Averages - Shortcuts & Tricks for Placement Tests, Job Interviews & Exams

Welcome to a quantitative aptitude video on averages from careerite.com. The problems on averages are extremely important for placement test, bank, MBA and all other entrance exams. In every exam, you can definitely find at least two averages sums. Being easily solvable, you can use them to increase your scores. In this video, we are going to show you some very easy tricks to quickly solve all questions from this topic. After this video, you can go to careerite.com which has thousand plus practice questions on quantitative aptitude and take the practice test over there. Let's get [Music] started. Let's start with the average formula. Now before going on to the formula directly, let us try to derive it or understand first what exactly average is. By this you'll be able to easily tackle all the sums as well as remember the formula very easily and quickly recollect it in the exams. Now let us say what let's go on first with what does average means. Average is nothing but equitable distribution. Okay, equitable distribution. What does that mean? That means if you have say 100 rupees, okay, if you have 100 rupees and there are say five children, right? Five children and you want to divide 100 rupees equally among five children. What you would do? You'll simply give 20 20 rupees to each child. Okay. So each child gets 2020 20 rupees equitable distribution. This total comes out to be 100, right? This is nothing but average. So what we can say is that 100 rupees all the five children get 20 20 20 rupees. Okay, each one gets same amount of rupees. So the average of money, okay, average amount with each child is nothing but 20 rupees. This is uh an example of average. We'll go on to more examples and the idea will become clearer. But you just need to remember that average is nothing but equitable distribution. Okay, let's go on to another uh example that would make the idea more clearer. Let us say there's a family which has a dad. Okay, there's the mom. Then there's the son and then there's the daughter. Okay? Right? Now the dad has 100 rupees. The mother has 200 rupees, the son has 25 rupees and the daughter has 75 rupees with them. Now what should we do first? Let us say if we take a box and collect all this money in the box. So how much money would be there in the box? 100 + 200 would be 300. 325 + 75 is 400. So now in the box there is rupees 400, right? 400 rupees. Now if I want to distribute this equally among all these four members, how much should each get? Dad will get 100 rupees, mother will get 100 rupees because that would be nothing but 400 divided by since there are four members. So divided by four. 400 is the total amount of money right with all the four and four is nothing but the total number of family members. So 100 rupees each. So each one must get 100 rupees. So son gets 100 rupees, daughter gets 100 rupees, right? All four of them have received the equal amount, right? 100 rupees. So this is nothing but average. The average amount with each person would be 100. Now did you see how did we calculate this 100 value? We simply added all the monies. Okay, with each ordered all the amounts with each of the person. Okay, this becomes the sum and we divided it by the total number of persons. Right? This is nothing but the formula for average. Average is nothing but sum of all the observations. Okay, observations means sum of all these values and divided by total number of values or total family members. Total members here there are four members. Dad, then there's the mother, then there's a son and the daughter. So four members are there. So right now over here we have four total members. Right? Total values are four. And how sum sum means how much value does each of them have? Dad has 100, mother has 200, son has 25, daughter has 75. So addition would be 400. So average comes out to be 100. What is this 100? 100 means on an average if we calculate the average amount of money with each person then it comes out to be 100 rupees. So this 100 okay this is the average amount of money with each person in the family right see how easily we calculated average there's nothing to it just add all the values that is nothing but the sum and divided by total number of values so how to remember average formula average is nothing but sum upon total sum upon total this is the only thing you want you should remember and this will solve all the sums related related to average. There's nothing to it. Average means equitable distribution and average is nothing but add everything divided by the total number of members or total number of things. Right? Now say if they say a person there's a batsman okay there's a batsman he is playing okay he has played say 20 matches and in each match he scored some runs and the total amount of runs which he scored is say 750 right. So this sum which is there that is addition of each match that we already have 750 and total number of matches we already have 20. So what would be the average of the batsman? What you'll do? 750 divided by 20 right? So average would be 750 divided by 2075 by 2 that is 3157.5 37.5 runs per match that becomes the average of the batsman cricket cricket innings or anything these are also favorite types of sums which are asked in averages right so just remember average is equitable distribution and average is nothing but sum divided by total this is very easy formula We'll see how to apply it in all the sums and once you go through this video entirely you'll be very much confident regarding averages sums and you'll be able to solve all the types of average sums. Okay. Now after this what you should remember is one very very small concept and that is nothing but total sum or sum. What it is we'll see. Okay. Now we have seen what exactly average is. Average is nothing but equitable distribution. Also we saw the formula for average that is nothing but the sum of all observations divided by total number of observations. Then what is total sum? Total sum is nothing but this sum can be written as average into total number of observations. Right? We just take it over here and multiply. This is nothing but total sum. See how easy it is. This concept will help you solve almost 80% of averages sums because most of the sums that are asked nowadays in exams are easily solvable through this total sum concept. We'll see some examples over here right now that would help you clarify this concept of total sum and how to use it and what exactly it means. Now over here let us say there are five people 1 2 3 4 and five. Let us say one has rupees 150 2 has rupees 100 three has rupees 50 four has rupees 200 and fifth one has say rups 1,000. Okay. So how much total money or does all these do these all these five people have? Okay. That would be 150 + 100 200 + 50 uh sorry 150 + 100 is 1 250 + 50 is 300 + 200 is 500 plus,000 that is nothing but 1,500 rupees let's see 150 200 400 500 yes 1,500 rupees okay total amount of money with these five people okay together is rups 1,500 this is nothing but the total Total sum say if they have given ages of say three people 1 2 and three. If they have given ages like 15 25 and say 40. Okay. Then what is the total ages or total sum of ages of all the three people? 15 + 25 is 40. 40 + 40 is 80. 80 years. Right? Say they have given say if they say that a batsman is playing for say seven innings. Okay. 1 2 3 4 5 6 7. If a batsman is playing for seven innings, okay, batsman plays for seven innings and each inning he scores like one run, 10 runs, 15 runs, two uh 12 runs, two runs, three runs and five runs. So what is the total amount of runs the batsman has scored? 10 + 11 10 + 1 is 11 + 15 26 38 40 43 48 runs. So this is the total sum of runs. So you understood what exactly the total sum concept is. It is nothing but the addition. Now addition right that is nothing but the sum. Now sum not every sum in not in every sum will directly have all these values and we just add them. Okay, here we added them because we had those values but not in every sum we'll have that. So whenever such sums come where there is where such values are not given there. How to find total sum very easily? By this formula or the formula of average both these formulas are same. We need total sum in such sums where all these individual values are not given. Simply take the average multiply it by total number of people or the total number of innings or total number of members family members something like that and you'll get the total sum of ages or total sum of in runs or total sum of uh say monies or the amount which is there right that so easily very easily we can get the total sum. Now why do we do that average into total? Because we have learned that equitable distribution means average. Average means equitable distribution. So total number of members if you multiply by average that means that each member gets this much amount. For example here it is 15 right 1500 is there. Total number of members is how much? Five. So what is the average amount of money with each member? 300 rupees right? So if there is 150 150 200 and 1,000 and say if there are these five people if there are other five people 1 2 3 4 and five and each one is given 300 rupees still the total sum would come out how much 300 + 300 + 300 5 that would be 1,500 only see so average which is there if it is average amount of money if it is equitably distributed 1 2 3 4 5 each one getting average amount of money 300 300 300 we get the proper sum right this is same as this this is same as this right so you need to remember that whenever you want to find total sum you just have to take the average multiply it with the total number of members you this is not a separate formula this is the same as the formula of average you are just taking this on the left hand side and multiplying there's nothing different you just have to remember one formula for averages there's There is no two formulas. There's only one formula for average. And only remember the concept that only remember two concepts that first one is average is nothing but sum upon total number of sum of observations upon total number of observations and other concept is generally try to find total sum. Now let us see when to find total sum and when to use the formula directly. Okay. Now moving on to the first question. Knowing that Vijay's expenditure for the first 3 days is rupees 100, rupees 125, rupees 85. What is his fourth day expenditure as his 4 days average expenditure is rupees 90. Now we know average is nothing but sum upon total. Right? This is the formula. Now what is exactly total? Total is nothing but total number of observations. Here they have given that four days average we need they have given four days average. Okay. And first three days expenditure they have given and we have to find the fourth day expenditure. So total number of days are nothing but four right. So average would be sum upon total number of days that is four. Now let us see what exactly would be the sum. Now the first day, second day, third day and fourth day. First day Vijay spends 100 rupees. Second day Vijay spends 125 rupees. Third day Vijay spends 85 rupees. Fourth day we do not know. So what is the total? What is the sum of all these? Plus plus this is 225 + 230 + 80 that would be 310 plus question mark. This is the sum right. So let's have 310 plus question mark. Now what they've given is that his 4 days average expenditure is 90. So they've given the average. So 90 is 310 + question mark divided by 4. Right? So what will we get? 4 into 90 is 360 = 310 plus question mark. Question mark equal to 50 rupees. So on the fourth day, okay, question mark is nothing but this fourth day. On the fourth day, Vijay spends rupees 50 only. See how easily with the formula and basic concept of addition, I'm getting the sum. We solve the we solve the numerical. Okay, moving on to next question. Question number two. What will be average price of all the goods bought if AJ buys 30 erasers for rupees 3 each, 35 chocolates for rupees 10 each and 25 clips at the rate of rupees 4 each. Okay. Now over here again what we see this is very easy. Average is nothing but sum upon total. Now what total number of observations? What is this total number? Total number is nothing but the amount of goods bought. What did he buy? He bought 30 erasers plus 35 chocolates plus 25 clips. So what is what is the total? 65 + 570 20 90 articles 90 goods were bought by him. What do we have to find? We have to find the average price, right? That is nothing but the cost. So we have to find average price which is there. We have one element that is t. Now what we need to find is the sum of everything. Now what is sum? Since this is the number of goods, this has to be price. Okay, the cost. This has to be the cost. Then only we'll get cost over here. Average price over here. Now what AJ did? How much total money did AJ spend? How to find it? Very easy. AJ buys 30 erasers for rupees 3 each. That means one eraser. one eraser. The cost of one eraser is rupees three. So 30 erasers how much would be the cost? Simply 30 into 3 is nothing but rupees 90. Right? Now again one chocolate cost is rupees 10. So 35 chocolates the cost would be rupees 350. Again over here one clip the cost is rupees four. 25 clips cost is rupees 25 into 4 is 100. So how much is the total amount spent by AJ? How much did AJ spend? What is the sum of the amount which AJ spent? 90 + 350 right plus 100 that is 350 + 100 is 450 + 90 is 540 rupees. So what would be average sum divided by total right that gives us six rupees. So average price of all the goods bought is six rupees. Understood? See here we all whenever we use formula just find two elements the sum and total number of goods or total number of family members. You will never have the amount or the total price in the denominator as total number of observations. that would always be in the numerator. Again, the runs scored by a batsman, that would be in the numerator. But the innings played, matches played, that would be in the denominator. Always remember that we just need to find the sum and the total number of observations. Right. Moving on. Question number three. Of the 20 cycles sold by AJ, average cost of 12 cycles is rupees 18,000. In total he earned rupees three lakhs. What was the average cost of remaining cycles? Now over here we do not have we know this is an average sum which is average is nothing but sum upon total. Now we know total that total a sold 20 cycles right. So we know total as 20. Now we but what the problem here is we do not know the individual values of each cycle. For what price did AJ sold? AJ sell each of the cycle. So finding total sum over here becomes little bit difficult. But if you look carefully, we can simply use another way to find the total sum which we have learned at the start. That is nothing but sum is nothing but average into total. Right? Now do we know average for 20 cycles? No, we do not know. But what do we know? Average cost of 12 cycles is rupees 18,000. So let us calculate over here for 12 cycles. Okay, this is for 12 cycles. What is the average cost for 12 cycles? 18,000. What does that mean? That means that AJ sold 12 cycles, right? And he collected some money. Okay, let the amount be XY Z. Some money he collected. Okay, rupees XY Z. He collected some money. And if he saw that whatever the money is there average amount of average money collected for each cycle is nothing but rups 18,000. That means each cycle if sold for 18,000 he'll get this much money. So he sort of sold every cycle for 18,000 rupees. These 12 cycles he sold for 18,000 rupees. So how much total money did he earn by these 12 cycles? Very easy. You know the average 18,000 over here. uh let's write it over here as a separate calculation. Okay, sum is average. Average is nothing but 18,000 multiplied by total. Total cycles are 12. So how much you'll get? You'll get over here 216 2A 16,000 rupees. So this is the amount of money he earned by selling 12 cycles. Right? So this is the total sum for 12 cycles. How much did he totally earn? He earned 3 lakh rupees totally. Okay, this is the total of 20 cycles, right? 12 cycles he for 12 cycles we'll write over here. For 12 cycles, he earned 2 lak 16,000 rupees. For 20 cycles, he earned 3 lak rupees. So you should see that remaining eight cycles which are there. Okay, for how much did they sell? Okay. How much money was collected by selling those remaining eight cycles? 3 lakh minus 2 lakh 16, right? 3 lakh minus 2 lakh 16,000 that is nothing but 84,000. So by selling these remaining eight cycles, AJ collected 84,000 rupees. So this is nothing but the sum for eight cycles. This is nothing but total number of cycles. Right? So what is the average for the remaining cycles? That is average for the eight cycles is sum of eight cycles upon total that is 84,000 upon number of cycles 8 that is nothing but 8 1 8 0 5 0 rupees 10,500 is the average cost of remaining cycles. Look, we'll go through this once again. This is very very easy. You might feel that this is confusing or something like that but it is not very easy. Take a look at this because once you understand this concept over here you can use it very easily in the other sums because this is very useful. Okay. Now what happens is that AJ has 20 cycles out of that he first sells 12 cycles and he gets some money and when he calculates he realizes that these 12 cycles whatever he has earned from these 12 cycles on an average for these 12 cycles for each cycle he has received 18,000 rupees on an average he might have sold one cycle for say 20,000 other cycle for say 11,000 only but when you calculate the average that average comes out to be 18,000 for a for these 12 cycles only. Okay. Then he say what does AJ do? AJ also sells the remaining eight cycles and in total AJ earns three lak rupees. Now we know average is sum total sum upon total number of cycles. We know total number of cycles as 20. But we do not know the average amount for each cycle or what is the average price for each cycle. We do not know that. So let's see what do we know. We know that for 12 cycles the average price is 18,000 right. So how much total money did AJ earn through these 12 cycles? Very easily we can count calculate that is total sum is nothing but average into total number of cycles. Average is 18,000. Total number of cycles is 12. So AJ earns 26,000 from 12 cycles. Okay. From 12 cycles AJ earns 2A6,000. But we know that when AJ sells 20 cycles he earns three lak rupees. So for 12 cycles AJ earns 2A6,000. For 20 cycles AJ earns three lakh rupees. So remaining eight cycles which are there to make 20 12 2A 16. So remaining eight cycles which are there how much did AJ in earn in that 3 lakh minus 2 lakh 16,000 that is nothing but 84,000 right? So 84,000 earned by AJ for these eight cycles. So total sum for eight cycles is 84,000. Total number of cycles is 8. So what is the average of the remaining eight cycles? Sum upon total 84,000 upon 8 that is nothing about 10,500. Right? So moving on to next sum. Question number four. Without considering the salary of the boss, the average salary reduces by rupees 1,000. What will be salary of the boss if average salary of 11 employees and their boss is rupees 18,000? Again over here we do not know individual salary. So it is much better to find the total sum. How to find the total sum? We know a formula for average is sum upon total. So total sum is nothing but average into total. Now we know how many how many people are there. What is what they have given in over there in the second sentence is that average salary of 11 employees and the boss is rupes 18,000. So 18,000 is the average. Now how many people are there? 11 employees plus their boss. So that would be 12 people. So average for 12 people is 18,000. So what is the total sum of salary or the total sum of money right for 12 people? 18,000 into 12. So what that what would that be? 1 lak 80,000 is 10 + 36 2 16 okay 2 lak 16,000 rupees is the total salary or the total sum for how many people this is for 12 people right now what they say in the first sentence without considering the salary of the boss. So if you exclude the salary of the boss average salary reduces by rupees 1,000. Average salary for 12 people that is 11 people plus the boss is 18,000. If you do not consider the boss, number of people would be only 11, right? 12 - 1. Boss is 1. So 12 - 1 is 11. Salary average salary reduces by,000. So 18,000 minus 1,000 that is nothing but 17,000. So average salary for 11 people is 17,000. Right? For the 11 employees it is 17,000. So what is the total amount of money or for these 11 people? Okay, total amount of salary. Okay, total sum of salary for these 11 people would be average into total total people are 11 because we are not considering the boss over here. So we have 11 plus the average also has decreased as 1,000 over here. Please remember right. So what we will get 11 7 are 77 11 1 + 7 is 1 lakh 87,000 right this is the salary of total sum of salary of 11 employees because we are not considering the boss over here and this is the total salary total sum of salary of 11 employees plus boss right see we simply have to subtract this from this to get the boss salary. Okay. 11 people plus boss minus the 11 people will give us the boss salary. So 2 lakh 16,000 minus 1 lak 87,000 would give us the salary of the boss. This is 16 by 7 9. Okay. 0 10 - 8 is 2. 29,000 is the salary of the boss. Right? See how total sum concept which is there, it was used to solve this numerical which had nothing in it. Only average and total number of observations was given. We used the total sum concept and solved the sum. Moving on, question number five. Average age of five people is 42 years. Another group has eight people who have average age of 81 years. When both groups are mixed, what is average age of all people? Again, another sum related to total sum. Okay, another numerical which is related to total total sum. We know average is nothing but sum upon total number of observations. Right? Over here do we know the total number of observations? They have as they have asked average age of all people. All people means both the groups are added. First group had five people, second group have eight people. So total number of people are 13. These are 13 people. This is nothing but t. We have the value of t. Do we have the value of sum of all the ages of all the 13 people? No. So let's see what exactly we have. Okay, we know average is sum upon total. So total sum is nothing but average into total number of observations. Right? Now what they given is that average age of five people is 42 years. So average age for five people is 42 years. That is nothing but sum of ages of all the five people divided by total number of people. We know the average as 42. Sum of ages of five people we do not know. And total number of people are five. Okay. Sum of the ages of these five people is 42 into 5 is 210 years. Similarly average age for eight people they have given some of the ages of these eight people divided by total number of people average age is given as 81 years. Sum of the ages we do not know. Total number of people are eight. So sum of the ages of all these eight people is 81 into 8 648 years. Right? So now we have some of the ages of five people and some of the ages of 18 people. We have 13 people over here. 13 is nothing but five a group of five people plus group of eight people. So 5 + 8 is 13. So if we want sum of the ages of 13 people that is same as sum of the ages of five people plus sum of the ages of eight people right. So that is nothing but 210 + 648 that would give us 858 years. So this is our value for s. We have the value for t. We have the value for s. Let's find out the average. Average would be 858 divided by 13. That is the total number of people. That is 13 into 6 is 78. Right? Again 7 78 66 years. So average age of all the people when both the groups are mixed is 66 years. Right? Moving on. Question number six. Three boxes have some average weight. When one box which weighs 89 kg is replaced by another box, the average weight increases by 5 kg. How much the new box weighs. Now this might look like a very uh different or a difficult sum but actually it is not. A very small and easy concept goes into solving such kind of sums. Pay attention to the concept because this is very important uh and as well as easy and this concept will help you solve almost 50 to 60% of average sums because they are asked in of because 50 to 60% average sums are asked by of this format. Okay. Now let's take a small example okay to explain what the sum is saying. Let us assume that there are three boxes 1 2 and three. And let us assume that the average weight of these three boxes is w kgs. So what does this mean? If the average weight of three boxes is w kgs, that means each box weight must be w and w, right? W kg, w kg, w kg. Now let us assume there are other three boxes 1 2 and three. And average weight of these three boxes is W + 5 kg or what we can say it is five more than the previous weight. So 5 kg more than the previous weight. That is nothing but W + 5 kg. Right? So what will be the individual weights of the boxes? W + 5 kg W + 5 kg and W + 5 kg. So what does this mean? If the average increases, average increases by five kgs then individual weights also increase by 5 kgs. Did you see over here it was www. When average became W + 5 kg, it became W + Y, W + Y and W + Y. Right? Now if initially this condition is there and finally this condition is there, then what does that mean? Say let us assume that this is box number three and we are replacing it with a new box. This box right this box new box a new box. We are replacing it with a new box. So what should be the weight of this new box? Let us see what should be the new weight of this new box. Now in this new box makes the average W + 5. When the average becomes W + 5, there is a increase of + 5 + 5 and + 5. Right? So initially these two box now in the final stage also these two boxes are the same. So their weights has to remain same. So this remains same. This remains same. Right? Same same. What changes? So this increase which is there + 5 + 5. Who will compensate for that increase? This is nothing but the new box. The new box that means should have this increase of + 5 for the first box. this increase of + five for the second blocks plus this increase of plus five for itself and plus the original weight. Now since we are replacing three okay box number three which has an weight of W with another box that is new box we do not know its weight. What does it should have? It should have this weight W plus it should have this increase in weight plus it should have this increase in weight right? All these three things should be compensated by this new box. So the new box. So over here in this what we have this third box has a weight of 89 kgs right they have given this is not the average please remember this is the individual weight we are removing this box. Okay we are removing this box we are putting in a new box and this new box will have what it does it do? It increases the average. Since it is increasing the average that means it is adding weight. So the new box weight has to be greater than old box weight. So what does the weight what should be the weight of new box? It should be 89 that is original box weight plus it should compensate for this increase of five right because average increases by five for the first box increase of five for the second box increase of five for itself also because when average increases by five five is added to each of the individual weights. So what would it be? This is nothing but 103 kgs. Okay it is very easy. You might think that this is difficult and very complicated but it is not. Again I'll explain it to you. Now let us assume there are three boxes. Okay. Box one, box two, box three. Now average weight of these three boxes is W kgs. So individual boxes will weigh W kg, W kg and W kg. Right? Now suppose take another three boxes 1 2 and 3. And if their weight is w + 5 kgs. Okay, average weight is W + 5 kg. What would be the individual weights W + 5 kg, W + 5 kg and W + 5 kg. Now let's come to our problem. Let us say that because of replacing one box, the average increases by 5. That means if average initially is W kg, it becomes W + 5 kg, right? It becomes W + 5 kg like this. Now since it becomes W + 5 kg, what would be the individual weights become? They would be confirmed. One will go from W to W + 5. Two will go from W to W + 5. And three will also go from W to W + 5. Okay. So there is increase of 55 kg in individual weights also. Remember once average increases individual weights also increase. But we know that two boxes are the same. Only the third boxes change. That means weight of the two boxes has to be the same. That is W and W. Right? W and W. So this extra weight of 5 kg and 5 kg from where should we compensate it or from where should we get that extra weight? They should get the extra weight from the new box. Right? So new box should have the extra weight of 5 kg for box number one. Extra weight of 5 kg for box number two. And also the extra weight of 5 kg for box number three that is itself. And what is the original weight of that box which was replaced? 89. Even that should be compensated because we are removing we are replacing. So 89 should be there. So when we add this we get 100 kg. So now when the new box is 103 kg then only the average increases by 5 kgs. Right? You can even calculate and see it. Now if they say that the average weight decreases by 5 kg then here instead of w + 5 it would be w - 5. And each of the boxes will have weight w - 5 w - 5 and w - y. And the third box which is getting replaced say if it is of 89 kg then what would be the weight of the new box? It would be 89 - 5 - 5 and - 5. Right? Now but here it is increasing. So we need not consider this. This is only for information or if some is asked regarding the weight decrease. Right? Now over here what we have written? We have written + 5 + 5 + 5. We can write it like this also. 5 kg for one box. For 3 kg, how much? Five. For three boxes, 5 kg for increase for one box. So for three boxes, how much increase is there? Five 5 kg into three boxes that is 15, right? 15 kg increase should be there. So the weight of new box should be 89 + 15 that is nothing but 103 kgs. Both are same, right? But remember how we expressed it because in some numericals you might have to calculate using this way. In some we have to calculate using this way. Right? Now moving on to next sum. Question number seven. How old will Raju be if the ratio of his age and one of his twin grandsons is 11 is to2 and average age of his and his both the grandsons is 50 years. Again this is a sum of total sum of finding total sum. What we know average is nothing but sum upon total right total number of members or total number of people. So what is do you know total number of people? Yes that is three because Raju and both his grandsons we know what is the average age of Raju and both his grandsons since they are twins. Okay that is nothing but 50. So what we have 50 equal to sum of the ages of all the three people Raju and both his grandsons divided by total number of people that is three. So sum of the ages of three people is 150 years. Right? Now what they saying is that ratio of Raju's age to one of his twin grandsons. Now both the grandsons which are they are twins actually. So both will be having same age. So ratio of one of the twins twin grandsons is say let us say G1 one grandson okay is nothing but 11 is to2 G1 is one of the twin grandsons and G2 would be the other grandson. Ratio of age of Raju to one of the grandson is 11 is to2. Let K be the common factor. So age of Raju would be 11K. Age of grandson would be 2K. Since they are twins, it would be same as G2 that is the second grandson would also be having the same age 2K. What is the total of their ages? 11K + 2K + 2K would be 15K. But we already know the sum of the ages of Raju and his grandsons is 150 years. We have calculated it over here. Okay. Equate this. What we get? 15K equal to 150 years. So K would be 10 years right? Since K is 10 years, what is the age of Raju? That is 11K. That is 11 into 10 110 years. See how easily using the concept of total sum, we calculated the answer. Moving to the next question. Question number eight. Had AJ scored 18 runs more in his third innings and four runs more in his seventh innings, his average would have become 66 66 runs but it is 64 runs. How many innings did he play? Now we do not know the number of innings over here. What we know is average is nothing but sum sum of all observations upon total. Now this is not a tricky sum or a complicated sum. We have to use the same concepts which we have done earlier and using those only we'll solve this especially the sum related to increase in weight of three boxes. Right? Let's see how to deal with such kind of sums. Now average is sum of all observations upon total of all observations or total number of observations. So sum of all observations would be nothing but average into total number of observations. So sum of all or sum of runs scored in all innings that is total runs would be same would be equal to average number of runs per innings multiplied by total number of innings right so let us say n is the total number of innings a played now let us what is the average for any innings 64 runs they have given now when he scores 18 more runs in third inning and four more runs in seventh inning then only his average becomes comes 66. That is a increase of two, right? So when he scores 18 + 4 is equal to 22 more runs. Okay, 22 more runs or 22 runs more then only the average increases. So increase in runs is 22 runs. But we also know that when average increases by two for each inning there is the addition of two runs in each inning. That is each inning runs or each inning score increases by two. We saw that if average weight of three boxes is W and on changing one box we get average weight as five more that is W by five. That means for each individual box the weight increases by five. Same way over here for each individual inning the runs increase or the score increases by 2 + 2 + 2 + 2. So how much is the total increase? We know total number of innings as n. For each inning the increase is two. So n into 2 is nothing but 2 n runs right. So 2 n runs is the increase in the total number of runs scored. But we already know that AJ scores 22 runs more then only this pos this is this becomes possible that is the average increases by two. That means these 22 runs is nothing but these two n runs both are the increase. So 2 n is 22. So n equal to 11. n is nothing but the number of innings. So the answer is 11. How many innings did AJ play? AJ played 11 innings. Moving on. Question number nine. In a group of people, the oldest and the youngest have an age difference of 100 years. If these two are left out of counting, then the average age of the remaining 40 people is 28. The average age of entire group being 30. How old is the eldest person? Now here we know average is sum upon total and total sum is nothing but average into total right. So sum of ages of all the people is nothing but average age of the group multiplied by total number of people in the group. Now what we know over here let us that old age difference between oldest and youngest is 100 years. So let us assume that the youngest is n years old. So how much would be the oldest 100 plus n years right? Then only the difference between both of them would be 100 years. Right? So this is the youngest, this is the oldest. They have given that if we leave these two out, average age of remaining 40 people is 28. That means if these two are excluded, how many people are left? 40 people. That means total people in the group are 40 plus the youngest plus the oldest. That is nothing but 40 + 1 + 1 is equal to 42 people. So 42 people are in the group. What is the average age of this group of 42 people? 30. That is nothing but sum of the ages of these 42 people divided by total number of people. Okay. So 30 would be equal to do we know the sum of the ages of 42 people? No. How to calculate what is the total number of people? 42. So sum of the ages of 42 people is 42 into 30 that is 1 2 6 0 years. Now what they given if you exclude the youngest and the oldest average age of the remaining 40 people is 28. So average age of the remaining 40 people is 28. Right? That is nothing but sum of the ages of these 40 people divided by total number of people. So 28 would be do we know sum of the ages of 40 people? know sum of the ages of 40 people upon total number of people are 40 because we are excluding the youngest and the oldest right. So sum of the ages of 40 people is 40 into 28 that is 4 4 8 are 32 4 2 8 + 3 11 1 2 0 years right? So sum of the ages of 40 people is 1120 years. Now, sum of the ages of 42 people is nothing but age of youngest plus age of oldest plus sum of the ages of 40 people. Right? So, what is the sum of the ages of 40 people? This we already know what is the age of youngest? N age of oldest 100 + n plus what is the sum of the ages of 40 people? 1120 and all this is equal to this 1 1260. Right? Sum of the ages of 42 people because this is 40 people. This is the 41st person and this is the 42nd person. Right? So 1 2 6 0. So what we will have over here? Let's calculate 2 n plus this is 1 220. Right? It goes over here. 1 2 6 that would be equal to 1 2 6 0 minus 1 220 would be 40. So I'll write over here 2 n is equal to 40 n = 20. What is the age of oldest person? 100 + n. Oldest person age is 100 + n that is 100 + 20 equal to 120 years. See how easily we have calculated the age of the oldest person. We used only the simple formula of average and total sum. These two concepts help you solve everything. Moving on to next question. Question number 10. A batsman played 11 innings and has a certain average. This average increases by two runs when his three innings of 32 runs, 33 runs and 34 runs are replaced by three other innings. Find the average of these three new innings. Now again this is easy one. We have our average formula that is sum upon total. Sum of all the observations is nothing but average into total. So sum of the total number of runs or the sum of runs scored in all the innings is nothing but average number of runs in one inning into total number of innings. Right? The batsman played 11 innings. Okay. Now in these 11 innings he has some average. When if three of his innings of are replaced by three new innings the average increases by how much? It increases by two runs since it is incre since average is increasing by two runs. Each individual inning must increase by two runs. So how much is the total increase? That is nothing but two into total number of innings is nothing but 11 that is nothing but 22 runs. Right? So 22 runs is the increase in the number of runs. This is nothing but the increase. Now what they say is that the three innings old innings are replaced by three new innings. Now whatever the new innings are there what must they cause the average to increase. Since they are causing the average to increase what the must they comp compensate for? They must compensate for old innings plus they must compensate for the increase in other innings also. Increase of two runs in other innings also like we have seen in the weights and boxes. Right? So how much did how much is the old innings score? 32 + 33 + 34 plus how much is the increase in every inning because of this? That is 22 runs. So how much will you get over here? That is 30 30 is 90 + 20 is 110 112 16 19 21 runs. So the new three innings which are there new three innings which are there in total they must provide 121 runs. So that is nothing but sum of three new innings. This is sum of three old innings. This is increase in the number of runs in all the 11 innings. This is sum of three new innings. What is the average of three new innings? Now sum of three new innings upon total number of innings. Sum of three new innings is 121 divided by 3. That would give us 41 by3 runs is nothing but the average of the three new innings. See how easily using only one formula and one small concept you can very easily solve almost all sums of averages. Right? With this we come to the end of this video. If you like this video, please give it a like and share it with your friends. You can leave your comments and suggestions for us in the comment box below. You can even tell us about any specific videos that you would like us to develop for you. We would be rolling out more such videos and tutorials. So subscribe to our channel and stay updated.