**Percent (%):**

When fraction is multiplied by hundred, itβs called percent. It actually, tells out of hundred how much or how many quantity of something or number of something are present. For instance,

When half of the total numbers of students are present in the class, what percent of the students are present in the class?

Let’s say 10 out of 20 students are present in the class. So students present as a fraction of total are:

If we multiply this fraction by hundred, it would give us percent of students who are present, i.e.,

In advance level study plan, we’ll discuss percents in much detail with different scenarios that usually come in exam. But here, we’ll cover some basic but important points.

Let’s learn how to convert a simple statement of percentage into an equation.

**Statement:**

*What is 20% of 400?* or *20% of 400 equals to what number?*

**Solution:**

** What** is converted to

*x*** is** is converted to

**=**

** 20** remains

*20***%** is converted to ^{1}⁄_{100}

** of** is converted to

**×**

** 400** remains

**.**

*400*

So,

*What is 20% of 400?* ⇒ *x* = 20(^{1}⁄_{100}) × 400

⇒ *x* = 20 × 4 **= 80** **Answer**

Similarly,

**Statement:**

*80 is what percent of 400?*

**Solution:**

** 80** remains

*80*** is** is converted to

**=**

** what** is converted to

*y***percent** is converted to ^{1}⁄_{100}

** of** is converted to

**×**

** 400** remains

**.**

*400*

So,

⇒ *80 is what percent of 400?* ⇒ 80 = *y*(^{1}⁄_{100}) × 400

⇒ 80 = 4y ⇒ y = ^{80}⁄_{4} **= 20%** **Answer**

**Percent greater vs percent less:**

**Statement:**

*15 is 50% percent greater than what number?*

**Solution:**

** 15** remains

*15*** is** is converted to

**=**

** 50% greater** is converted to

**(1 +**

^{50}⁄_{100})(**Note:** Here instead of the word ** of**, a word

**is used.)**

*greater than*** than** is converted to

*×*** what number** is converted to

*m*

So,

⇒ *15 is 50% percent greater than what number?* ⇒ 15 = (1 + ^{50}⁄_{100}) × *m*

⇒ 15 = (1 + ^{1}⁄_{2}) × *m*

⇒ 15 = (^{3}⁄_{2}) × *m*

⇒ 15 × (^{2}⁄_{3}) = *m* (Multiplying both sides of equation by ^{2}⁄_{3})

⇒ *m* **= 10** **Answer**

Similarly,

**Statement:**

*15 is what percent greater than 10?*

**Solution:**

** 15** remains

*15*** is** is converted to

**=**

** what percent greater** is converted to

**(1 +**

^{n}⁄_{100})(**Note:** Here instead of the word ** of**, a word

**is used.)**

*greater than*** than** is converted to

*×*** 10** remains

*10*

So,

⇒ *15 is what percent greater than 10?* ⇒ 15 = (1 + ^{n}⁄_{100}) × 10

⇒ 1.5 = 1 + ^{n}⁄_{100}

**Important note:** Never do mistake here by multiplying 100 on both sides of equation. Because on left hand side, 100 is only dividing by *m*, rather than by (1 + *m*). Many beginners commit an arithmetic mistake here. So be careful! π

⇒ 0.5 = ^{n}⁄_{100} (By subtracting 1 on both sides of equation)

⇒ *n* **= 50** **Answer**

Now, let’s learn this for *‘percent less than’*

**Statement:**

*15 is 50% percent less than what number?*

**Solution:**

** 15** remains

*15*** is** is converted to

**=**

** 50% less** is converted to

**(1 β**

^{50}⁄_{100})(**Note:** Here instead of the word ** of**, a word

**is used.)**

*less than*** than** is converted to

*×*** what number** is converted to

*a*

So,

⇒ *15 is 50% percent greater than what number?* ⇒ 15 = (1 β ^{50}⁄_{100}) × *m*

⇒ 15 = (1 β ^{1}⁄_{2}) × *m*

⇒ 15 = (^{1}⁄_{2}) × *m*

⇒ 15 × 2 = *m* (Multiplying both sides of equation by 2)

⇒ *m* **= 30** **Answer**

Similarly,

**Statement:**

*15 is what percent less than 30?*

**Solution:**

** 15** remains

*15*** is** is converted to

**=**

** what percent less** is converted to

**(1 β**

^{b}⁄_{100})(**Note:** Here instead of the word ** of**, a word

**is used.)**

*greater than*** than** is converted to

*×*** 30** remains

*30*

So,

⇒ *15 is what percent less than 30?* ⇒ 15 = (1 β ^{b}⁄_{100}) × 30

⇒ ^{15}⁄_{30} = 1 β ^{b}⁄_{100}

**Important note:** Never do mistake here by multiplying 100 on both sides of equation. Because on left hand side, 100 is only dividing by *m*, rather than by (1 β *b*). Many beginners commit an arithmetic mistake here. So be careful! π

⇒ ^{1}⁄_{2} = 1 β ^{b}⁄_{100}

⇒ ^{b}⁄_{100} = 1 β ^{1}⁄_{2} (By rearranging)

⇒ ^{b}⁄_{100} = ^{1}⁄_{2}

⇒ *b* = ^{100}⁄_{2} (Multiplying both sides of equation by 100)

⇒ *b* **= 50** **Answer**

##### Takeaway:

Greater than and less than has only difference of signs i.e +ve and βve respectively.

**Basic Notations:**

There are some statements in word problems that usually tease test takers while understanding correctly. For instance,

*“There are twice as many apples as Mr. A has as Mr. B”*

Many students confused about this statement, and started over time-wasting habit of thinking on whether it would be:

2A = B or A = 2B

You have to be quick in your exam, also have to be accurate. Now, I hope you understood why I’ve said you that these exams do not test your knowledge, rather these test your decision making skill. A quick as well as accurate decision making skill is the potential that universities are seeking in their applicants.

In such situations, you have to be confident while converting these statements into equation. The confidence will push you by simply answering to the pinpoint decision. i.e

Whatever variable come after *‘as many … as’*, just multiply that variable with twice or thrice etc and then equate this to other variable. In above case, Mr. B come after second *‘as’* (i.e in *‘as many … as’*), so we should multiply B by 2. And equate it with the other variable A. Thus,

A = 2B is correct translation. This method will never confuse you. And from onward, you can translate such statements confidently. π

**Other basic notations:**

**Statement:**

*z is x less than y*.

**Solution:**

** z** remains

*z*** is** is converted to

**=**

** x** remains

*x*** less than** converted to

*βve sign*** y** remains

*y*

So,

*z is x less than y* ⇒ *z* = *y* β *x***Answer**

**Statement:**

*If n is greater than m, the positive difference between twice n and m.*

**Solution:**

** n greater than m** means that positive difference between them is

**.**

*n*β*m*Now, you must be clear about the difference between the two statements as below:

*The positive difference between twice n and m.* | *Twice the positive difference between n and m.*

The above two statements will give different results. As we are asked the left sided statement, so

*The positive difference between twice n and m* ⇒ 2*n* β *m* **Answer**

**Statement:**

*The ratio of 4q to 7p is 5 to 2.*

**Solution:**

Remember that ratio of *a* to *b* = ^{a}⁄_{b}

Similarly,

*The ratio of 4q to 7p is 5 to 2.* ⇒ ^{4q}⁄_{7p} = ^{5}⁄_{2}**Answer**

**Statement:**

*The product of a decreased by b and twice the sum of a and b.*

**Solution:**

Notice on the first part of the statement *‘a decreased by b’*. It means the variable a is decreased i.e something will be subtracted from *a*, rather than from *b*. i.e

First part of the statement: *a decreased by b* ⇒ *a β b*

Now, the second part also requires careful attention. There is a difference between below two statements:

*Twice the sum of a and b.* | *Sum of twice a and b*.

Second part of the statement: *Twice the sum of a and b.* ⇒ *2(a + b)*

Now, question is asking the product of both parts of the statement. i.e

Product of ** a decreased by b** and

**⇒**

*twice the sum of a and b***(**

*a β b*) ×*2(a + b)*

**Answer**You may further simplify this expression as below:

⇒ = 2 (*a + b*)(*a β b*) = 2(*a ^{2} β b^{2}*) {As we know, (

*a + b*)(

*a β b*) =

*a*}

^{2}β b^{2}So,

⇒ **2( a^{2} β b^{2})**

**Answer**

**Statement:**

*A quarter of the sum of a and b is 4 less than a.*

**Solution:**

First part of the statement: *quarter of the sum of a and b* ⇒ ^{(a + b)}⁄_{4}

Remember that *‘a quarter’* means one-fourth (i.e ^{1}⁄_{4}). So a quarter of means ^{1}⁄_{4} times or simply divided by 4.

Second part of the statement: *4 less than a* ⇒ *a β 4*

Now, question states that first part of the statement **is** the second part of the statement. As we know ** ‘is’** is converted to

**=**.

Therefore,

⇒ ^{(a + b)}⁄_{4} = *a β 4***Answer**

Now, you are ready to attempt a mini Quiz. Click on *‘Mini Quiz’* button below: