Solving Proportions

Solving proportions by using cross product to find unknown terms is what this lesson is about.





We will also show somes principles, special techniques or shortcuts that can be used to quickly solve a proportion.

Terms to know:

x, y, or any other letter is used to stand for an unknown number.

Unknown term: The missing or unknown number in a proportion.

We have seen in the lesson about proportions that we can use cross product to determine if the fractions or ratios are in proportions.

Cross products can also be used to find an unknown term in a proportion. Here is how!

If  
a / b
=
c / d
  then, a × d = b × c
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:

First cross product: 5 × 16 = 80

Second cross product: 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 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:

equivalent-proportion

The above can be useful if you solving

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




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:

First cross product: 5 × 16 = 80

Second cross product: 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 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:

equivalent-proportion

The above can be useful if you solving

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


Solving proportions quiz. Can you get 100% on this quiz?



Use this proportion calculator to solve proportion word problems





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