Math Formulae: Ratio And Proportion

 Some important Ratio and Proportion formulae:

	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

To, readily access formula book on your mobile, you can open tools section in Tricky Maths App.

Math Formulae: Profit, Loss and Discount

 Some important Profit, Loss and Discount formulae:

Terms Used: CP = Cost Price,     SP = Selling Price,     , MP = Marked Price

1. Profit = SP - CP
   
   
2. Profit % = 

   	(SP - CP)
   	CP

     × 100  = 

   	profit
   	CP

     × 100  
   
   
3. Loss = CP - SP
   
   
4. Loss % = 

   	(CP - SP)
   	CP

     × 100  = 

   	loss
   	CP

     × 100
   
   
5. SP = 

   	100 + profit %	× CP
   	100

      or,
   SP = 

   	100 - loss %	× CP
   	100

   
   
   
6. CP = 

   	100	× SP
   	100 + profit %

      or,
   CP = 

   	100	× SP
   	100 - loss %

   
   
   
7. SP = (100 + p) % of CP; where profit is p % of CP.
   
   
8. SP = (100 - x) % of CP; where loss is x % of CP.
   
   
9. When two articles are sold at same price, but one at profit of x % and other at loss of x % then there will be a loss, and
   loss % = 

   	x ²
   	100

   
   
   
10. MP = CP + Markup, i.e.
    MP = CP +(% markup on CP)
    
    
11. Discount = % discount on MP
    
    
12. SP = MP - Discount
    

To, readily access formula book on your mobile, you can open tools section in Tricky Maths App.

Math Formulae: Percentages

 Here are some important formulae related to Percentages

1. Fraction to percentage: Multiply fraction by 100 and put % sign.
   
   
2. Percentage to Fraction: Divide percentage by 100, reduce fraction to simplest form and remove % sign.
   
   
3. Decimal to percentage: Move decimal point 2 places to the right and add % sign.
   
   
4. Percentage to Decimal: Move decimal point 2 places to the left and remove % sign.
   
   
5. Fraction-Percentage values to remember:
   

   	1
   	2

      =  50 %, 

   	1
   	3

      =  33.33 %, 

   	1
   	4

      =  25 %,
   
   

   	1
   	5

      =  20 %, 

   	1
   	6

      =  16.66 %, 

   	1
   	7

      =  14.28 %,
   
   

   	1
   	8

      =  12.25 %, 

   	1
   	9

      =  11.11 %, 

   	1
   	10

      =  10 %,
   
   

   	1
   	11

      =  9.09 %, 

   	1
   	12

      =  8.33 %, 

   	1
   	13

      =  7.69 %,
   
   

   	1
   	14

      =  7.14 %, 

   	1
   	15

      =  6.66 %, 

   	1
   	16

      =  6.25 %,
   
   

   	1
   	17

      =  5.88 %, 

   	1
   	18

      =  5.55 %, 

   	1
   	19

      =  5.25 %,
   
   

   	1
   	20

      =  5 %, 

   	1
   	25

      =  4 %, 

   	1
   	50

      =  2 %,
   
   Examples:
   1. 6.25 % of 320 = 

   	320
   	16

      =  20
   
   2. 20 % of 630 = 

   	630
   	5

      =  126
   
   
6. Quantity X as percentage of quantity Y = 

   	X
   	Y

    × 100
   
   
7. What percentage of a is b = 

   	b
   	a

    × 100
   
   
8. Percentage increase or decrease in a quantity = 

   	Actual increase/decrease
   	Original quantity

    × 100
   
   
9. What percentage is X more than Y = 

   	(X - Y)
   	Y

    × 100
   
   
10. What percentage is X less than Y = 

    	(Y - X)
    	Y

     × 100

To, readily access formula book on your mobile, you can open tools section in Tricky Maths App.

Math Formulae: Interest

Here are some interest related formulae that can be handy in exam time:

Terms Used: SI = Simple Interest,    CI = Compound Interest,    , A = Amount

P = Principle,    t = Time,    r = rate of interest,    I = Interest

• A = P + I
  
  
• Simple Interest:
  
  1. SI = 

     	P × r × t
     	100

     
     
     
  2. A = P + 

     	P × r × t
     	100

        =  P 

     ( 1 +	r.t	)
     	100

     
     
     
• Compound Interest:
  
  1. CI = A - P
     
     
  2. A = P 

     ( 1 +	r	)	t
     	100

        , when interest rates are compounded yearly.
     
     
  3. A = P 

     ( 1 +	r/2	)	2t
     	100

        , when interest rates are compounded half-yearly.
     
     
  4. A = P 

     ( 1 +	r/4	)	4t
     	100

        , when interest rates are compounded quarterly.
     
     
• For first year, CI = SI
  
  
• For two years,
     CI - SI = P 

  (	r	)	²
  	100

  
  
  
• For three years,
     CI - SI = P 

  (	r	)	²
  	100

   

  (	r	+ 3	)
  	100

  
  
  
• Population: Total population (P) after t years,
     when population rises, P = P_i 

  (	1 +	r	)	t
  		100

  
     and, when it declines, P = P_i 

  (	1 -	r	)	t
  		100

  
  where P_i is initial population of area, and r is rate of increase or decrease of population.
  
  
• Depriciation: Depriciation in value of a product after t years,
      V_f = V_i 

  (	1 -	r	)	t
  		100

  
  where, V_f is final value of product,
  V_i is initial value, and
  r is rate of depreciation in value of product.

These formulae can be found in formula book under tools section of Tricky Maths App.

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Math Formulae: Averages

Some need to remember Average Formulae:

1. Average is the arithmetic mean of given data.
   
   
2. Average lies between highest and lowest values in th given data.
   
   
3. If X is added to all the values in the given set of data, then average of the data increases by X.
   
   
4. If X is subtracted from all the values in the given set of data, then average of the data decreases by X.
   
   
5. If X is multiplied to all the values in the given data, then new average X times the old average.
   
   
6. If all the values in the given data is divided by X, then new average of the data is old average divided by X, provided X is not 0.
   
   
7. If given set of data is in arithmetic series,
   then, if number of values in data is odd, then average is middle value
   and, if number of values is even, then average is mean of mid two values,
   provided that data is arranged in increasing or deceasing order.
   For Example: average of 1, 3, 5, 7, 9 is 5,
   and that of 1, 3, 5, 7 is 8
   
   
8. Weighted Average:

   Aw =	(k1.A1 + k2.A2 + ..... + kn.An)
   	(k1 + k2 + ...... + kn)

   where, k₁, k₂, .... , k_n are weights associated with A₁, A₂, ... ,A_n respectively.

These formulae can be found in tools section of Tricky Maths App.

Math Formulae: Algebraic Identities

Here are some important algebraic Identities: 

1. (a + b)² = a² + 2ab + b²
   
   
2. (a - b)² = a² - 2ab + b²
   
   
3. (a + b)³ = a³ + b³ + 3ab(a + b)
          = a³ +3a²b + 3ab² + b³
   
   
4. (a - b)³ = a³ - b³ - 3ab(a - b)
          = a³ - 3a²b + 3ab² - b³
   
   
5. a² - b² = (a - b).(a + b)
   
   
6. (a + b)² + (a - b)² = 2a² + 2b²
   
   
7. (a + b)² - (a - b)² = 4ab
   
   
8. a³ + b³ = (a + b).(a² - ab + b²)
   
   
9. a³ - b³ = (a - b).(a² + ab + b²)
   
   
10. a³ + b³ + c³ - 3abc = (a + b + c).(a² + b² + c² - ab - bc - ca)
    if a + b + c = 0,
    then a³ + b³ + c³ = 3abc
    
    
11. (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
    
    
12. 1 + 2 + 3 + ....... + n = 

    	n(n+1)
    	2

    
    
    
13. 1² + 2² + 3² + ....... + n² = 

    	n(n+1)(2n+1)
    	6

    
    
    
14. 1³ + 2³ + 3³ + ....... + n³ = 

    	(	n(n+1)	)2
    		2

    
    
    
15. sum of 1st n odd numbers = n²
    
    
16. sum of 1st n even numbers = n(n+1)
    
    
17. if aⁿ = a^m, then n = m
    
    
18. if aⁿ = bⁿ,
    then a = b, (if n is odd), and
    a = +b or -b (if n is even)
    

To, readily access formula book on your mobile, you can open tools section on Tricky Maths App.