Word Problems
(Fractions & Percents)
Let’s take another scenario where percentage change formula is used,
If x is 60% greater than y, then y is what percent less than x?
Solution:
As x is 60% greater than y
⇒ x = y (1 + 0.60) = 1.6y
As we know,
As question ask y is what percent less than x so x would be reference, because y is what percent less than x means y is how much less from (i.e start from) x. So whenever less than or greater than words come again see what comes after this, it would be the reference.
⇒ Percentage Change = {(x – y)⁄x} × 100
By putting x = 1.6y
⇒ = {(1.6y – y)⁄1.6y} × 100
⇒ = {0.6y⁄1.6y} × 100
⇒ = {0.6⁄1.6} × 100
⇒ = 37.5% Answer
Alternate Method:
Suppose y = 100 (as question is ask about percent not value, so we shouldn’t suppose 100x)
Many people may ask, why we suppose y = 100 rather than x; well because if we’ll suppose x to be 100, then as it would be hard to get y from this. As x is 60% greater than y, it doesn’t mean y must be 60% less than x. So y wouldn’t be 40. It’s complicated, as in this case, equation would become x = 1.6y, so 100 = 1.6y and therefore y = 100⁄1.6 = 62.5.
Very Important Concept & Point: 60% greater than 100 is 160 (i.e difference makes 60, an easier one to find) but 60% greater than 40 is not 100 (i.e difference would no more 60, as here’s percent greater than which gives value of reference ’40’ whose 60 percent is not 60. So it is suggested to suppose reference 100 so you may find your ways easy to calculate things in absence of calculator.
Now as we assumed y to be 100, so
⇒ x = 100 (1 + 0.6) = 160
Now we know that,
As question ask y is what percent less than x so x would be reference, because y is what percent less than x means y is how much less from (i.e start from) x. So whenever less than or greater than words come again see what comes after this, it would be the reference.
Many students would definitely confuse over the reference on this question. Well you should confuse at this stage but never later after getting clear after this. Yes, I told you to take reference y to be 100, but please look at the complete question, that has two parts. First part: if x is 60% greater than y. Here, reference is y that comes after ‘greater than’ so we assume y = 100 as it was reference. Second part: y is what percent less than x. Here, reference is x that comes after ‘less than’ so we take x to the denominator (place of reference) in answering this question. Note that first part is the information, while second part is the question that need to be answered, so in solving the question, we will take reference of the second part where question is asked. But while assuming 100 as reference, we assume this on basis of the information given to us, therefore we’ll take reference of the first part while assuming to assign 100 value to reference. I hope you got clear. If still not, you may ask me by clicking ‘leave a message’. I would automatically know that you are in this specific page, just ask the question, you’ll receive further detailed answer.
As x is the reference in question part, so percent change would be:
⇒ = {(x – y)⁄x} × 100
By putting x = 1.6y
⇒ = {(160 – 100)⁄160} × 100
⇒ = {60⁄160} × 100
⇒ = 37.5% Answer
Let’s take another type and tricky scenario of percents,
In a country M, the capital city is C. If the number of registered students at graduate level in city C is 25% of the registered students at graduate level in rest of the registered student population in country M, than graduate registered students at city C are what percent of the graduate registered students population at country M?
Take 2 minutes and try answering this question by your own. After that see solution below:
Solution:
As it’s mentioned that C is the capital city of country M, also that registered graduate at C is 25% of the rest of the graduate in country M. So let’s suppose for our convenience registered students in rest of the cities of country M (i.e except city C) is 100 (again not 100x because we don’t need to find numbers, rather we need to find answer in percents).
Now according to the given condition, it can be drawn:
Sky blue region indicates registered graduate population in other part of the country M except in City C, while gray area indicates registered graduate population in city C.
From this figure, we can easily conclude that total registered graduates in country M are (100 + 25 =) 125.
Now the question said to answer registered graduates in city C, as a percent of total registered graduates in country M
As we learned from Study Blan for Beginners, that whatever comes after percent or fraction of, its reference that always placed in denominator, so total registered graduates in country M will come in denominator and registered students in city C will come in numerator as follows:
⇒ 20% Answer
At this stage, we should try a high difficulty level scenario,
In a pound of some fishes, 20% are tagged fishes. If there are 500 tagged fishes, then what percent of un-tagged fishes must be tagged to make 50% tagged fishes in the pound?
Solution:
According to given condition,
Tagged fishes = 20% of Total Fishes in the pound.
⇒ 500 = 20⁄100 × Total fishes
⇒ 500 = 1⁄5 × Total fishes
⇒ Total fishes in the pound = 500 × 5 = 2500
Now, If 20% of fishes in the pound are tagged, then 80% of the fishes must be un-tagged.
So untagged fishes = 80% of total fishes
⇒ = 80⁄100 × 2500
⇒ = 80 × 25
⇒ = 2000
Alternatively, untagged fishes would be (total fishes – tagged fishes) = 2500 – 500 = 2000
Now question asks that what percent of untagged fishes must be tagged so that there would 50% (i.e half) of the total fishes become tagged.
So according to given condition,
We need first to find reference that would come to denominator place, and that is untagged as bold faced words ‘percent of’ indicates that untagged comes after ‘percent of’.
Now what variable would come to numerator place. It’s the point where many students may have conflicts. Well you must have conflict over this point because this point makes high difficulty level of this question. Many students would suggests total tagged fishes (50% of total fishes would come). But some students would consider this logic to be wrong, and would suggest others please focus on each wording of question, especially the bold faced portion as bellow:
“What percent of untagged fishes must be tagged so that there would 50% (i.e half) of the total fishes become tagged.
It clearly tells the untagged fishes that needed to be tagged in future (i.e newly tagged fishes require to tag in future), are what percent of present untagged fishes. Still not understood, let’s split question into two steps:
Step 1:
“What percent of untagged fishes (means present untagged fishes) must be tagged so that there would 50% (i.e half) of the total fishes become tagged.
Step 2:
“What percent of untagged fishes must be tagged (means out of untagged fishes that would be tagged in future i.e newly tagged fishes) so that there would 50% (i.e half) of the total fishes become tagged.
Now, I’m sure you’ve been cleared in such ambiguity of what would come in numerator place. It’s those proportion of untagged fishes that requires to be tagged in future to increase tagged fishes to 50%, which are 20% at present.
Important note: Remember that there is a big difference between ‘increase by’ and ‘increase to’. If it’s increase from 20% to 50% that means the increment amount is 30%. On other hand, from 20%, if it’s ‘increase by’ 50%, that it means the increment amount is 50%, as it’s increase by 50%.
Let’s learn how to find this,
Now, present tagged fishes = 500
Required target to get number of tagged fishes = 50% of total fishes = 1⁄2 of 2500 = 2500⁄2 = 1250
So additional required new tagged fishes from untagged fishes = 1250 – 500 = 750
This thing must be place in numerator as below:
⇒ = {750⁄2000} × 100
⇒ = {75⁄200} × 100
⇒ = {75⁄2}
⇒ = 37.5% Answer
Percentage topic is almost always tested in exams, but in GMAT and GRE, it many times tested with Venn diagram, Rate, mixtures and many other topics. We’ll further learn it’s scenarios together with other topics in later lectures.
Here, let’s discuss both fraction and percent scenario combined, in further hard level question type.
Data Sufficiency question (Only for GMAT):
In an examination hall, 60% of the boys and 1⁄3 of the girls are good in Math. What percent of girls who are good in Math are boys who are good in Math.
(1) There are total 270 students in the examination hall.
(2) Number of girls are 50% less than the number of boys in the hall.
A) Statement (1) ALONE is sufficient, statement (2) alone is not sufficient.
B) Statement (2) ALONE is sufficient, statement (1) alone is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient.
This question is among the most tricky questions. GMAT students should try to answer by their own, then we’ll discuss below:
Here majority of students most probably would select choice C. And only few would select B and E. But right answer is B. After completion of the whole course, you will come to know that data sufficiency questions are easier than problem solving questions. The best thing here is that you don’t need to find the answer, rather you need to see whether can we answer? That’s it.
At this level you have to learn how to become smart enough to decide it well. Data sufficiency will be your time savor in quantitative section of GMAT. So you have to take maximum advantage from this to complete quantitative section well in time in order to get 99 percentile on it. Let’s solve the question given above.
Firstly, data about boys and girls, who are good in math, are given in percentage or fraction from.
Always we start from statement (1) whether it’s sufficient or not.
Statement (1) gives total number of students, which clearly is not sufficient to answer the question. Because 60% is from total boys, and one-third is from total girls. And we have no information about boys and girls separately. Total number of students do not tell this.
As statement (1) is not sufficient, we must start analyzing statement (2) to see whether it’s sufficient to answer the question.
Statement (2) imply number of girls are half as many as number of boys. That’s what we needed. Now, why many students quickly made decision to select choice C is because they though from statement (2) alone, we cannot find number of boys and number of girls. Ok they are right. Both of the statement may also answer the question, but you are making decision in narrow level. Think little bit more about statement (2). You’ll realize that in question we are not asked the number of boys and that of girls who are good in Math. Rather, we are asked to tell ratio of boys to girls who are good in Math. So that hint to the point that if we would have same multiple of boys and girls (or in simple words the ratio of boys to girls), we can right the number of boys and girls in term of a common variable (let’s suppose x), and then by taking required percent or fraction of these number of boys and girls would give required numbers of boys and girls who are good in Math, in term of common variable x. By dividing these, the variable would cancel out and would provide the answer. That’s it. Whole process you would do in mind, which perhaps you’ve learned in verbal lecture that mind is a miracle whose processing speed is million times faster than the fastest Super Computer. But you should do little-bit rough work to ensure accuracy. Below is the total working for those who still not understand.
From statement (2), we have,
Total Girls = 1⁄2 × Total Boys
Let’s suppose Total Girls = x
⇒ Total Boys would = 2x
So,
Boys who are good in Math = 2x (0.6) = 1.2x
Girls who are good in Math = x⁄2
Now, if you divide the ratio of these two to get percent of girls who are good in math the variable x would cancel out. And you’ll able to solve it. That’s it. Note again that what ever come after ‘percent of’, would be the reference that should put in denominator.
Remember that “what percent of the girls who are good in Math are boys who are good in Math”; this statement is similar to the statement “Boys who are good in Math are what percent of girls who are good in Math”. So don’t be confuse. Be clear while you interpret any statement.
Now let’s discuss a tricky type question in quantitative comparison that is only for GRE. Note that quantitative comparison of GRE is much easier and even more less time taking than data sufficiency of GMAT.
In library, 20% of the books are chemistry, 30% of the remaining books are Math books, 25% of the remaining are History books, 50% of the remaining are Biology books and the rest are religious books. If two-third of the religious books were removed from the library.
Quantity A Quantity B
Remaining religious books 7%
as a percent of total books
(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.
Try this question by your own, and come to learn below for explanation after you answered.
As I said this question is not as easy as you thought. During the whole course you’ll learn from your mistakes if you do. And doing mistakes is not a thing to worry. But repeat those mistakes is a thing to worry. Those who answered this carefully well have perhaps learn the importance of the word that I advised you in start. i.e BE CAREFUL!
The answer that majority of the people have selected is choice C, which is incorrect! Unfortunately the correct answer is choice A. let me just tell you why?
The reason for choice C that most of the people have selected is because they didn’t kept my advise in consideration as repeated above. Therefore, they ignored a variable change that can also change the whole thing. For instance, in above question as you have supposed total number of books to be 100 (not 100x, because we have to answer in percentage, rather than in numbers). So you must have correctly find number of religious books as 21. Also after removing two-third (i.e 2⁄3) of the religious books, you got correctly 7 number of remaining religious books. But the mistake is you forgot the reduction in religious books will also reduce total number of books from the library by same numbers. Therefore, as you removed 14 religious books (i.e 2⁄3 of 21), so remaining total number of books must be 86 (i.e 100 – 14). When the denominator has decreased so result would become greater than 7%. Please BE CAREFUL! from onward. Those who answered this correctly, well done! But never be overconfident! These exams really kick out those who are over confident.