Very Important formulae, tricks and shortcuts of Ratio And Proportion:
Ratio is a quantity which represents the relationship between two similar quantities. It expresses a magnitude by which quantity is multiple of another one. Ratio is represented as 2:3 or 2/3. Here, numerator i.e. 2 is known as "ANTECEDENT" and denominator i.e. 3 is known as "CONSEQUENT".
If antecedent is more than the consequent, then it is known as improper ratio and if less ,then it is known as proper ratio.
Some formulae, tricks and shortcuts of Ratio And Proportion:
1) If ratio is written as A:B, it is said to be a linear form and in case it is written as A/B, it is said to be fractional form.
2) Ratio does not have any unit. It is mere number.
3) The equality of two ratios is known as proportion i.e. a/b = c/d
If a/b = c/d , then it is also equal to a+c/b+d
Invertendo : If a/b = c/d , then b/a = d/c
Alterendo : If a/b = c/d , then a/c = b/d
Componendo : If a/b = c/d , then a+b/b = c+d/d
Dividendo : If a/b = c/d , then a-b/b = c-d/d
Componendo and Dividendo : If a/b = c/d , then a+b/a-b = c+d/c-d
4) If a/b = b/c = c/d =...... so on, then a,b,c,d... are in G.P.
Proof: Let a/b = b/c = c/d =k
c= dk, b= ck, a= bk
Therefore, b= dk^2 and a= dk^2
All are in G.P.
5) If a>b and same positive number is added to each term, then ratio decreases.
For example: a/b = 4/3 = 1.3, If 2 is added to each term, then a/b = 4+2/3+2 = 6/5 = 1.2
Therefore, ratio decreases.
6) If a For example: a/b = 3/4 = 0.7, If 2 is added to each term, then a/b = 3+2/4+2 = 5/6 = 0.8
Therefore, ratio increases.
7) If we multiply or divide any number, there will be no effect on ratio.
8) Let a:b is a ratio
a^2:b^2 is duplicate ratio of a:b
a^3:b^3 is triplicate ratio of a:b
a^1/2:b^1/2 is sub-duplicate ratio of a:b
a^1/3:b^1/3 is sub-triplicate ratio of a:b
9) Proportions i.e. a:b = c:d
a and d are known to be extremes
b and c are known to be means.
10) In a:b :: c:d, d is fourth proportional to a,b and c.
11) If x is third proportional to a,b then it is written as a:b :: b:x.
Lets Solve a problem with simple trick and shortcut to save time:
Example: A bag contains 1rupee, 50 paise and 10 paise coins in the ratio 3:4:10 amounting to Rs.102. Find the number of 10paise coins?
Denomination of coins = 50p, 10p, 1 rupee
Total amount = Rs.102
Ratio of coins = 3:4:10
Let the number of 1rupee (100 paise), 50p, 10p coins be 3x, 4x and 10x respectively
Since the total amount is given in Rupees (102), the 50p and 10p has to be converted into rupees
1 rupee = 1 rupee
50p = [50/100] = ½ rupee
10p = [10/100] = [1/10] rupee
Multiply the converted rupees with the given ratios so,
1×3x + (½ ×4x) + [(1/10)×10x]
We will get,
à3x + 2x + 1x = 102
=> 6x = 102
=> x = [102/6] = 17
Now, the number of 10p coins are asked in the question, the ratio term of 10p =10x
Therefore, no. of 10p coins in the bag = 10x = [10×17] =170 coins
SHORTCUT METHOD with Best Trick:
3×100p: 4×50p: 10×10p
Sum of all parts 6= Rs.102
Single part 1=?
x = Rs. 102 /6 = 17
There for no. of 10p = 10×17 = 170coins
For solved problems on above formulas please visit below sections: