# Important Formulas - Ratio and Proportion - Tricks & Shortcuts

1. Ratio:
The ratio of two quantities a and b in the same units, is the fraction a/b and we write it as a : b.

In the ratio a : b, we call a as the first term or antecedent and b, the second term orconsequent.

Eg. The ratio 5 : 9 represent  5/9 with antecedent = 5, consequent = 9.

Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.

Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.

2. Proportion:
The equality of two ratios is called proportion.

If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.

Here a and d are called extremes, while b and c are called mean terms.

Product of means = Product of extremes.

Thus, a : b :: c : d (b x c) = (a x d).

3. Fourth Proportional:

If a : b = c : d, then d is called the fourth proportional to a, b, c.

Third Proportional:
a : b = c : d, then c is called the third proportion to a and b.

Mean Proportional:
Mean proportional between a and b is √(ab).

4. Comparison of Ratios:
We say that (a : b) > (c : d) a/b > c/d  .

Compounded Ratio:
The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

5. Duplicate Ratios:
Duplicate ratio of (a : b) is (a2 : b2).

Sub-duplicate ratio of (a : b) is (√a : √b).

Triplicate ratio of (a : b) is (a3 : b3).

Sub-triplicate ratio of (a : b) is (a1/3 : b1/3).

If    a/b = c/d , then  (a + b)/( a - b)  =  (c + d)/( c - d).     [componendo and dividendo]

6. Variations:
We say that x is directly proportional to y, if x = ky for some constant k and we write,xy.
We say that x is inversely proportional to y, if xy = k for some constant k and
we write, x 1/y.