math lecture 1

1.WHY PERCENTAGE? Percentage is a concept evolved so that there can be a uniform platform for comparison of various things. (Since each value is taken to a common platform of 100) Example: To compare three different students depending on the marks they scored we cannot directly compare their marks until we know the maximum marks for which they took the test. But by calculating percentages they can directly be compared with one another. 2.PERCENTAGE The word percent can be understood as follows:Per cent => for every 100. So, when percentage is calculated for any value, it means that you calculate the value for every 100 of the reference value. When you see the word "percent" or the symbol %, remember it means 1/100. E.g.:-

(a) 25% means = 25/100 = 1/4

(b) if I invest 30% of my income in mutual funds then it means I invest Rs.30 out of every Rs. 100 of my income.

Some important facts:-

(a) There is neither any unit of percentage nor itself is a unit.

(b)If X% is required to be converted into fraction, it is X/100. e.g 75% means = 75/100 = 3/4

(c) If X/Y is required to be converted into percentage, it will be X/Y × 100 e.g 2/5 to percentage 2/5*100=40%

Before starting any type of tricks practice, I will ask you some questions.

(a) What percent of 75 is 45? OR 45 IS WHAT PERCENT OF 75? = 45/75 × 100

(b) What percent of 95 is 76?

= 76/95 × 100

3.Use of conversion table:

• I know u r thinking that , what is the benefit of learning

Fraction to Percentage to Fraction conversions, when we already know the old method to calculate

percentages. To show the advantage of the above conversions. Let us take an example.

Q1: If population of hisar City is 9600.If population of hisar City grows by 37.5 % in a year. Calculate the

population after one year.

Solution:

Old method

Population after one year = 9600 + 37.5.% of 960

= 9600 + 37.5/100 * 960

= 9600 + 3600 = 13200 Ans.

In old method we need to do more calculation, but our conversion makes it so easy.

Using Conversion Method: In above section we have seen that 37.5% is equivalent to 3/8 in fraction. I have

earlier discussed that calculating 37.5 % of any number is exactly same as calculating 3/8 of that number.

Population after one year = 9600 + 3/8 X 9600 = 9600 + 3600 = 13200 Ans

let’s see how you should calculate mentally.

37.5% increase = 3/8 increase

3/8 of 9600 = 3600

Total Population after a year = 9600 + 3600 = 13200

This question can be done mentally without using pen. Remembering these conversions are very essential to

succeed in entrance examination.

Now you have seen the magic of these conversions, Now do one work, I am providing a conversion table below

note it down in a A4 sheet of paper & paste it in front of your study table, glance through it once a day. It will

make your life easier.

Examples of using CONVERSION TABLE:

Ex1: What is 28.56 % of 1428.

Solution: 28.56 % of 1428 = 2/7 of 1428 = 408.

Ex2: If 16.67 % of some value is 32. Calculate 37.5 % of that value?

Solution: 16.67 % (1/6) of X = 32 => X = 32 x 6

37.5 % of 32 x 6 = 3/8 of 32 x 6 = 3 x 4 x 6 = 72

Thus, we see that calculations become very easy once we adopt this technique.

4.HOW TO FIND PERCENTAGE OF A NUMBER :

With this trick you can mentally find the percentage of any number within seconds

This is one of the coolest trick which makes maths fun.

Let us understand a simple concept on percentages here

100% of a number will be the number itself

ex:100% of 360 will be 360

50% of a number will be half of the number

ex:50% of 360 will be 360/2=180

25% of a number will be quarter of the number or half of 50% of the number

ex:25% of 360 will be 360/4 or 180/2=90

10% of a number can be found by shifting the decimal point by 1 place to the left.

ex:10% of 360 will be 36.0

5% of the number is half of 10% of a number

ex:5% of 360 will be 36/2=18

15% of the number will be the sum of 10% and 5% of a number

ex:15% of 360 will be (10% of 360)36+(5% of 360)18=54

1% of a number can be found by moving the decimal point by 2 places to the left.

ex:1% of 360 will be 3.60

Now that we know the concept let us see how we can apply this trick in the below examples

Example 1: 11% of 550

Step 1:Split the number into 2 numbers that we can easily find the percentage of.

We can split 11% into 10% and 1%

Step 2:First we find 10% of 550 by shifting the decimal point to the left by 1 place.

We get,

10% of 550=55.0

Step 3:Now we need to find 1% of 550 by shifting the decimal point to the left by 2 places.

We get,

1% of 550=5.50

Step 4:Adding both the results

we get 10%+1%=55.0+5.50=60.50

Ans 11% of 550=60.50


Step 1:21% can be split into 10%+10%+1%

Step 2:First we find 10% of 250 by shifting the decimal point to the left by one place

we get,

10% of 250=25.0

Step 3:Another 10% of 250 we get

10% of 250=25.0


We get,

1% of 250=2.50

Step 5:Adding all the results we get,

10%+10%+1%=25+25+2.50=52.50

Ans 21% of 250=52.50


Step 1: 61% can be split into 50%+10%+1%

Step 2:First we find 50% of 754 which is half of the number

we get,

50% of 250=377

Step 3:First we find 10% of 754 by shifting the decimal point to the left by one place

we get,

10% of 754=75.4


We get,

1% of 754=7.54

Step 5:Adding all the results we get,

50%+10%+1%= 377+75.4+7.54=459.94

Ans 61% of 754=459.94

1.62% of 420 would be 50%+10%+2% which is 210 + 42 + 8.4 = 260.4

2. 22% of 324 would be 10%+10%+2% i.e. 32.4×2 +3.24×2 = 64.8 + 6.48 = 71.28

3. 16% of 42 would be 10%+5%+1% i.e. 4.2 + 2.1 + 0.42 = 6.72.

5.A% OF B IS SAME AS B%

Statement: Is A% of B same as B% of A ?

Let us verify above statement.

A% of B = A/100 * B= A*B/100 -- (1)

&

B% of A = B/100 * A= B*A/100 -- (2)

So, From equation 1 and 2 it is clear that A% of B has exactly same value as B% of A .

The above statement reduces our calculation in many percentage problems. Let us take an example.

Ques: Calculate 74 % of 50.

Solution: Since we know, A% of B and B% of A are same.

So, 74% of 50 = 50% of 74.

Now there is nothing left to calculate as you all know,

50% = 1/2

So, 50% of 74= 1/2*74= 37.

.

Ques1: what is the value of 55% of 280 + 28% of 550.

Solution: Look at the second term,

28% of 550 can be written as 55% of 280.

Now, our expression becomes

55% of 280 + 55% of 280= 110% of 280= (100% of 280)+( 10% of 280)= 280+28= 308 Ans.

Question2: 52% of 250 + 25% of 520.

Solution:

Look at the second expression.

25% of 520 = 52% of 250

Now our expression becomes.

52% of 250 + 52% of 250=104% OF 250 =260.

6.CONCEPT OF INCREMENT AND DECREMENT….

Understanding INCREMENT AND DECREMENT is very important. Since INCREMENT AND DECREMENT used

while solving problems related to Profit,Loss, Discount &SIMPLE INTEREST, Compound Interest problems AND IN

DATA INTREPRETATION ALSO.

Let us understand first concept of Multiplying Factor (M.F.)

Multiplying Factor (M.F.)

Case 1: When a quantity is increased by certain percentage.

A) Suppose we have to increase a value 1200 by 20%. What will be the final value?

Final Value= Initial Value + 20% of Initial Value

= 1200 + 20% of 1200

=120 0(1+ 20%)

=120(1+20/100)

=1200(1.20)

=1200 x 1.20=1440

So now we can say for 20% increase if we multiply initial value by 1.20 we will get our final value.

So, here we call 1.20 as Multiplying Factor (M.F.)

Note: For 20% increase Multiplying Factor =1.20

B) Suppose we have to increase a value 1200 by 40%. What will be the final value?

So now we can say for 40% increase if we multiply initial value by 1.40 we will get our final value.


Note: For 40% increase Multiplying Factor =1.40

Let us generalize the above case.

So, if we increase our initial value by x%. What will be our final value?

Final Value= Initial Value + x% of Initial Value

= Initial Value (1+ x %)

= Initial Value (1+ x/100)=INITIAL VALUE*[(100+X)/100]

So, if we increase a value by x%.

To get Final Value we need to Multiply Initial value by Multiplying Factor .

Multiplying Factor=( 100+X)/100

Case 2: When a quantity is decreased by certain percentage.

B) Suppose we have to decrease a value 1200 by 20%. What will be the final value?

Final Value= Initial Value - 20% of Initial Value

= 1200 - 20% of 1200

=1200 (1 - 20%)

=1200(1-20/100)

=1200(1-0.2)

=1200 x 0.8=960

So now we can say for 20% decrease if we multiply initial value by 0.8 we will get our final value.


Let us generalize the above case.

So, if we decrease our initial value by x%. What will be our final value?

Final Value= Initial Value - x% of Initial Value

= Initial Value (1- x %)

= Initial Value (1 - x/100)

So, if we decrease a value by x%.

To get Final Value we need to Multiply Initial value by Multiplying Factor .

Multiplying Factor= (100-x)/100.

Summary:

Case 1: for x% increase Multiplying Factor= (100+ x)/100

Case 2: for x% decrease Multiplying Factor= (100- x)/100

Note: So, from now onwards your thought process should be

20% increase ==> multiply by 1.20

20% decrease==> multiply by 0.80

4% increase ==> multiply by 1.04

4% decrease ==> multiply by 0.96

E.G. 1: Population of a town is 10,000. If it increased by 20 % in two consecutive years. Then it is decreased by 20

% in the third year. Find the population after 3 years.

Solution: The question can be solved easily with the help of Multiplying factor.

I Year ——->II Year ———->III Year —————–>IV Year

10000 10000 x 1.2 10000 x 1.2 x 1.2 10000 x 1.2 x 1.2 x 0.8 = 11520

2: If in the above question, the initial value is not given then calculate % increase/decrease in population after 3

years.

Solution: Trick : We can calculate these types of % Increase/Decrease with the help of Multiplying factor assuming

initial value as 100 OR 100x. By assuming initial value as 100, we can calculate the % Increase/Decrease by

subtracting 100 from the final value.

I Year ——->II Year ————->III Year ————–>IV Year

100 100 x 1.2 = 120 120 x 1.2 = 144 144 x 0.8 = 115.2

Therefore % Increase = 115.2 – 100 = 15.2 % (Ans.)

math lecture 1

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