##### Ratio and Proportion

# Ratio and Proportion Formulas

**Important Formulas**

**1.Ratio:**

The ratio of two quantities a and b in the same units, is the fraction 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 or consequent. Eg. The ratio 5 : 9 represents 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:**

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 (*a*^{3} : *b*^{3}).

Sub-triplicate ratio of (*a* : *b*) is (*a*^{1/3} : *b*^{1/3}).

[componendo and dividendo]

**6.Variations:**

We say that x is directly proportional to y, if x = ky for some constant k and we write, x ∝ y.

We say that x is inversely proportional to y, if xy = k for some constant k and

we write,