1. | In ratio a : b (or | a | ), a (numerator) is called antecedent and b (denominator) is called consequent. |
b |
2. | Qualities to be expressed as ratio should have same units (like both should be in cm or in kg, but not one in kg and other in gram). |
3. | Ratio has no units. |
4. | Value of ratio (a : b) is not changed when both a and b are multiplied (or divide) by same number. |
5. | a/b | = | a | × | d |
c/d | b | c |
6. | Compounded ratio is result of multiplication of two or more ratios. |
7. | Duplicate ratio of (a : b) = ( | a | )2. |
b |
8. | Sub-Duplicate ratio of (a : b) = ( | a | )1/2. |
b |
9. | Triplicate ratio of (a : b) = ( | a | )3. |
b |
10. | Sub-triplicate ratio of (a : b) = ( | a | )1/3. |
b |
11. | a | = | (c + a.n) | , if and only if, | a | = | c |
b | (d + b.n) | b | d |
12. | (a + c) | > | a | , if, | a | < | c |
(b + d) | b | b | d |
13. | (a + c) | < | a | , if, | a | > | c |
(b + d) | b | b | d |
14. | If | a | > 1, then, | (a + x) | < | a |
b | (b + x) | b |
15. | If | a | < 1, then, | (a + x) | > | a |
b | (b + x) | b |
16. | If | a | > 1, then, | (a - x) | > | a |
b | (b - x) | b |
17. | If | a | < 1, then, | (a - x) | < | a |
b | (b - x) | b |
18. | If a : b = a1:b1 |
and b : c = b2:c1 | |
then, a : b : c = (a1.b2) : (b2.b1) : (b1.c1) |
19. | If a : b = a1:b1 |
and b : c = b2:c1 | |
and c : d = c2:d1 | |
then, a : b : c : d = (a1.b2.c2) : (b2.b1.c2) : (b1.c1.c2) : (b1.c1.d1) |
20. | Proportion is equality of two ratios, i.e. a : b :: c : d. a and d are called extremes and b and c are called means. |
Here, a is 1st proportional, b is 2nd proportional, | |
c is 3rd proportional, and d is 4th proportional. |
21. | Product of extremes = Product of means, i.e. a × d = c × d |
22. | If a : b = b : c then a, b, c are said to be in (continued) proportion. |
23. | If a : b = b : c, then b2 = a × c. Here, b is said to be mean proportional between a and cand c is called third proportional. |
24. | Invertendo: |
If, | a | = | c | , then, | b | = | d | |
b | d | a | c |
25. | Alternando: |
If, | a | = | c | , then, | a | = | b | |
b | d | c | d |
26. | Componendo: |
If, | a | = | c | , then, | (a + b) | = | (c + d) | |
b | d | b | d |
27. | Dividendo: |
If, | a | = | c | , then, | (a - b) | = | (c - d) | |
b | d | b | d |
28. | Componendo and Dividendo: |
If, | a | = | c | , then, | (a + b) | = | (c + d) | |
b | d | (a - b) | (c - d) |
29. | Direct Variation: If X ∝ Y, then |
X = k.Y, or, | X | = k | |
Y |
Where, k is constant of proportionality |
Also, | X1 | = | X2 | |
Y1 | Y2 |
30. | Inverse Variation: |
If X ∝ | 1 | |
Y |
then, | X = | k | , or, X.Y = k |
Y |
Where, k is constant of proportionality |
Also, X1.Y1 = X2.Y2, or, | X1 | = | Y2 | |
X2 | Y1 |