Weighted Averages: Favourite GRE/GMAT Trick?

I've been saving one of my favorite

trick

s so far and it has to do with

weighted

averages

. I call it the middle number

trick

and it really is one of my favorites. When students see it, they often feel overwhelmed, happy and grateful and say Philip, how did you not tell me this before? Is incredible. In fact, I learned it from a student. I used to teach things algebraically, then I learned this method and I think it's much better. I've been teaching it for years. Find this trick useful, leave a like and a comment and watch until the end because there are multiple facets to this

weighted

average or mean number trick.

This whole trick depends on you detecting a number midway between two other numbers or so to speak. In more elegant words, detecting a weighted average between two other terms, that is the key you must keep in mind to use this trick. Here's a classic question and we'll discuss why this is an average number or a weighted average trip and then I'll go. to show you the trick and you will be amazed a portrait museum has a collection made up only of paintings and sketches 30% of the paintings are on display while 15% of the sketches are on display if 26% of the collection of the museum is on display, what is the proportion between the collection of paintings and sketches?

Can you identify the middle number? Yes, the weighted average is 26 percent. Okay, so we have 30 percent of the paintings on display on one side and 15 percent of the sketches on display on the side. other side and an overall weighted average of 26 percent of the total collection on display, so this is the perfect question to use the next trick and it starts with a number line where we simply draw the number on the left and the number on the right. and the middle number on a number line at this point you might say to me oh I have an algebraic method and I know that method and I used to teach that way but I think this way is superior other of you would say oh well I can In a way I say you should there will be more paintings because that's what the weighted average of the middle number is closest to and you're right, but this method will give us an accurate ratio between paintings and sketches.

What is the first step in the trick? find the distance between the number on the left and the number in the middle and also the number in the middle and the number on the right and write those two distances at the top of the number line, this will be four to the right, you don't have to write four percent only four is fine and eleven to the left those are the two distances the distance between 15 and 26 is 11 and between 26 and 30 is 4 now this is the first key moment we have to reverse those distances to the other side and let me try to give you a little clue as to why we do this.

You don't actually need to know why we do it, but I thought I would give you an idea of ​​why we do it just for your satisfaction since the distance is smaller. The right indicates that there are more than 30% because the average is skewed towards 30%, so it wouldn't make sense, but that smaller distance remains to the right because the larger proportion should be to the right. 30% is much more common than 15% in fact it is inversely proportional the smaller the distance to the right the more than 30% we have and that is why that greater distance that is to the left really belongs to the right the 11 that is a huge distance indicates that we have a A much smaller amount than 15% is inversely proportional and inversely means that it belongs to the other side of the right side, so the small distance travels to the left, 11 the large distance travels to the right and now we have a ratio. that makes sense, the much larger number belongs to the 30% because the average is much closer to 30%.

In summary, we now have a relationship between the number of sketches, which is 4, and the number of paintings, which is 11, there are many more paintings than sketches. so note that the question is the ratio of paintings to sketches, so it is 11 - 4, not 4 to 11, that would be the ratio of sketches to paintings. The left side, 15% are sketches, the right side, 30% are paintings. Now I know. What you're thinking about is great for proportion questions, but what if they want a fraction? For example, time for my next amazing example. If you want, you can pause the video and try it yourself.

The trick is so effective that you might be able to do it. do it only with that first example only here we go a theater cells only matinee and evening tickets the matinee tickets are priced at five dollars and the evening to be surprised at nine dollars if the theater sold 120 tickets for an average price of seven dollars fifty how many of Were those tickets night tickets? Can you find the middle number? We have a matinee for five dollars, an evening for nine dollars and an average price of seven dollars. Fifty in the middle, so we put on the number line like before, we find the distance on the left and the distance on the right I didn't need the dollar sign because it's essentially a point five on the right is the distance between seven dollars fifty and Nine dollars and two point five is a distance to the left, what do we do? reverse those distances so that one point five goes left and two point five goes right quickly.

We don't like ratios to be decimal so multiplied by two to eliminate the decimals the ratio of 1.5 to 2.5 becomes three to five but essentially we have a reason, we don't have a fraction as I wrote under two from that we add the ratio to find a total three plus five is eight, this then becomes the denominator that three and five now become 3 over 8 and 5 over 8 in other words three eighths of the tickets were matinee five eighths of the tickets were boom night We just converted a proportion into a fraction How do we do it? Remember that we added the ratio 3 plus 5 to get 8. and then we made that denominator of both sides to be 3 over 8 and 5 over 8 3 eighths and five eighths finally we can answer the question if the theater sold 120 tickets and they want to know how many of those entries were night entries. the tickets for the night were 5/8 of the total, so we make 5/8 of 120, which is 75.

How did I calculate it? You make 120, divide it by 8, which you can use long division and that's 15 times 5 75, so 75 of the tickets. We left in the evening, I think 45 tickets for the matinee, but again we used this amazing middle number trick by swapping the distances creating a ratio and then finishing by adding the ratio to turn the ratio into a fraction. An incredible trick, one of my favorites that we are going to do. I finished with two more examples, the first is what to do if they ask for a percentage and the second is more difficult, well you should focus even more on the middle number because there will be multiple variables, both topics could easily appear on the GRE and GMAT if Do you want to pause the video and try it yourself, otherwise I'll review it.

The Stafford Charity are either volunteers or employees. Charity staff work on average 20 hours a week if volunteers work. for 10 hours a week and employees work four times as many hours as volunteers, what percentage of staff are volunteers? Can you identify the middle number this time? We haven't been explicitly given two numbers in a middle number, but we can calculate one quickly. volunteers work ten hours employees work four times as many hours as 40 hours and they say that on average the staff works 20 hours a week, so 20 is the middle number between 10 and 40 Bob's Runkle we create the diagram again on the left . label it as volunteers on the right, we can label it as employees, find the distances as 20 and 10, reverse those distances and we get the ratio of 20 to 10 from left to right, the left side is volunteers and the right side is employees. simplifying that ratio we get 2 to 1 to convert this to a percentage we do the same thing we did with fractions we convert the ratio into a fraction which is a great first step so 2 plus 1 is 3 so the ratio becomes 2/3 to 1 /3 now we would probably have labeled the left side as volunteers and the right side as employees, so the question is about the percentage of staff who volunteer that are on the left, that's 2/3 at this point simply we convert. the fraction in percentage by long division you may also know by heart that two thirds is 66.6%, that's good to memorize, just like 1/3 is 33.3% recurring 2/3 is 66.6% recurring, so 66.6% of the staff are volunteers, it's time for a final example and this one is much more difficult.

Focus on the average. What makes it more difficult, as you will see, is that they give us an average number, but it is not that simple. You can try to do it yourself or else I will explain. Class A is 30% water and 70% lemonade. Glass B is 50% cola and 50% lemonade if the two glasses are poured into a container and the resulting 500 milliliter mixture is 65% lemonade. How many milliliters of the mixture is Cola, the first challenge with this more difficult question, you need to spot the middle number, but this time you may be distracted by the different numbers there.

We have water, lemonade, Cola, which one, what ingredient did they give you? you an average number because they gave us an average a weighted average of lemonade 65% lemonade when they were mixed the average was 65% lemonade they didn't use that word average but they gave us the resulting proportion that was lemonade so draw the number line just for lemonade don't worry about the water don't worry about the cola for now this time we're going to label it correctly glass A is 70% lemonade glass B is 50% lemonade and the resulting mixture is 65% lemonade so that's it The first challenge overcome we focus only on the ingredient for which we were given an average most of the rest of the question is as you have seen before we find the distance to the right and to the left we turn create a proportion simplify the proportion add the proportion 3 plus 1 is 4 and creates 2 fractions 3/4 and 1/4, so we can say from that mixture that 3/4 was from glass A and 1/4 was from glass B, but then they try to mess with It comes to us head because they say how many milliliters of the mixture is Cola but Cola only appeared in glass B and this is the second reason why this question is more challenging How do we find the number of milliliters of the mixture that is Cola when glass B ? was the only one who had Cola first, let's only think in fractions, as we can see here on the right, 1/4 of the mixture is glass B, so of those 500 milliliters we know that 1/4 is glass B and 50%. of glass B is Cola, thinking of that as a fraction that is 1/2 half of glass B is Cola, so you could calculate the quantities one after another, as if you could calculate a quarter of 500 milliliters and that is 125 milliliters is the glass B. and then find half of what is Cola or we can simply multiply the fractions B is 1/4 of the mixture and B is 1/2 Cola 1/4 times 1/2 is 1/8, so 1 /8 of the total mixture is Cola, that's what I prefer to do because imagine they asked us for a general fraction, it's Cola instead of an actual number of milliliters, this would be the preferable method: you multiply the fraction of the total that is Class B with the fraction of glass B which is Cola because Cola only appears in glass being 1/4 times 1/2 is 1/8 since this time they asked us for a real amount all we have to do is find 1/8 of 500 milliliters which is 500 milliliters divided by 8 you can use long division for that and I think it would be 60 to 0.5, which has proven that even in the most difficult circumstances, if you can spot the middle number and follow the trick carefully , which works because of the inverse proportionality of distances to the proportion we can solve these questions in 30 seconds or less without a single moment of algebra.

Hope you like. I love this trick. It is one of my favorites. I have been teaching for many years. If you like it so much. Please leave a comment or, if any of the examples were a little confusing, watch them again. Thanks again for watching. See you in the next video.