What is Percentage?
When we say "Percent" we are really saying "per 100", written as x%.
To express x% as a fraction:
We have, x% = x/100
Thus, 10% = 10/100 = 1/10
To express a/b as a percent:
We have, a/b = (a/b X 100)%.
Thus, 1/5 = (1/5 X 100)%. = 20%.
Percentage Increase/Decrease:
If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is:
[R/(100 + R) X 100]%
If the price of a commodity decreases by R%, then the increase in consumption so as not to decrease the expenditure is:
[R/(100  R) X 100]%
Results on Population:
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 n years ago =P/(1 +R/100)^{n}
Results on Depreciation:
Let the present value of a machine be P. Suppose it depreciates at the rate of R% per annum. Then:
1. Value of the machine after n years = P[1  R/100]^{n}
2. Value of the machine n years ago = P/(1  R/100)^{n}
3. If A is R% more than B, then B is less than A by = [R/(100 + R) X 100]%
4. If A is R% less than B, then B is more than A by = [R/(100  R) X 100]%
1. 

Let, last year the strength was 100.
This year it is 110.
Last year, it was (10/110) X 100% = (1/11) X 100% = 100/11%
i.e., 91/11% , less than this year.
2. 

Let B's weight be 100
A's weight is 90
C's weight is 110
So, A's weight is (20/110) X 100%
A's weight = (2/11) X 100% = (200/11)%
A's weight = (182/11)% less than that of C.
3. 

Case 1:
New scheme income  old scheme income = 400
3000 + 5/100(S  10000)  9/100(S) = 400
2500  0.04S = 400
S =2100/0.04
S = 52500
Case 2:
Old scheme income  new scheme income = 400
9/100(S)  [3000 + 5/100(S10000)] = 400
(0.04)S  2500 = 400
0.04S = 2900
S = 72500
Since we have two possible answers satisfying the required conditions, we cannot determine the answer uniquely.
4. 

The teachers contribution = 4 X 100 = 400
Students contribution = 5 X 30 = 150
The contribution of nonteaching staff = 2 X 25 = 50
Total contribution = Rs 600
Contribution of each student = (30/600) X 100%
Contribution of each student = (1/20) X 100% = (100/20)% = 5%
5. 

Suppose originally he had x apples.
Then, (100  40)% of x = 420
(60/100)x = 420
x = (420 X 100)/60 = 700 apples.
6. 

Let the cost of 50 papayas or 30 watermelons be Rs 150.(money with Hari)
So, each papaya costs Rs 3 and each watermelon costs Rs 5.
Let him buy x papayas and 9 watermelons for Rs 135 (90% of his money)
3x + 45 = 135
x = 30
7. 

Let the amount taxable purchases be Rs. x.
Then, 6% of x = 6/100
x = [30/100 X 100/6] = 5
Cost of tax free items = Rs. [25  (5 + 0.30)] = Rs. 19.70
8. 

10/100a = b/100 (200)
a/b = 20
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