Ratio, Proportion and Variation - Algebra

* Ratio

            - When a number x is to y denoted as x:y where y is not equal to zero. 
            - Simply the fraction of two numbers.

            x:y = x/y

Examples: 

1. What is the ratio of 5 ft to 85 in?

If 1ft = 12 in, then 5 ft is equal to 60. Hence,

60ft/85ft = 12/17 = 12:17

2. What is the ratio of the mass of the blood to the mass of the body of a 75 kg man? If the mass of blood of the man is 5 kg.
5kg/75kg = 1/15 = 1:15

* Proportion
            - is the equality of two ratios or fraction.

a:b = c:d or  a/b = c/d 

Where:

- a and d are called extremes
- b and c are called means

We have Laws in Proportion and here are the following:

1. The product of extremes is equal to the product of means.

From a/b = c/d we will get,

ad = bc

Example:

1/4 = 2/8  

(1)(8) = (4)(2)

        8 = 8

2. The reciprocals of both ratio in a proportion are equal.
b/a = d/c

Example:

1/4 = 5/20

4 = 20/5

4 = 4

3. Interchanging the mean of the first ratio with the extreme of the other will not affect the proportion and vice versa.
a/c = b/d     or    d/c = b/a 

Example:

1/2 = 4/8 by interchanging the mean and extreme,

1/4 = 2/8

1/4 = 1/4

4. Adding the mean on the extreme of the left-side of the equation while adding the extreme on the mean of the right-side of the equation (Vise-versa) will not change the equality.  

(a+b)/b = (c+d)/d 

Example:

1/3 =  5/15

(1+3)/3 = (5+15)/15

4/3 = 20/15

4/3 = 4/3

5. Subtracting the mean on the extreme of the left-side of the equation while subtracting the extreme on the mean of the right-side of the equation (Vise-versa) will not change the equality.

 
(a - b)/b = (c - d)/d 

Example:

1/5 = 5/25

(1-5)/5 = (5-25)/25

- 4/5 = -20/25 

-4/5 = - 4/5

6. (a+b)/(a-b) = (c+d)/(c-d)

Example:
2/7 = 6/21

(2+7)/(2-7) = (6+21)/(6-21)

-9/5 = -27/15

-9/5 = -9/5 

* Variation

3 Principles of Variation:

1. If y varies directly as x then, y = kx.

2. If y varies inversely as x then, y = k/x

3. If y varies jointly as x and z then, y = kxz

Note:

 
- k is the constant of variation.

Example:

* If y varies inversely as x and directly as z and if y = 4 when x = 3 and z = 5. Find z when x = 4 and y = 6.

First, solve for k:(Substitute the first given where x,y and z are known)

y = kz/x

k = xy/z

k = (3 * 4)/5

k = 12/5 or 2.4

Now that we have the value of k, we can solve for z in the second set of values of x and y.

y = kz/x

z = xy/k

z = (4 * 6 )/ 2.4

z = 24/2.4

z = 10

Hence, the value of z is 10 when x is 4 and y is 6.