Averages and Mixtures - Formulae, Shortcuts

Formulae:

1.       Average = sum of quantities/number of quantities

2.       Sum = avg * no. of quantities

3.       If the numbers are in A.P. (Arithmetic Progression), then the avg of numbers is given by

Avg = sum/n =[ (n/2)(a+l)]/n

                = (a+l)/2

Where  n – no. of quantities

                a – first term or value

                l – last term or value

4.       Weighted Average:

Section1:

                No. of quantities: m

                Avg of section1: p

Section2:

                No. of quantities: n

                Avg of section1: q

Avg of section1 and section2 = (m*p + n*q)/(m +n)

Points to remember/Shortcuts

1.       Average lies between minimum and maximum values.

a.       Avg > minimum value

b.      Avg < maximum value

2.       If every quantity is increased/decreased by ‘k’ value, then the avg also get increased/decreased by the same ‘k’ value.

Mixtures

This is the extension of Averages which we have discussed earlier.

Formulae and shortcuts used to solve the following problems are discussed in the previous post.

Solved Problems
1. Let the cost of 2 quantities of rice be Rs.15 and Rs.19 per kg. Find the ratio of mixture which cost Rs.18 per kg?
Soln.:
          Method1:
          Let the quantity of rice of quality A (Rs.15/kg) be 'x'
          Let the quantity of rice of quality B (Rs.19/kg) be 'y'
          As per the problem, 15x+19y = 18(x+y)
          y = 3x => x/y = 1/3
          Therefore, the ratio of quantities mixed to make the quality of Rs.18 is 1:3.

          Method2: Using shortcut,
         
              Therefore, ratio is 1:3  
 
2. Let the cost of 2 quantities of rice be Rs.15 (quality A) and Rs.19 (quality B) per kg. Find the quantity of rice of quality A that has to be mixed with 27kg of quality B to make the cost of rice as Rs.18?
Soln.:
          Method1:
          Let the quantity of rice of quality A (Rs.15/kg) be 'x'
          Let the quantity of rice of quality B (Rs.19/kg) be 'y'
          As per the problem, 15x+19y = 18(x+y)
          y = 3x => x/y = 1/3
          Therefore, the ratio of quantities mixed to make the quality of Rs.18 is 1:3.
          => 3 parts of quality B is to be mixed with 1 part of quality A 
          => 27kg of quality B is to be mixed with 9kg of quality A.
          Ans: 9kgs

          Method2: Using shortcut,
         
              Therefore, ratio is 1:3
               => 3 parts of quality B is to be mixed with 1 part of quality A 
               => 27kg of quality B is to be mixed with 9kg of quality A.
               Ans: 9kgs

3. Let the cost of 2 quantities of rice be Rs.15 and Rs.19 per kg. These 2 qualities of rice are mixed and sold at Rs.27 per kg of profit 50%. Find the ratio in which 2 qualities of rice mixed?
Soln.:
          Method1:
          Let the quantity of rice of quality A (Rs.15/kg) be 'x'
          Let the quantity of rice of quality B (Rs.19/kg) be 'y'
          Given Selling price, SP = 27 and the profit % is 50%
          We know that the profit%, p% = (SP-CP)/CP * 100
          => 50/100 = (27 - CP)/CP
          => CP = 18
          As per the problem, 15x+19y = 18(x+y)
          y = 3x => x/y = 1/3
          Therefore, the ratio of quantities mixed to make the quality of Rs.18 is 1:3.

          Method2: Using shortcut,
          Given Selling price, SP = 27 and the profit % is 50%
          We know that the profit%, p% = (SP-CP)/CP * 100
          => 50/100 = (27 - CP)/CP
          => CP = 18
         
              Therefore, the ratio of quantities mixed to make the quality of Rs.18 is 1:3.

4. A man purchased TV and Washing machine for Rs.30000. He sold the TV at 30% profit and washing machine at 60% profit. He makes overall profit of 50%. Then find for how much did he purchased TV and washing machine?
Soln:
          Method1:
          Let the cost price of TV be 'x'
          Let the cost price of washing machine be 'y'
          As per the problem,
          x + y = 30000 ------> Eq. 1
          SP of TV = 1.3x
          SP of washing machine = 1.6y
          1.3x + 1.6y = 30000 * 1.5
          1.3x + 1.6y = 45000 --------> Eq. 2
          On solving equations 1 and 2, x = 10000 and y = 20000
          Therefore, cost price of TV and washing machine are Rs.10000 and Rs.20000 respectively.

          Method2: Using shortcut,

         
              Therefore, ratio is 1:2. 3 parts is equivalent to 30000
              => Cost of TV is 10000 (1 part) and Cost of washing machine is 20000 (2 parts)