Average, Mixture and Alligations – Concepts for Aptitude

1. Averages

Imagine you went to a café three times and spent ₹100, ₹150, and ₹200. If your friend asks you how much you spend each time, you will say on an average 150. Mentally you are trying to give a number which is closer to your spend and not extreme. If you say 200, you feel you said a little high. If you say 100, it might sound low. So you end up saying 150. Average is also called “Mean” or “Mean Value”

$Average = \frac{Sum \ of \ value \ of \ all \ items}{Number \ of \ items \ in \ group}$

$A_n = \frac{x_1 + x_2 + x_3 + ….. + x_n}{n}$

Let’s take slightly difficult but interesting example which oyu might have seen in splitwise as well

Q. Three friends, John, Emily, and Sarah, went on a road trip. John spent ₹3,000, Emily spent ₹4,500, and Sarah spent ₹2,500. If they want to split the total expenses equally, how much does each person need to contribute? Who owes whom and how much?

Sol: Total Expenses = ₹3,000 (John) + ₹4,500 (Emily) + ₹2,500 (Sarah) = ₹10,000

Number of People = 3

Average (Equal Share per Person) = 10,000/3 = ₹3,333.3

  • John spent ₹3,000, so he owes ₹333.33.
  • Emily spent ₹4,500, so she should be paid back ₹1,166.67.
  • Sarah spent ₹2,500, so she owes ₹833.33.
  • John pays ₹333.33 to Emily.
  • Sarah pays ₹833.33 to Emily.
2. Weighted Averages

Now, weighted average is like saying, “Some things are more important than others.”

Let’s say you have two tests: One is worth 70% of your grade, and the other is worth 30%. If you score 90 in the first test and 70 in the second, you will be more happy than if you score 70 in the first test and 90 in the second test.

$ A_w \ [weighted] = \frac{x_1n_1 + x-2n_2 + x_3n_3 + ….. +x_kn_k}{n_1 + n_2 + n_3 + …. + n_k}$

Q. You are calculating your CGPA for a semester with three subjects. Each subject has a different credit weight, and your grades are as follows:

  • Subject 1: Grade = 8 (Credits = 4)
  • Subject 2: Grade = 7 (Credits = 3)
  • Subject 3: Grade = 9 (Credits = 2)

What is your CGPA for the semester?

Sol: In a CGPA system, your final grade is influenced by the credits of each subject. To calculate CGPA using the weighted average formula, follow these steps:

Total Weighted Grade Points = Sum of (Grade × Credits):

(8×4)+(7×3)+(9×2)=32+21+18=71

Total Credits = Sum of all credits:

4+3+2=9

Weighted Average (CGPA):

CGPA=$ \frac{\text{Total Weighted Grade Points}}{\text{Total Credits}} = \frac{71}{9} = 7.89 $

3. Some properties of average
  • The average will always be greater than the smallest value and less than the largest value in the group
  • If the value of each item is increased or decreases by the same value p, then average of the group or items will also increase or decrease by the same value, p respectively.
  • If value of each item is multiplied or divided by the same value p, then the average of the group will also get multiplied or divided by the same value p respectively.
  • If the average age of a group of persons is x-years today then after n years their age will be (x+n) years. Also n years ago, their average age would have been (x-n)
4. Mixtures
  • Mixing of two or more qualities of things produces a mixture.
  • Mixtures is basically application of averages and weighted averages.
  • If x1 is the quantity (number of items) of one particular item of quality A1, and x2 is the quantity (number of items) of the second item of quality A2, and then both are mixed together to give a new mixture. The Weighted average value (p) of the quality of mixture is given by $p = \frac{x_1A_1 + x_2A_2}{A_1 + A_2}$
  • If there are more than two groups of items mixed, the weighted average formula can be applied in a similar manner.
  • A mixture can also be a solution (a liquid mixed with another liquid). The concentration of the solution is expressed as the proportion (or percentage) of the liquid in the total solution.
5. Alligations

Alligations is nothing but a faster technique of solving questions based on the weighted average situation.

$\frac{n_1}{n_2} = \frac{A_2 – A_w}{A_w – A_1} and A_w = \frac{n_1A_1 + n_2A_2}{n_1 + n_2}$

Q. A coffee shop owner has two types of coffee beans:

  • Type A costs ₹400 per kg
  • Type B costs ₹600 per kg

She wants to mix them in a way that the final blend costs ₹500 per kg. In what ratio should she mix Type A and Type B?

Sol: Use the alligation method:

Ratio= (Higher price−Mean price)/(Mean price−Lower price

$\text{Ratio} = \frac{600 – 500}{500 – 400} = \frac{100}{100} = 1:1$

She should mix Type A and Type B in a 1:1 ratio

6. Diluting solution

If there is P volume of pure liquid initially and in each operation, Q volume is taken out and replaced by Q volume of water, then at the end of n such operations, concentration (k) of the liquid in the solution is $k = {\LARGE[} \frac{P – Q}{P} {\LARGE]}^n$

Q. You have 20 litres of juice with 25% fruit concentration. You want to dilute it to 15% concentration by adding water. How much water should you add?

Sol: Fruit content before dilution = 25%×20 litres=5 litres of pure fruit

Let the amount of water to be added be x litres

Total volume after adding water = 20+x 

We want the final concentration to be 15%, so: $\frac{5}{20 + x} = 15\% = \frac{15}{100}$

Cross-multiply and solve for x: ≈13.33

You can also refer following videos to get more perspective on the topic:

Practice

Refer Topic: Time and Work: https://www.learntheta.com/placement-aptitude-time-and-work/

Refer more Aptitude Questions with Solutions on Averages and Alligations: https://www.learntheta.com/aptitude-questions-averages-mixture-alligations/

Practice Aptitude Questions on Averages and Alligations with LearnTheta’s AI Practice Platform: https://www.learntheta.com/placement-aptitude/

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