Formulae and Questions of Percentage

1.    Concept of percentage or Percentage formula

To express x% as a fraction: We have, x% x/100 Thus, 30%=30/100= 3/10

In simple terms we can conclude that percentage symbol "%" means 1/100

In case we are having a fraction and we have to calculate its percentage then we will multiply it      by 100.

that is, 

a/b=(a/b)×100)%

2.    Increase or decrease in percentage with price 
(1) If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure will be = [R/ (100+R)×100]%

( 2) If the price of a commodity decreases by R%, then the increase in consumption so as not to decrease the expenditure will be = [R/(100−R)×100]

3.    Results on population
Formula's on result of population are very important when we have to calculate the population n years after or n years before,
Let the population of a town be P now and suppose it increases at the rate of R% per annum, then
(1) Population after n years =P(1+R/100)ⁿ

(2)  Population before n years =P/(1+R/100)ⁿ

4.    Results on Depreciation
We know that value of a machine depreciate with time, so it will decrease with the time. To calculate the value of machine after n years or before n years, we use these formula's

Let the present value of a machine be P. Suppose it depreciates at the rate of R% per annum. (1) Population after n years   = P(1−R/100)ⁿ

(2) Population before n years =P/(1−R/100)ⁿ

5.    Important results for Percentage
The formulas we are going to mention below is same as of increase or decrease in consumption with increase or decrease in the commodity price, just here we are in a bit different context.
(1) If A is R% more than B, then B is less than A by = [R/(100+R)×100]%

(2) If A is R% less than B, then B is more than A by = [R/(100−R)×100]%

Some Questions:- 

1.    Two numbers are less than third number by 30% and 37% respectively. How much percent is the second number less than by the first
(1) 20%
(2) 30 %
(3) 10%
(4) 40%

2.    In an examination, 34% of the students failed in mathematics and 42% failed in English. If 20% of the students failed in both the subjects, then find the percentage of students who passed in both the subjects.

(1) 40%                 (2) 41%

(3) 43%                 (4) 44%

3.    In the new budget , the price of kerosene oil rose by 25%. By how much percent must a person reduce his consumption so that his expenditure on it does not increase ?

(1) 20%                 (2) 30%

(3) 22%                 (4) 24%

4.    The value of a machine depreciates at the rate of 10% per annum. If its present is Rs.1,62,000.? What was the value of the machine 2 years ago ?

(1) 200000             (2) 20000

(3) 2000                 (4) None of these

5.    A student multiplied a number by 3/5 instead of 5/3. What is the percentage error in the calculation?

(1) 60%                 (2) 62%

(3) 64%                 (4) 65%

 ANSWERS WITH EXPLANATIONS:

1.    (3) : Let the third number is x.
then first number = (100-30)% of x
= 70% of x = 7x/10

Second number is (63x/100)
Difference = 7x/10 - 63x/100 = 7x/10
So required percentage is, difference is what percent of first number
=> (7x/100 * 10/7x * 100 )% = 10%

2.    (4) : Failed in mathematics, n(A) = 34Failed in English, n(B) = 42

n(AUB) = n(A) + n (B) – n (A∩ B )

= 34 + 42- 20 = 56
Failed in either or both subjects are 56

Percentage passed = (100−56)%=44%

3.    (1) : Reduction in consumption = [((R/(100+R))*100]%? [(25/125)*100]%=20%.

4.    (1): Value of the machine 2 years ago
= Rs.[162000/(1-(10/100)²)]=Rs.[162000*(10/9)*(10/9)]= Rs.200000

5.    (3) : Let the number be x

Then, error =  (5/3)x – (3/5)x =(16/15)x

Error% = [(16x/15) /(5x/3)] * 100% = 64%

Give your answer in the comment section along with your timing and judge where you stand in the competition. 

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Important Quant Quiz - Pipes and Cisterns

1.  Two pipes A and B can fill a tank in 36 hours and 45 hours respectively. If both the pipes are  opened simultaneously, how much time will be taken to fill the tank?
      (1) 20 hours                       (2) 30 hours

(3) 15 hours                       (4) 25 hours

2.    Two pipes can fill a tank in 10hours and 12 hours respectively while a third, pipe empties the full tank in 20 hours. If all the three pipes operate simultaneously, in how much time will the tank be filled?

(1) 8 hrs 30 min                 (2) 7 hrs 30 min

(3) 8  hrs 38 min                (4) 7 hrs 38 min

3.    If two pipes function simultaneously, the reservoir will be filled in 12 hours. One pipe fills the reservoir 10 hours faster than the other. How many hours does it take the second pipe to fill the reservoir?

(1) 10 hours                       (2) 40 hours

(3) 30 hours                       (4) 20 hours

4.    Two pipes A,B can fill a tank in 24 min. and 32 min. respectively. If both the pipes are opened simultaneously, after how much time B should be closed so that the tank is full in 18 min.?

(1) 8 min                            (2) 12 min

(3) 15 min                          (4) 20 min

5.    Two pipes A and B can fill a tank in 36 min. and 45 min. respectively. A water pipe C can empty the tank in 30 min. First A and B are opened. After 7 min, C is also opened. In how much time, the tank is full?
(1) 39 min                          (2) 40 min

(3) 42 min                          (4) 38 min

6.    Two pipes can fill a cistern in 14 hours and 16 hours respectively. The pipes are opened simultaneously and it is found that due to leakage in the bottom it took 32 minutes more to fill the cistern. When the cistern is full, in what time will the leak empty it?
(1) 112 min                        (2)  115 min

(3) 120 min                        (4) 110 min

7.     An electric pump can fill a tank in 3 hours. Because of a leak in , the tank it took 3 1/2 hours to fill the tank. If the tank is full, how much time will the leak take
to empty it?

(1) 28 hours                       (2) 25 hours

(3) 21 hours                       (4) 20 hours

8.    A cistern has two taps which fill it in 12 minutes and 15minutes respectively. There is also a waste pipe in the cistern. When all the 3 are opened, the empty cistern is full in 20 minutes. How long will the waste pipe take to empty the full cistern?

(1) 20min                           (2) 40 min

(3) 30 min                          (4) 10 min

9.    Three pipes A, B and C can fill a tank in 6 hours. After working at it together for 2 hours, C is closed and A and B can fill the remaining part in 7 hours. The number of hours taken by C alone to fill the tank is:

(1) 10                                 (2) 14

(3) 12                                 (4) 12

10.  A tap can fill a tank in 6 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

(1) 3 hours 15 min             (2) 3 hours 45 min

(3) 3 hours 40 min             (4) 3 hours 50 min

ANSWERS WITH EXPLANATIONS:

1.    (1)

Explanation :

Part filled by A in 1 hour = (1/36);
Part filled by B in 1 hour = (1/45);
Part filled by (A + B) In 1 hour =(1/36)+(1/45)=(9/180)=(1/20)
Hence, both the pipes together will fill the tank in 20 hours.

2.    (2)

Explantion:

Net part filled In 1 hour =(1/10)+(1/12)-(1/20)=(8/60)=(2/15).
The tank will be full in 15/2 hrs = 7 hrs 30 min.

3.    (3)

Explanation:

let the reservoir be filled by first pipe in x hours.
Then ,second pipe fill it in (x+10)hrs.
Therefore (1/x)+(1/x+10)=(1/12)

 (x+10+x)/(x(x+10))=(1/12).
x^2 –14x-120=0

(x-20)(x+6)=0
x=20 [neglecting the negative value of x]
so, the second pipe will take (20+10)hrs. (i.e) 30 hours to fill the reservoir

4.    (1)

Explantion:

let B be closed after x min. then ,
Part filled by (A+B) in x min. +part filled by A in (18-x)min.=1
Therefore x*((1/24)+(1/32))+(18-x)*(1/24)=1

(7x/96) + ((18-x)/24)=1.
7x +4*(18-x)=96.
Hence, be must be closed after 8 min.

5.    (1)

Explanation:

Part filled in 7 min. = 7*((1/36)+(1/45))=(7/20).
Remaining part=(1-(7/20))=(13/20).
Net part filled in 1min. when A,B and C are opened=(1/36)+(1/45)-(1/30)=(1/60).
Now,(1/60) part is filled in one minute.
(13/20) part is filled in (60*(13/20))=39minutes.

6.    (1)

Explanation:

Work done by the two pipes in 1 hour =(1/14)+(1/16)=(15/112).
Time taken by these pipes to fill the tank = (112/15)hrs = 7 hrs 28 min.
Due to leakage, time taken = 7 hrs 28 min + 32 min = 8 hrs
Work done by (two pipes + leak) in 1 hour = (1/8).
Work done by the leak m 1 hour =(15/112)-(1/8)=(1/112).
Leak will empty the full cistern in 112hours.

7.    (3)

Explanation:  

work done by the leak in 1 hour=(1/3)-(1/(7/2))=(1/3)-(2/7)=(1/21).
The leak will empty .the tank in 21 hours.

8.    (4)

Explanation:

Work done by the waste pipe in 1min
=(1/20)-(1/12)+(1/15) = -1/10 [negative sign means emptying]
therefore the waste pipe will empty the full cistern in 10min

9.    (2)

Explanation: (A+B+C) - (A+B) can give you the answer.
(A+B+C) =1/6 and (A+B+C) in 2hrs=2/6 and remaining part 1-2/6=2/3.
So (A+B) in 7 hrs is 2/ (3*7) =2/21.
1/6-2/21=1/14 so answer is 14.

10.  (2)

Explanation:

A tap can fill a tank in 6 hours.
After half the tank is filled i.e. after 3 hrs.
three more similar taps are opened i.e.
no. of taps to fill remaining half tank = 4 taps
when 1 tap is used = (3 hrs/1 tap)
when 4 taps are used = (3 hrs/4 taps)=45min
total= 3hrs+45min

Give your answer in the comment section along with your timing and judge where you stand in the competition. 

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