A quick way to compute weighted average of large numbers

Hi

Weighted average is an easy concept that gets test quite often in the quantitative section of the TANCET test.

Let us take a look at a question that requires you to compute the weighted average and then evaluate an alternate shorcut approach to computing the same.
Question
If the average marks scored by 30 students of school A in Class XII board exam is 1072 and that by 20 students of school B is 1084, then what is the average marks scored by the students of the two schools taken together?
1. 1078.6
2. 1076.8
3. 1075.9
4. 1080.2
5. 1077.

Correct Answer
Choice (2). 1076.8

Explanation
The conventional way to solve this question is to find the weighted average of the two marks as show below.
(30*1072 + 20*1084)/(30 + 20)
= (32160 + 21680) / 50
= 53840 / 50
= 1076.8.

As you realize, conceptually there was nothing difficult with solving this question. The only pain point was the calculation. And it might not be possible for most of us to solve this calculation in under 1.2 minutes.

Now let us look at an alternate, shortcut approach to this question.
Step 1: Subtract 1072 from both 1072 and 1084.
Step 2: Compute the weighted average of the resulting numbers. i.e. ,0 and 12
The weighted average of 30 students scoring 0 and 20 students scoring 12
= (30*0 + 20*12) / (30 + 20) = 240 / 50 = 4.8
Step 3: Add 1072 to the weighted average obtained in step 2. i.e., 1072 + 4.8 = 1076.8

Therefore, whenever you have to compute the weighted average of two large numbers, subtract the smaller of the two numbers from both the numbers and then compute the weighted average and then add back the subtracted number to the result.

TANCET Prep Qn: Ratio Proportion

Ratio proportion is a fundamental topic from which you should expect anything from 2 to 3 questions out of the 20 questions in the Math section of TANCET MBA test.

The following question is a ratio question and tests your ability to express one term in terms of another, given the ratio between the two terms.

Question
Sum of the ages of three friends is 98 years. If the ratio of the ages of the oldest to the middle is 5 : 3 and that of the middle to the youngest is 5 : 3, how old is the oldest?
1. 49 years
2. 50 years
3. 30 years
4. 32 years
5. 42 years

The correct answer is 50 years. i.e., Choice (2).

Explanatory Answer
One of the easiest approach to solve this question is outlined below

Let A be the oldest, B the middle one and C the youngest amongst the three friends.
Ratio of the ages of A and B, A : B :: 5 : 3
i.e., A/B = 5/3
Therefore, A = (5/3) B

Ratio of the ages of B and C, B : C :: 5 : 3
i.e, B/C = 5/3.
Hence, C = (3/5)B

We also know that A + B + C = 98

Rewriting the equation in terms of B, we get (5/3)B + B + (3/5)B = 98.
Taking 15 as the common denominator and adding we get (25B + 15B + 9B)/15 = 98
Or (49/15)B = 98
Or B = (98*15)/49 = 30.

Hence, age of A = (5/3)B = (5/3)*30 = 50 years.

Various alternative formats of the same question exist. Some oft repeated questions are given at the Ratio Proportion Practice Question Set.

TANCET Math prep question - Linear Equations

Linear Equations is one of the fundamental topics from which you could expect one or two questions in the TANCET test.

A Linear Equation is one in which the power or the exponent of the variable is 1. An equation of the form 3x + 2y = 17 is a linear equation.

A linear equation in two variables x and y represents a straight line when plotted on a x - y plane. The equation of the x-axis is y = 0 and that of the y-axis is x = 0.

The following question is a word problem in linear equation. You will be presented some information as a statement and you will be expected to fram linear equation based on the information provided there and solve the linear equation to get the answer.

Question
QuestionRam's father is thrice as old as Ram. 6 years hence, his father will be 24 years older than Ram. How old is Ram?
1. 24 years
2. 18 years
3. 12 years
4. 36 years
5. 42 years

Correct Answer and Explanation
Let Ram's present age be R.
So, his father's present age is 3R.

6 years hence, Ram will be (R + 6) years old
6 years hence, Ram's father will be 3R + 6 years old.
The question states that 6 years hence, Ram's father is 24 years older than Ram.
i.e., 3R + 6 = (R + 6) + 24

Or 3R + 6 = R + 30
or 2R = 24 or R = 12.
Ram's present age is therefore 12 years.

Shortcut Approach
You should actually solve this question by going back from the answer choices. That way, you will save a lot of time. Anyone who is serious about cracking the TANCET should commence your preparation for the TANCET exam not later than the day your fifth semester exams are over if you are in a 3-year degree program or after your seventh semester exam gets over if you are in a 4-year degree program.

Work - Time questions

Work Time is topic that gets tested in the quant section of both TANCET MBA and TANCET MCA tests. It is an easy topic, if one had understood the basics of the Work Time questions.

Let us say Ram completes a task in 10 days. And Lakshman takes 15 days to complete the same task. A typical question will expect us to find out the time taken when both Ram and Lakshman work together.

The first temptation is to just add the two values 10 and 15 and say 25. However, that does not make sense. If Ram alone could complete the work in 10 days, then when Lakshman helps him, it should take lesser than 10 days.

So, we have to find an alternative approach to answer this type of questions in TANCET.

When Ram and Lakshman work together the amount of work done in a day will be the sum of the work done by Ram in a day and that done by Lakshman in a day.

If Ram takes 10 days to complete a task, then he will complete 1/10th of the task in a day.
Similarly if Lakshman takes 15 days to complete a task, then he will complete 1/15th of the task in a day.

Now, if Ram and Lakshman worked together they will complete 1/10 + 1/15 of the task in a day.

Now, that translates to 5/30th or 1/6th of the task getting completed in a day.
This certainly makes sense as higher amount of work gets done when the two of them work together.

If Ram and Lakshman complete 1/6th of the work in a day, then the entire task will be over in 6 days (the reciprocal of the amount of work done by them in a day).

Now let us summarize the method to solve work time question:

1. The reciprocal of the time that the first person takes to complete a task is the amount of task completed by the first person in a day.

2. Find the amount of work done by the second person in a day in the same way.

3. Add the amount of task completed when the two of them work together. i.e., add the result of step 1 and step 2.

4. To find the amount of time that the two will take when they work together, find the reciprocal of the answer obtained in step 3.

Try this question

If 'A' can complete half the work in 12 days and 'B' can complete half the work in 18 days, how long will they take to complete the entire task if they worked
together?

1. 30 days
2. 60 days
3. 7.2 days
4. 14.4 days
5. 21.6 days

Answers and explanation to this work time question.

Note that the same method can be adopted to solve questions on Pipes and Cisterns. You could get additional practice questions on the work time and pipes cisterns at the following link.

Weekly Math prep question for TANCET MBA

I will periodically post interesting questions similar to those that are tested in the TANCET MBA and TANCET MCA tests.

As you are aware a fifth (i.e.,) 20 questions out of the 100 questions in the TANCET MBA paper is the math or the quant section. And about 60 questions out of the 100 questions in TANCET MCA paper is the math or the quant section.

The math that is tested in both the TANCET MBA and TANCET MCA sections are 10th standard level math.

Be assured that you will not see any of the 12th standard level math questions such as Calculus, Matrices, Vectors etc., as this exam is open to even students who would not have taken math in class 12.

In effect, what TANCET math tests is Arithmetic, Basic Algebra, Geometry topics. For good measure get yourself acquainted with Modern math topics such as Permutation Combination and Probability and Set theory if you want to score high in this section.

TANCET Exam is conducted in the second or third week of May every year. If you are keen on cracking the test, then you should start your TANCET Preparation about 3 to 6 months before the test date.