Level 2 Percentage Problem

Level 2 Percentages Problem

Dell computer has two branches: one in ohio and second in texas: The total no. of employees in ohio office grew this year by 25% to 750 but the ratio of male to female employee is same as in the previous year. The no. of employees in the Texas office grew this year by 9.09% to 1200. The ratio of male to female employees last year in the texas office was 5:6 and the no of male employees in the ohio office was 20% less than that of texas office.

The total no of female employees this year in both the office is:

Solution:

                        Ohio                            texas

Last year: 600<-(2:1)                          (5:6)-> texas

                        (M:F)                           (M:F)

This year : 750<-(2:1)              unknown -> 1200

Since we don’t know the number of female employees in the texas office this year so we cant determine

The total no of employees in both the offices last year was:

Solution:

1100 + 600 = 1700

A shepherd had n goats in the year 2000. In 2001 the no of goats increased by 40%. In 2002 the no of goats declined to 70%. In 2003 the no of goats grew by 30%. In 2004, he sold 10% goats, then he had only 34,398 goats. The percentage increase of the no. of goats in the duration was:

Solution:

There is no need to use the no. of goats i.e., (34, 398) let initially there be 1000 goats then

1000à 1400 à 980 à 1274à 1146.6

Thus the % increase = 1146.6 -1000/1000 x 100 = 14.66%

 In the above question in which year the no of goats was minimum?

Solution:

In 2002 (980 goats) as per the flow chart

Optional	Science	Commerce	Engineering	Total
	5000	3000	8000	16000
Finance HR Marketing	1000 1600 2400	1200 720 1080	680 1040 6280	2880 3360 9760

Level 2 Percentage Problem

Level 2 Percentages Problem

In the awadh school gomti nagar, there are 500 students 60% of the students are boys, 40% of whom play hockey and the girls don’t play hockey, 75% of girls play badminton. There are only two games to be played. The number of students who don’t play any game is:

Solution:

Total students = 100

Boys = 60    &  Girls = 40

Boys => Hockey = 24  & Badminton ( don’t know about badminton)

Girls => Badminton = 30  & Hockey = 0

Since we do not have information that whether the rest of the boys playing badminton or not. So we can not determine the total no. of students who are not playing any of the two games.

A fraction in reduced from is such that when it is squared and then its numerator is increased by 25% and the denominator is reduced to 80% it results in 5/8 of the original fraction. The product of the numerator and denominator is:

a)      6          b) 12    c) 10    d)7

Solution:

Go through option. Let us assume option (C)

10= 2 x 5 = 5 x 2 = 1 x 10 x 10 x 1

Consider the proper fraction = 2/5

Since the given percentage value are 25% and 20% that’s why we have picked up option C

2/5 = 4/25 = 5/20

2/5 x 5/8 = 1/4= 5/20

Hence presumed option is correct.

In the chidambaram’s family the ratio of expenses to the savings is 5:3. But his expenses is increased by 60% and income increases by only 25% thus there is a deficit of rs.3500 in the savings. The increased income of Mr. Chidambaram’s is:

Solution:

Income = expenditure + savings

8 x = 5x + 3x   => 1

10x = 8x + 2x  => 2 ( equation 1 = -x to get equation 2)

Now the deficit = (3x – 2x) = x = 3500

And therefore the new salary = 10x = 35,000

In this presidency college two candidates contested a presidential election. 15% of the voters did not vote and 41 votes were invalid. The elected contestant got 314 votes more than the other candidate. If the elected candidate got 45% of the total eligible votes, which is equal to the no. of all the students of the college. The individual votes of each candidate are:

a       a)      2250 and 1936       b) 3568 and 3254    c) 2442 and 2128       d)2457 and 2143

Solution:

Go through options

2457 – 2143 = 314

Again ( 2457 + 2143 ) + 41 = 4641

Now  4641/0.85 = 5460

Again   5460 x 45 /100 = 2457

Hence the presumed option is correct

The annual earning of mr.sekar is rs. 4 lakhs per annum for the first year of his job and his expenditure was 50% later on for the next 3 years his average income increases by rs.40,000 per annum and the saving was 40%, 30% and 20% of the income. What is the percentage of his total savings over the total expenditure if there is no any interest is applied on the savings for these four years:

Solution:

Income => 4    4.4       4.8       5.2]      18.4     lakh

Saving => 2     1.76     1.44     1.04]    6.24     lakh

Exp =>    2      2.64     3.36     4.16]    12.16   lakh

So,       6.24/12.16 x 100 = 51 6/19%

  

Percentages Problem (Level 1)

Percentages Problem (Level 1) 

On the april 1, 2005 my salary increased from rs.10,000 to rs.16,000. Simultaneously the rate of income tax decreased by 37.5%, so the amount of income tax paid be me remains constant what is the value of income tax paid be me:

Solution:

Since we don’t have sufficient data. Further any value is possible as the required income tax.

The average of a set of whole numbers is 27.2. When the 20% of the elements (ie numbers) are eliminated from the set of numbers then the average becomes 34. The number of elements in the new set of numbers can be: 
a)27      b)35        c)52          d)63

Solution:

Only C is correct since it is divisible by 4.
Let the original number of the element be x then the new no. of elements will be
4x/5 = K
So K must be divisible by 4
Since,   X = K * 5/4

In a class, the no. of boys is more than the no. boys is more than the no. of girls by 12% of the total strength. The ratio of boys to girls is:
a) 15:11            b) 11:14         c)14:11          d)8:11

Solution:

Go through option
Boys                Girls                  Total
14                     11                       25
56                     44                       100
------(12)--------
Since out of 100 students no. of boys are greater than the no. of girls by 12 i.e 12% hence it is correct.

The population of a village is 5000 and it increases at the rate of 2% every year. After 2 years, the population will be:

Solution:

Population after 2 years = 5000 ( 1 + 2/100)^2
5000 * 51/50 * 51/50 = 5202

A customer asks for the production of x number of goods. The company produces y number of goods daily. Out of which z% are unfit for sale. The order will be completed in:

Solution:

Daily supply = (100 –z)% of y = (100-z)y/100
Required no. of days = x * 100/(100-z)y

Level 1 Problems Percentage

Level 1 Problems Percentage

In our Singapore office there is 60% female employee. 50% of all the male employees are computer literature. If there are total 62% employee s computer literature out of the total 1600 employees, then the no. of employees who are computer literature:

Solution:

                        F                                  M

                        60x                              40x

                        42x                              20x

                        --------------(62x)---------------

Therefore female comp. literature 1600 x 42/100  = 672

The price of a car depreciates in the first year by 25% in the second year by 20% in the third year by 15% and so on. The final price of the car after 3 years, if the present cost of the car is rs.10,00,000:

Solution:

(10,00,000) x 0.75 x 0.80 x 0.85 x = rs. 5,10,000

A shopkeeper chargers sales tax of x% upto rs.2,000 and above it he charges y%. A customer pays total tax or rs.320, when he purchases the goods worth rs. 6000 and he pays the total tax of rs.680 for the goods worth rs.12,000. The value of (x-y) is:

Solution:

2000 x X/100 + 4000 x Y/100 = 320

And 2000 x X/100 + 10,000 x Y/100 = 680

Therefore  x= 4  and y = 6

x-y = -2

40% of a number when added to the square of the same number, then it is increased to 4040% of itself the actual number is:

Solution:

Go through option (40 x 0.3) + x2 = x * 4040/100

ð  X^2 = 40x = > x = 40

In my office there are 30% female employees and 30% of these earn greater than rs.8000per month and 80% male employees earn less than rs.8000 per month. What is the percentage of employees who earn more than rs.8000 per month?

Solution:

 Cant be determined .we don’t that whether there are some male employees who have exactly rs. 8000 per month as their salary or not. 

Percentage Problems

Percentage Problems

Remember: Percentage change

= (difference between original value and new value / original value) x 100

Percentage increase = (increased value - original value / original value) x 100

Percentage decrease = (original value - decreased value / original value ) x 100

Note

1) There is a huge difference between "decreased value" and "decrease in value" and between "increased value" and "increased in value".

for example : Initial value = 70

                       Final value = 90

it mean increased value is = 90
but increase in value is 20
and % increase = 90 -70 / 70  x 100  = 28.57%


2) If there is increased  of x/y in any value P then the increased value will be 
       P (1 + x/y)

3) If there is decrease of x/y in any value, then the decreased value will be
       P(1 - x/y)


Example 1  

Nishith is now 20 years old. Some years later his age will increased by 50% of himself. What will be the new age at that time?

Solution:

20(1 + 1/2 ) = 30 years        ( therefore 20 + 20 x 1/2 = 20 (1 + 1/2) )

Example 2

The average salary of purushottam in infosys is 20% less than that was in microsoft. If the salary of prurushottam in microsoft be rs 80,000 per month then what is the salary of purushottam in infosys? 

Solution:

20% = 1/5

so new salary = 80,000 ( 1 - 1/5)

                     = 80,000 x 4/5 = rs.64,000

Example 3

Which one of the following is greater 3/4, 7/8, 16/19, 13/15 ?

Solution:

3/4 = 75%

7/8 = 87.5%

16/19 = 84.21%

13/15 = 86.66%

so, 7/8 is the greatest fraction ( or rational number) 

Problems on Percentage

Problems on Percentage

Example 1

Shweta is a very expert in bargaining. Once she went to a nearby shop. When shweta asked the price of shampoo sachet the shopkeeper told her the price by increasing 15% of the original cost. But shweta insisted to decrease the price by 15% so the shopkeeper sold it by decreasing the price by 15% what is the loss or profit of shopkeeper and by how much percent?

Solution:

Let the actual price be 100

100 ---(+15%)-----> 115----(-15%)--------> 97.75

loss of 2.25%

Example 2

If the length and breadth of a rectangle are changed by +20% and -10% what percentage change in area of rectangle?

Solution:

l x b = area

1 x 1 = 1

1.2 x 0.9 = 1.08

so there is 8% increase in the area of rectangle.

Example 3

The salary of A is 20% lower than B's salary and the salary of C is 56.25% greater than A's salary. By how much percent the salary of B is less than the salary of C.

Solution:

A                          B                               C

80<----(-25%)------------100                           125

------------------(+56.25%)------------------------->

the required value = 25/125 x 100  = 20%

Percentage Population Related Mathematical Problems

Percentage Population Related Mathematical Problems

If the original population of a locality (i.e. region) be P and the annual growth rate be r%. The Population after n years

= P(1 + r/100)^n

change ( or increase) in the population

= P((1 +r/100)^n - 1)

if there is decrease in population be r% then

total population after n years = P ( 1 - r/ 100)^n

and decrease in population = P ( 1- (1- r/100)^n )

Example 1

If the annual increase in the population be 20% and the present population be 10,000. What will be the population after 3 years hence?

Solution:

10,000 (1+ 20/100)^3  = 10,000 (6/5)^3

= 10,000 x (1.2)^3 = 17,280

Example 2:

The Population of a town in the first year increase by 10% in the next year it decrease by 10% once again in the third year it increase by 10% and in the fourth year it decrease by 10%. If the present population be 20,000 then the population after four year will be:

Solution:

20,000 (1+ 10/100) (1- 10/100) (1+ 10/100) ( 1- 10/100)

= 20,000 (1.1) (0.9) (1.1) (0.9) 

= 20,000 x 121 x 0.81 x 

= 19,602

Percentage Concept of BY and TO - problems

Percentage Concept of BY and TO - problems

Due to fall in manpower, the production in the factory decreases by 60% by what per cent should the working hour be increased to restore the original production in the factory? 

Solution:

Manpower x working hours = production

  (-60%) = 3/5 -------> 3/2 = (+150%)                        (3/5 -->  3/5-3  = 3/2)

hence by 150% working hours will be increased. It means the new working hours will be 2.5 times (not 1.5 times) of the original time.


Two numbers are respectively 25% and 40% less than a third number. What per cent is the second of the first?

Solution:

consider A,B,C three numbers and assume C = 100 ( as a base)

A                 B                    C
75                60 <-----------100
                             (-40%)

<------------(-25%)------------

Now  60/75 x100 = 80%


A person gives 10% to his wife 10% of the remaining to a hospital (as a donation) again 10% of the remaining to prime minister's relief fund. Then he has only 7290rs. with him. What was the initial sum of money with that person?

Solution:

Since he gives 10% so he is left with 90% of the original sum and since he does the same with the remaining (or left) amount. so it forms a chain.

therefore remaining amount = (x) x 0.9 x 0.9 x 0.9
                                             = 0.729x = 7290

=> x = 10,000, where x is supposed to be initial amount.

Initially a shopkeeper had n chocolates. A customer bought 10% chocolate from n then another customer bought 20% of the remaining chocolates, after that one more customer purchased 25% of the remaining chocolates. Finally shopkeeper is left with 270 chocolates in his shop. How many chocolates were there initially in his shop?

Solution:

n x 0.9 x 0.8 x 0.75 = 270

n = 270 x 10,000 / 9 x 8 x 75

n =500

Concept of 'BY' and 'TO' Percentage

Concept of 'BY' and 'TO' Percentage

Please note that there is a clear difference between "by" and "to" e.g., the income is reduced by 40% it means the new income is 60% of the original and the income is reduced to 40% means the new income is 40% of the original value.  Thus "by" represents difference and "to" represents final value.

e.g., The income of sarika is increased by 20% means its new income is 100+20 = 120% of the original income. The income of sarika is increased to 120% means the new income of sarika is 120% of the original income.

Example 1: 

In an election between two candidates, the candidate who got 57% valid notes won by a majority of 420 votes. Find the total no. of valid votes.

Solution:

winner                          loser

0.57x                             0.43x                                   (100-57=43)

       -----------------------                                            (57-43 = 14)

                 0.14x

0.14x = 420

x = 3000

Hence total valid votes = 3000

Product Constancy Conditions

Product Constancy Conditions

i) When one factor of a product is increased by p% then the other factor will be decreased by (p/100+p X 100)%

 It means when one factor of a product is increased by n/d then the other factor is decreased by n/(d+n)

ii) When one factor of a product is decreased by p% then the other factor will be increased by
(p/100-p x 100)%

 It means when one factor of a product is decreased by n/d then the other factor will must be increased by n/(d-n)

Example 1:

If the price of a commodity be raised by 20% then by how much per cent a house holder reduce his consumption of the same commodity so that his expenditure does not increase.

Solution:

since here product (i.e., expenditure) is constant rate x consumption = expenditure

initially ------> 1 x 1 = 1

after change   1.2 * X = 1

X = 0.833 . therefore decrease in value = 16.66%


Example 2:

If the price of petrol falls down by 20% by how much per cent must a person increase its consumption, so as not to decrease the expenditure on this items?

Solution:

since product is constant
decrease by                              increase by

20% = 1/5 ------------------>       1/4 = 25%                ( n/d -----> n/(d-n))


Example 3:

Due to 50% increase in the price of rice. We purchased 5 kg less rice with the same amount of rs.60. What is the new price of rice?

Solution:

Increase in price                           Decrease in amount
50% = 1/2           ---------------->       1/3 = 33.33%

since the new quantity of rice decreased by 33.33% which is equal to 5kg it means initially there was 15kg rice to be used.

so, the initial price = rs.4           (60/15 = 4)

and final price = rs.6                   (60/10=6)


Example 4:

The length of a plot is decreased by 33.33%. By how much % the breadth of the plot will be increased so that the area remains constant?

Solution:

(Decrease)                             (Increase)

33.33=1/3 --------------->    1/2 = 50%

Concept of Product Constancy

Concept of Product Constancy

It is the same as we know the inverse proportion in the chapter of ratio, proportion and variation.

eg., when the rate of pencil is rs.1.25 then we can purchase 16 pencils by paying rs.20. If the rate of a pencil is decreased  by rs.0.25 then we can purchase 20 pencils by paying rs.20.

Explanation: 

Rate x No. of Pencils = Price

1.25 x 16 = 20
1.00 x 20 = 20

so you can see that here product 20 is constant in both the cases. Thus it is clear that if we reduce the price of a pencil to rs.0.50, then we can purchase 40 pencils in rs.20.

some more examples of product constancy:

speed x time = distance
rate x time = cost
efficiency x time = work
length x breath = area
average x no of elements = total value
rate x quantity = price

eg., The price of sugar is increased by 25% then by how much per cent should a customer reduce the consumption (ie., quantity used) of sugar so that he has not to increase his expenses on sugar.

price x quantity = expenditure

100 x 100 = 10,000
125 x X = 10,000

x = 10,000/125 = 80. Therefore, % reduction = 20%

or

1 x1 = 1
 1.25 x k =1 = > k = 0.8

thus there will be 20% decrease in the consumption of sugar in order to maintain to same expenditure on sugar.

Problems on Percentage

Example 1:

Height of arun is 50% greater than the height of amir khan. Height of amir khan is how much per cent less than that of arun?

Solution:

                             +50%  = 1/2

Amir khan --------------------------->  Arun

                 <----------------------------

                     -33.33% = 1/3 = 1/2+1

thus the height of amir is 33.33% less than arun

Example 2:

Due to irregular working habits of ramachandran his salary was reduced by 20% but after same month his salary was increased to the original salary. What is the percentage increase in salary of ramachandran? 

Solution:

Let initial salary be rs.100  

                -20%

100 -------------------> 80

      <-------------------

               +25%

Example 3:

Kajolu usually wear saree, which is 16.66% less than the actual length of the saree. By how much per cent the actual length of the saree is greater than the length of saree which kajol usually wears?

Solution:

Actual length                                                              Usual length

                                    -16.66% = 1/6

     -------------------------à A---------------------àUß-------------------       

                                          +20% = 1/5 = (1/6-1)

So the actual length of the saree is 20% greater than the usually used saree   

Example 4:

Initially mr. rakhi sawant has rs. 200 in her wallet then she increased it by 20%. Once again she increased her amount by 25%. The final value of money in her wallet will be how much percent greater than the initial amount.

Solution:

            +20%                                       +25%

200 --------------------à 240 -----------------------à 300

So the required % increase = 300-200/200 x 100 = 50%

Example 5:

The age of B is 50% greater than the age of A. The age of C is 20% less than the age of B. By how much percentage the age of C is greater than the age of A.

Solution:

            +50%                                       -20%

A -----------------------à B-----------------------------àC

100                              150                                          120

So the required percentage change = 20/100 x 100

                                                       = 20%

       

Advanced Concept of Percentage Change

Advanced Concept of Percentage Change

(A) If a value p is increased by x%, then we have to decrease the resultant value by
 (x/x+100 x 100)% to get back to the original value p.

original value p ------------->  p * x /100 --------> ( p + px/100) = p (100 + x/100)

Now the percentage decrease

= (p(100 + x /100) - p) / p (100 + x/100 ) x 100 = ( x/100+ x  * 100) %

In other words (i.e., in terms of fraction) if a value is increased by n/d then to get back the same number p from the resultant value, we have to decrease the increased value by ( n/ d+n).

Example 1:

Salary of arun in 2001 was 100rs per day and his salary in 2002 was 125rs per day. Again in 2003 his salary was rs 100 per day.

i) what is the percentage increase in his salary in 2002?
ii) what is the percentage decrease in his salary in 2003 over 2002?

Solution:

i) 125 - 100 /100  x 100 = 25%

ii) 125 - 100/125 x 100 = 20%

In fact is same absolute change but percentage change is different due to different denominator (i.e., base change). So the base change is as much important as the numerator or absolute change in quantity.


Example 2: 

Age of arun is 20 years. If the udhay age is 25% greater than that of arun then how much per cent arun age is less than udhay age?

Solution:

Age of arun                                    age of udhay

20 -------------+25%------------------->     25
     <-------------(-20%)-----------------

Solution using percentage change formula
here x = 25, then

percentage decrease (or less)  =  25/125 x 100 =20%

Solution using fractional change formula

since increase = 1/4                        ( 25% = 1/4)

Therefore decrease = 1/4+1 = 1/5 = 20%


Example 3

Arun has 33.33% more pencils than udhay has. By how much per cennt less pencils udhay has than that of arun?

Solution:

arun --------33% = 1/3-------------------> udhay
        <------(-25% = 1/4 = 1/3+1 )--------------

so udhay has 25% less pencils than arun has.

Percentage Change and Percentage Point change

Percentage Change and Percentage Point change

            Last year arun slaray was rs.10,000 and aruna huberta salary was rs.8,000. This year arun salary is rs. 12,000 while huberta salary is rs.10,000.

1.      I) What is the percentage increase of arun salary?

Ii) What is the percentage increase of huberta salary?

iii) Percentage increase in huberta salary is how much percent greater than the percentage increase in arun salary?

iv) What is the percentage point change in the salary of huberta and arun?

Solution:

i)                    12,000-10,000/10,000 x 100  = 20%

Or 2/10 --à 1/5 = 20%

ii)                  2/8 = 1/4 = 25%

iii)                Percentage increase of huberta salary = 25

Percentage increase of arun salary = 20

So the required percentage = 25-20/20 x 100 = 25%

It means percentage increase of huberta salary is 25% greater than the percentage increase of arun salary.

iv)                Percentage point change = (Percentage increase in huberta salary – Percentage increase in arun salary)

= 25-20 = 5 percentage point

Infact percentage point change is the difference between two percentage values.

Percentages (increase / decrease in quantity)

Some Most Important Values

1/2 =50%                    2/5=40%

1/3 = 33.33%              3/5=60%

2/3 = 66.66%              1/6=16.66%

1/4=25%                     5/6 =83.33%

3/4 =75%                    1/5 = 20%

 Example 1:

The height of arun some times ago was 110cm. Now his height is 120cm. Find the percentage change in his height.

Solution:

120-110/110 x 100 = 9.09%

Shortcut 

10/110 = 1/11 = 9.09%

Alternatively : 12/11 = 109.09%, so increase = 9.09%

Example 2

The total expenses of a hostel were rs.8000 per month. Some students left the hostel due to which the new expenses come down by rs.1000. Find the percentage decreases in expenses of the hostel.

Solution:

1000/8000 = 1/8 = 12.5%

Example 3

Salary of raja is rs.9000 per month and salary of rani is rs. 10,000 per month.

             i) What per cent is the salary of rani to that of raja?

            ii) What per cent is the salary of raja to that of rani?

Solution:

i) 10,000/9,000 x 100 = 111.11%

Alternatively:

10/9 = 9/9 + 1/9 = 100% + 11.11% = 111.11%

ii) 9/10 = 90%                                ( 1/10 = 10%)

Hence, salary of raja is 90% to the salary of rani.