1. If four quantities are in proportion, then Product of Means = Product of Extremes. In the proportion a:b::c:d, we have bc = ad

2. If a:b::c:x, x is called the fourth proportional of a, b, c. a/b = c/x or, x = bc/a.

3. If two numbers are in a:b ratio and the sum of these numbers is x, then numbers will be ax/(a+b) and bx/(a+b) respectively

4. If three numbers are in the ratio a:b:c and the sum of these numbers is x, then these numbers will be ax/(a+b+c), bx/(a+b+c) and cx/(a+b+c) respectively

5. The ratio of two numbers is a : b. If n is added to each of these numbers, the ratio becomes c : d. The two numbers will be given as an(c-d)/(ad-bc) and bn(c-d)/(ad-bc) respectively

6. The ratio of two numbers is a : b. If n is subtracted from each of these numbers, the ratio becomes c : d. The two numbers are given as an(d-c)/(ad-bc) and bn(d-c)/(ad- bc) respectively

7. If the ratio of two numbers is a: b, then the numbers that should be added to each of the numbers in order to make this ratio c:d is given by (ad-bc)/(c-d)

8. If the ratio of two numbers is a:b, then the number that should be subtracted from each of the numbers in order to make this ratio c:d is given by (bc-ad)/(c-d)

9. The CP of the item that is cheaper is CPcheaper and the CP of the item that is costlier (dearer) is CPDearer. The CP of unit quantity of the final mixture is called the Mean Price and is given by ?????? ????? = ???ℎ????? − ?????? ????? ?????? ????? − ???ℎ????�