=    a / b

Proof:

Cross multiply:

b × c = a × d

bc = ad

Add ab to both sides of the equation

ab + bc = ab + ad

Factor b from the left side. Factor a from the right side.

b(a + c) = a(b + d)

Rewrite the above as a proportion. It is like undoing a cross multiplication.

a + c / b + d

=    a / b

Why is principle #3 useful when solving proportions?

Say you have   x + 2 / 8 + 4

=    x / 8

It is equivalent to   x / 8

=    2 / 4

Again, the last format has a friendly look and it can be solved faster.

Just remember these 3 principles when solving proportions and it will ease the proportion exercise for you. Thanks for reading!