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 or consequent.
Eg. The ratio 5 : 9 represents 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.
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.
a : b = c : d, then c is called the third proportion to a and b.
Mean proportional between a and b is
4. Comparison of Ratios:
We say that
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]
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, x ∝ 1/y
Types of variation:
(i) Direct Variation
If A is in direct variation with B, then increase or decrease in A will lead to proportionate increase or decrease in B.
A ∝ B
A = KB
(ii) Indirect variation
If A is in inverse variation with B, then increase in A will lead to
Proportionate decrease in B and vice versa.
A ∝ 1/B
A = K/B
(iii) Joint Variation
Let us consider the area of triangle, which is dependent on both, the height as well as the base of the rectangle.
When both the dimension of the triangle changes then the area also changes.
When the area of the triangle varies with the change in the base of the triangle.
A ∝ b
When the area of the triangle varies with the change in the base of the triangle
A ∝ h
This is called the joint variation of the area of the triangle with respect to its base and height.
A ∝ 1/2 x b.h
7. The LCM process gets very cumbersome when we have to find the ratio out of multiple ratios.
We have the following simple method for that for chain of ratios of any length.
Suppose, you have the ratio train as follows:
A : B = 1 : 2
B : C = 2 : 3
C : D = 5 : 6
D : E = 7 : 8
If we were to find A : B : C : D : E, then the LCM method would have taken quite a long time which is infeasible in examinations of limited hours.
The short cut is as follows:
A : B : C : D : E can be written directly as:
1 × 2 × 5 × 7 : 2 × 2 × 5 × 7 : 2 × 3 × 5 × 7 : 2 × 3 × 6 × 7 : 2 × 3 × 6 × 8
70 : 140 : 210 : 252 : 288
The thought algorithm for this case goes as:
To get the combined ratio of A : B : C : D : E, from A : B, B :C, C :D, and D : E
In the combined ratio of A : B : C : D : E. A will correspond to the product of all numerators (1 × 2 × 5 × 7).
B will take the first denominator and the last 3 numerators (2 × 2 × 5 × 7). C on the other hand takes the first two denominators and the last 2 numerators (2 × 3 × 5 × 7),
D takes the first 3 denominators and the last numerator (2 × 3 × 6 × 7) and E take all the four denominators (2 × 3 × 6 × 8).