Word Problems
(Venn Diagram)
In matriculation/O-Level mathematics, you’ve studied overlapping sets or Venn diagram. This topic also tested in exams like this. So let’s start with two variable Venn diagram.
Suppose,
In a class of 100 students, 70 students study Math and 80 students study English. How many students are studying both of the subjects if at least one student is studying either Math or English?
Solution:
As there are total 70 Math students and 80 English, but total students are 100; Also it says that at least one student is taking one course, which means no student is there who is neither taking Math nor taking English.
So, draw the venn diagram as follows:
In above diagram, also in every two variable venn diagram, use this equation to solve for the required thing:
Only Math + Only English + Both + Neither = Total
⇒ (70 – x) + (80 – x) + x + 0 = 100
⇒ 70 – x + 80 – x + x = 100
⇒ x = 50 Answer
Now let’s discuss some tough scenario in two variables venn diagram.
In a class of 60 students, if 40 Students play basket ball and 35 students play table tennis. What is the maximum number of students who play both of the games? Also What is the maximum number of students who play neither of the games.
After that, tell what is the minimum number of students who play both games. Also how many minimum possible number of students who play neither of the game?
Solution:
Here the information of ‘at least one student play either of the game’ is not mentioned, which means there may or may not be be few students who play neither of the game.
In such case, always start form maximum possible value of both. The maximum both students are almost always the small circle, i.e if small circle (i.e table tennis students) would come completely inside the larger circle (basket ball students) as below:
So, maximum value of both = 35 Answer
And remember, in situation that gives maximum value of both sets, the same situation gives maximum value of neither. Because the smaller set is completely merged inside the larger set, so the outer region of the sets become maximized that gives maximum value of neither set.
So, maximum value of neither set = 20 Answer
Important:
Now, in order to find the minimum values, you need to make the two circles as much far from each other as possible. But remember in order to find the minimum value, you must start with the neither rather than both region. And because the minimum value of neither is not given in question, so minimum value of neither is 0 (cannot be negative) in that case always. And after this we’ll find the minimum value of both by using the equation as discussed earlier in this lecture. So the venn diagram to find minimum both would be:
Only Basket ball students + only table tennis students + both + neither = Total
⇒ 40 – x + 35 – x + x + 0 = 60
⇒ x = 15 Answer
And minimum value of neither = 0 Answer
Minimum value of neither would always zero if it’s minimum limit is not mentioned in the question through words like ‘at least’.
Now, let’s learn some tricky concept with two variables.
In a class of 165 students, each student are taking GMAT preparation course or GRE preparation course. 80 students are taking GMAT course and 120 students are taking GRE course. If at least one student is taking either GMAT nor GRE course, how many maximum number of students who are taking neither GMAT nor GRE course. Also find how many minimum possible number of students who are taking neither GMAT nor GRE. Similarly, how many maximum as well as minimum number of students who are taking both of the courses?
Solution:
Draw the above information in venn diagram as below:
Now from the diagram above, as smaller circle is 80, that could have marge in larger circle of 120 students. So 80 would be maximum both exam preparing students.
Maximum possibility of both GMAT and GRE students = 80 Answer
Remember that when there are maximum both students, at that situation there would also be maximum possible neither students. Because there’s only one circle in the universal set. So, the outer region of the two circles would maximized when one circle fully marge to another one.
⇒ Maximum possible of neither GMAT nor GRE students = 165 – 120 = 45 Answer
Now, as I said when there is maximum both (i.e x), there would also be maximum neither in that situation. Similarly, when there’s minimum both, there would also be minimum neither. As the two circles would cover maximum possible area in universal set, leaving minimum outer area for neither (i.e n). Now minimum both can only find by using the basic equation that we’ve learned before, i.e:
⇒ Only GMAT + Only GRE + Both + Neither = Total
⇒ (80 – x) + (120 – x) + x + n = 165
⇒ 200 – x + n = 165
⇒ x = 35 + n
Now, minimum neither is always 0 unless stated in the question. So
Minimum Neither = 0 Answer
And hence,
Minimum both = 35 Answer
Let’s discuss three variable overlapping sets scenario that is usually tested in GMAT or GRE exams.
For instance, in a certain class, 80 students are studying Physics, 120 are studying Chemistry and 150 are studying Math. 20 students are studying both Physics and Chemistry, 80 students are studying both Chemistry and Math, and 30 students are studying both Math and Physics. Moreover, 15 students are studying all the three courses, how many students are in the class if each of the students is studying at least one of the courses?
Solution:
Consider the venn diagram below:
In figure above, you can see that how we find the areas that consists of students studying exactly two courses. Now, we can find the area that consists of students exactly one courses i.e, only Physics (P), only Chemistry (C) and only Math (M) as below:
(Only P) + (Only C) + (Only M) + (Only P & C) + (Only C & M) + (Only M & P) + (P, C & M) = Total Students
⇒ {80 – (15 + 15 + 5)} + {120 – (65 + 15 + 5)} + {150 – (65 + 15 + 15)} + 5 + 65 + 15 + 15 = Total Students
This equation is formed easily with the help of above venn diagram. This is basically the original equation that we’ve learned in two variable scenario. Here you have an advanced version of that equation. First only Physics students is obtained by subtracting the sum of region that do not fall in only Physics region from the total Physics circle. Similarly only Chemistry and only Math is obtained. After that only Physics & Chemistry is region that is not all, but only P and C (i.e 5), which is obtained by subtracting both P and C region (i.e 20) from all P, C and M (i.e 15). And other things is obtained in similar passion. If anybody still not understood this, he/she may ask me on whatsApp. So let’s solve above express to find total Studnts.
⇒ {80 – 35} + {120 – 85} + {150 – 95} + 70 + 30 = Total Students
⇒ Total Students = 45 + 35 + 55 + 100 = 235 Answer
Remember that the three variable scenario is very important especially for GMAT exam. It is most frequently test in GMAT now-a-days. So you have to be quick as well as accurate in calculating for the required answer.
Similarly, in three variable scenario, some further complex question can be formed, but trust me the method of solving would be same as discussed above. Some times, total students would be given and students who are taking exactly two courses are required to answer. In that case, you should be smart enough to assume (let’s say only P and C) as a, (only C and M) as b, and (only M and P) as c. And then remaining variable would be given or you may find. So the method would remain the same as discussed above, but difference would be that the assumed values of a, b and c would be cancelled out in some instances. While leaving a + b + c that would give the value of this on other side of the equation through applying the method discussed above. This type of question has came to me when I appeared last time in GMAT. It’s definitely of harder difficulty level. Remember that when you’ll see a harder difficulty level question in GMAT, it’s not a thing to worry, but to more confident and happy. AS you are on absolutely right track. 🙂
You’ll see such complex scenario type of questions in practice questions and quizzes. I’m repeating you again, don’t ignore accuracy while attempting any question with your best speed.
Let’s discuss mixed matrix scenario with four variables involved:
Remember that in four variable scenarios, almost always the two horizontal variables not overlap, and similarly, the two vertical variable wouldn’t overlap. For instance the two variables Males and Females cannot overlap. Also MBA qualified students and not qualified students cannot overlap. For instance,
60% of Male employees and 70% of female employees are juniors. If 20% of the employees are female, what percent of senior are junior females?
Solution:
In first step, you must draw as below:
And then,
That’s it.
Now, among variables, assign each circle to consider as variable: Male (M), Female (F), Senior (S), and Junior (J).
It’s given that 20% of the employees are female, therefore 80% of employees must be male. So suppose there are 100 employees. (As we need to answer in percentage, so we don’t need to assume 100x).
Now, as 60% of male employees are juniors, so 40% must be seniors. Also 70% of females are juniors, so 30% are senior females. By applying this to our assumed value, we’ll get senior employees to be the sum of senior males and senior females. Similar in case of junior employees we’ll find as below:
Seniors = 32 + 6 = 38
So Juniors = 100 – 38 = 62
In diagram above, note that the shaded regions consist of no employee, so it’s zero. In short,
Total employees = a + b + c + d
Now you can clearly answer i.e
Junior females as percent of senior employees = {d⁄(a + b)} × 100 = {14⁄38} × 100 ≈ 37% Answer
If a, b, c, and d are not given, and only seniors and juniors are given you would again use this diagram to answer question by finding a, b, c and d. Finally , let’s do this type in four variable scenario:
At a certain university, 60% of the professors are women, and 70% of the professors are tenured. If 90% of the professors are women plus tenured then what percent of the men are tenured?
Solution:
As 60% of professors are women, so 40% must be men. Also it’s given that 70% of professors are tenured, so 30% must be un-tenured.
Again here, suppose total professors are 100.
Now, it’s given that 90% of the professors are women plus tenure, and
Women + Tenured = a + b + d
Remember that women plus tenured doesn’t mean both women and tenured (i.e b only). As you know the word ‘or’ is translated to + sign in math, and the word ‘and’ is translated to × sign. So women plus tenured means either women only, tenured only or both women and tenured. Understanding the word statement is the key to answer word problem questions. Word problem covers 70% of quantitative section. So it’s importance is shown by its weightage.
So, clearly, a + b + d = 90
⇒ c = 10 (As a + b + c + d = 100)
Now, things are quite easy for you. As a + c = 40 ⇒ a = 30
Also, a + b = 70 ⇒ b = 40
And, b + d = 60 ⇒ d = 20
Now, we are required to find percent of men are tenured. Again this is tricky. Here percent of men means (a + c) will come on denominator. But who are tenured doesn’t mean total tenured (i.e 70), but rather only tenured men is asked (i.e a).
⇒ Percent of men are tenured = {a⁄(a + c)} × 100 = (30⁄40) × 100 = 75% Answer