Word Problems
(Fractions & Percents)
Fractions & Percents are one of the most common type of word problems that always come in such exams. The reason of being important is that many professionals have to face such problems in their daily routine while dealing with clients or vendors. So let’s learn this concept in detail.
Fractions:
As you’ve learned little-bit about fractions in beginners refresher, Right!
Let’s discuss application of this concept.
This is used in word problems as well as in arithmetic. But as currently we are doing word problems, we’ll discuss word problem type of fractions here only. Arithmetic part of fraction you’ll learn later on while doing arithmetic topic of quantitative section.
Let’s consider a father, who has 3 kids, needed to distribute all of his candies among his three kids. He distributed those candies in such a way that the youngest kid received 1⁄2 of the candies while the middle kid receive 3⁄5 of the remaining candies. If the eldest kid receive 10 candies, how many total candies the father distributed among his kids?
Solution:
Remember that in all word problem type fractions, you must always take L.C.M (Least Common Multiple) of all the fractional parts given in the questions. i.e always take L.C.M of the denominators of all fractions. In above fractions, there are two fractions i.e 1⁄2 and 3⁄5. So the L.C.M of 2 & 5 is 10. (You will learn the shortest and quickest way of taking L.C.M in Arithmetic lecture later on).
Now, suppose there are 10x total numbers of candies that we need to find. Now its very easy to split this supposed amount into different parts in terms of x as shown below:
So, in term of x eldest kid received 2x candies.
As eldest kid received 10 candies, so
⇒ 2x = 10
⇒ x = 5
Now, we need to find total number of candies distributed by the father which is 10x that we supposed at start of the calculation, by putting value of x here, we’ll get
Total candies distributed by the father = 10 (5)
⇒ Total candies = 50 Answer
Let’s move on to a higher difficulty level scenario and consider a class of few students, that include 1⁄3 girls. If 1⁄5 of the boys have laptops and there are twice as many girls having laptops as boys having laptops, then what fraction of all students in the class don’t have laptops?
Solution:
In the same way as I mentioned before,
Let’s take L.C.M of the two given fractions first. We have two denominators 3 and 5, whose L.C.M is 15.
Important Note: Remember that if in any question of fractions or percents, if it’s ask answer to be in form of percent or fraction, then you don’t need to put x in supposed value of total numbers. But if somethings value is given or it’s ask about how many numbers or value of something rather than percent or fraction of something, then you must suppose x with the L.C.M.
So, suppose there are total 15 students in the class; here we’ll not have need to suppose 15x as it’s ask fraction rather than number/value.
As it’s mentioned that 1⁄3 of the students are girls, so this implies that remaining (1 – 1⁄3 =) 2⁄3 of the students are boys. Also it’s mentioned that 1⁄5 of the boys have laptops, therefore this implies that remaining (1 – 1⁄5 =) 4⁄5 of the boys don’t have laptops.
Additionally, given that there are twice as many girls having laptops as boys having laptops. How to convert this statement into mathematical equation.
To do so, let’s take a different but an easy scenario of same thing.
Mr. A has twice as many apples as Mr. B has.
What you think?
B = 2A ?
or
A = 2B ?
Well, to answer such things you have to be 100% clear and confident while answering.
It’s very easy, if you follow this principle.
Always see the words like “as much as” or “as many as”. Then see what variable or value comes after second ‘as’. Then just multiply by 2 with this in case of ‘twice’ or by 3 in case of thee times etc. And then the second variable comes on other side of equality sign alone.
So clearly Mr. A has twice as many apple as Mr. B has, would be written mathematically as follows,
Apples that Mr. A has = 2 (Apples that Mr. B has)
⇒ A = 2B.
Similarly, in our actual question, which says ‘there are twice as many girls having laptops as boys having laptops’, after second ‘as’ Boys having laptops come, so be confident to write equation as follows
Girls having laptops = 2 (Boys having laptops)
Don’t consider girls having laptops as total girls, keep in mind that both are different variables.
Now, we can solve the question with much better ease as fellows,
From the above tree, we see that number of boys who don’t have laptops are 8, while number of girl who don’t have laptops is 1. So there are total 9 students who don’t have laptops if total students are 15.
So we need to find the number of students not having laptops as a fraction of total number of students. Converting word problems into equation is also a key to answer correctly. So I want you to remember again, as you read previously in Study Plan for Beginners, that what ever comes after words like ‘fraction of’ or ‘percent of’, always place that variable or value in denominator because that is our reference point; reference always placed at denominator.
So according to given scenario, we need to find,
⇒ = 9⁄15
⇒ = 3⁄5 Answer
Important Point: Many of you may be confuse at this level to see very long and time consuming answer explanation. Never confuse from this, its just for those who are not very well with understanding things quickly. Different people have different level of intellect, so never be so fast. I’ve seen many of my students, who were so excellent in understanding, but in actual exam, they made very stupid silly mistakes only due to over confidence and proving themselves as extravagant. Big masters may loose in such exams, so the key to get perfect score is never take anything easy, and remain alert of trap answer choices in each questions.
Percents:
As far as the percents are concerns, in any quantitative or math subject, this topic is always tested in any exams level.
As you already read about this in Study Plan for Beginners Level. Percents means proportion out of 100. Let’s learn this topic in detail and advance level here.
When a fraction is multiplied by 100, it gives percent, and that percent always tells numerator is what percent of denominator. We have discussed this in Beginners Level, so we’ll not do this again. Here we’ll discuss this in detail level and in harder perspective.
After learning Study Plan for Beginners Level, you can now transform statement like ’20 is 80% of what number?’ into mathematical equation. So at this level, you must move forward to broaden scenarios in this topic.
Percentage Change:
When its given that a variable (let’s say price of shirt) decrease from original value by 20%, then how much percent the price must be increase to get back to original value. See below,
Remember that in any question, it is asked about by what percent the price or anything increase or decrease, it simply ask percentage change, whose formula to find is given below,
As I’ve mentioned when the value is not given in any question of fractions or percents, and it’s ask to find percent or fraction; always suppose 100 as initial or starting value, and then solve for the answer.
Similarly, in above question, it required percentage change, so we must suppose 100 as original value. After 20% decrease it would move on to 80. Now we need to find by what percent the price from 80 must be increase to reach to its original value 100. In short what is the percent change from 80 to 100.
So let’s use formula of calculating percentage change
The reference point is 80 rather than 100 here, because we need to find the price changes from 80 to 100. But in previous step the reference was 100, because price was changed from 100 to 80.
⇒ = {(100 – 80)⁄80} × 100
⇒ = (1⁄4) × 100
⇒ = 25% Answer
Important Note: Keep in mind percent change from 100 to 80 is not the same as percent change from 80 to 100.
Let’s suppose a rice dealer buy 1000kg rice at some cost price, and then sell this at 25% profit. If the dealer sells those rice at PKR 50000, what would be his cost price per kg?
First try yourself, then move on below. Take 2 minutes to try.
Many people do mistake in such question. They mistakenly take 25% of selling price (50000) and then subtract it from selling price to find the cost price. Well this method would give wrong answer. See how,
As we know that,
Sale = Cost + Profit
Always profit is taken from the cost price, if given in percentage, unless it’s not stated clearly i.e 25% of sale price and in case nothing mentioned just as in this question we must take it as percent of cost price. As in this scenario, 25% profit is not 25% of sale price (50000), rather its 25% of cost price (that we need to suppose x).
Now according to the given condition,
Sale = Cost + Profit
⇒ 50,000 = x + 25% of x
⇒ 50,000 = x + 0.25x
⇒ x = 50,000⁄1.25
By multiplying & dividing by 100, we’ll get
⇒ x = (50,000 × 100)⁄(1.25 × 100)
⇒ x = (50,000 × 100)⁄125
⇒ x = (50,000 × 4)⁄5
⇒ x = 10,000 × 4
⇒ x = 40,000
As we need to find cost price per kg, so we need to divide this total cost by number of kg (units)
⇒ Cost price per kg = 40,000⁄1000
⇒ Cost price per kg = PKR 40 Answer
Alternate Method:
25% profit means the dealer has buy the rice and then sold this at higher/increased price. That means we can use formula of percentage change here as well to find the answer as fellows,
As we know that
Suppose, x be the cost price at which the dealer buy 1000kg rice,
⇒ 25% = (50,000 – x)⁄x (As reference is cost price from where price started increasing)
⇒ 25⁄100 = (50,000 – x)⁄x
⇒ 25x = 100 (50,000 – x)
⇒ 25x = 50,00,000 – 100x (By cross multiplication)
⇒ 125x = 50,00,000
⇒ x = 40,000
As we need to find cost price per kg, so we need to divide this total cost by number of kg (units)
⇒ Cost price per kg = 40,000⁄1000
⇒ Cost price per kg = PKR 40 Answer
Percent vs Times:
There is a very important relationship between Percent vs Times. This relationship would ease your calculation and save your time. So I would recommend to get benefit as much as you can while solving for percentage problems.
Let’s suppose if value of x is increased by 20%, how much times of x would the result becomes?
Now, let’s solve this step by step to make you understand it clearly,
First, place 1 with x times i.e
⇒ x ( 1 )
Now in case of percent increase (or increase by), place ‘+’ sign after 1, i.e,
⇒ x ( 1 + )
Now, look at what percent must it be increased. As it’s 20% given in required question, so add 20% after ‘+’ sign. i.e,
⇒ x ( 1 + 20%)
as we know, % = 1⁄100, so 20% = 0.20
⇒ x ( 1 + 0.20)
⇒ x (1.20)
⇒ 1.2 x Answer
So when x is increased by 20%, it would give 1.20 times x
Now, you don’t need to follow all steps after this concept, just do this
Let’s say when x is increased by 250%, than what’s the result?
Solution:
⇒ x (1 + 2.5) (As 250% = 250⁄100 = 2.5)
⇒ 3.5 x Answer
Similary, in case of percent decrease, we’ll subtract from 1 as follows,
If x is decrease by 25%, what is the result?
Solution:
⇒ x (1 – 0.25) (As 25% = 25⁄100 = 0.25)
⇒ 0.75 x Answer
Important Note: Remember the difference of 20% and 2%. 20% = 0.20, while 2% = 0.02. Many students have done silly mistakes in such minor things in their actual exam, so take care of such minor things.