This material prepared and compiled by Akula. Praveen
Kumar, SPM, Papannapet Sub Office502302, Medak
Division, Andhra Pradesh, Mob.no. 9849636361,
8019549939
Disclaimer: All Material/Questions/Information provided in this post are Compiled by A. Praveen Kumar for in good faith of Departmental Employees. The types of questions, number of questions and standard of questions may be vary in actual examination. This is my predictions only. Author of blog does not accepts any responsibility in relation to the accuracy, completeness, usefulness or otherwise, of the content
PERCENTAGE
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Percent literally, means ‘for every 100’ and is derived from the French word ‘cent’ for 100. The basic utility of Percentage arises from the fact that it is one of the most powerful tools for comparison of numerical data and information. It is also one of the simplest tools for comparison of data. In the context of business and economic performance, it is specifically useful for comparing data such as profits, growth rates, performance, magnitudes and so on.
The concept of percentage mainly applies to ratios and the percentage value of a ratio is arrived at by multiplying by 100 the decimal value of the ratio.
Let us understand the concept of percentage by an example. If a student scores 20 marks out of a maximum possible 30 marks, his/her marks can then be denoted as 20 out of 30. That is,
20 out of 30 = (20/30) = (20/30) * 100% = 66.66%
The process for getting this is perfectly illustrated through the unitary method:
Marks got

Out of
 
20

——–>

30

x

——–>

100

Then the value of x * 30 = 20 * 100
x = (20/30) * 100 —> the percentage equivalent of a ratio.
Expression of percent in different ways
(a) Percent as a fraction
To covert a fraction into percentage we have to multiply the fraction by 100 and put the percent sign (%).
Thus, 2/3 = (2/3) * 100 % = 66 2/3%
(b) Percent as a decimal
To convert a given percent in decimal form, we express it as a fraction with denominator as 100.
Thus, 47% = 47/100 = 0.47.
(c) Percent as a ratio
A percent can be expressed as a ratio with its second term 100 and the first term equal to the given percent.
For example, 7% = 7/100 = 7 : 100.
Expressing one quantity as a percentage of another quantity
Let x & y be two numbers and we have to find what percent of x is y. Let a% of x be equal to y. Then,
x * (a/100) = y => a = (y/x) * 100
Thus, y is ((b*y)/x) * 100 of a.
Multiplying Factor
Whenever we have to find the final/new quantity with the help of initial/old quantity we use the term Multiplying Factor.
Final Quantity = Initial Quantity * Multiplying Factor
1.

Concept of Percentage :
By a certain percent, we mean that many hundredths. Thus, x percent means x hundredths, written as x%.
I. To express x% as a fraction: We have, x% = x / 100.
Thus, 48% = 48 / 100 = 12 / 25.
II. To express a / b as a percent :
We have a / b = (a / b × 100)%.

3.

2. 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) ×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) ×100]%.

Results on Population :
4. 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 :
5. 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}.

SHORT CUT METHODS PERCENTAGES
1. If x% is deducted on tax and y% of the remaining is spent on
education and still there is a balance, the formula is :
Balance * [ 100/(100x) ] * [ 100/(100y) ] * [ 100/(100z) ]
2. The population of a town is 'P'. It increased by x% during Ist year,
increased by y% during II nd and again increased by z% during IIIrd. The
population after 3 years will be,
P * [ (100+x)/100 ] * [ (100+y)/y ] * [ (100+z)/100 ]
3. % of effect
(i) Inc of x% Dec of x% xy[(x*y)/100]
(ii) Inc of x% Inc of y% (x+y)+[(x*y)/100]
(iii) Dec of x% Inc of y% [x2/100]
(iv) Dec of x% Dec of y% (xy)+[(x*y)/100]
(v) Inc of x% Dec of x% [x2/100]
(vi) Inc of x% Inc of x% 2*x+[x2/100]4. (i) If the sides of the triangle, rectangle, square, circle, rhombus etc
is increased by x%. Its area is increased by
2x+(x2/100)
(ii)If decreased x%.Its ares is decreased by,
2x+(x2/100)
5. In an examination x% failed in Hindi and y% failed in Science, if z%
of the candidates failed in both of the subjects. The percentage of
students who passed in both of the subjects is,
100(x+yz)
6. If A's income is r% more than B's income, the B's income is less
than A's income by
(r/100+r) * 100%
7. If A's income is r% less than B's income, then B's income is more
than A's income by
(r/100r) * 100
8. (i) If the price of commodity increases by r% then reduction in
consumption so as not to increase the expenditure is
(r/100+r) * 100
(ii)If the price of commodity decreases by r% then,
(r/100r)*100
9. If the population of town (or) length of a tree is 'p' and its annual
increase is r% then,
(i)population (or) length of a tree after 'n' years is,
p[1+(r/100)]
(ii)population (or) length of a tree 'n' years ago is,
p/[1+(r/100)n]
10. If the population of town (or) value of a machine is 'p' and annual
decrease is r% then,
(i)population (or) value of machine after 'n' years is,
p[1(r/100)n]
(ii)population (or) value of a machine 'n' years ago is,
p/[1(r/100)n]
11. If 'A' is x% of 'C' and 'B' is y% of 'C' then 'A' is
(x/y) * 100% of 'B'.
12. If two values are respectively x% and y% more than a third value,
then the first is
[(100+x) / (100+y)] * 100%
of second
13. Total no. of votes =
(Difference in votes/Difference in %) * 10014. Maximum marks =
[(pass marks/pass %) * 100]
15.Total marks =
(Difference in marks / Difference in %)*100
16. (i)Reduced rate =
[(Amount/Quantity more) * (Reduction % /100)]
(ii)Original rate (or) previous rate =
[(Amount/Quantity more) * (Reduction % /100reduction%)]
17. (i)Increased rate =
[(Amount/Quantity less) * (increase % /100)]
(ii)Original rate (or) previous rate =
[(Amount/Quantity less) * (Increase % /100increase%)]
18. If the numerator of fraction is increased by x% and its denominator
is diminished by y% ,the value of the fraction is A/B. Then the original
fraction is, (A/B) * [(100y) / (100+x)]
education and still there is a balance, the formula is :
Balance * [ 100/(100x) ] * [ 100/(100y) ] * [ 100/(100z) ]
2. The population of a town is 'P'. It increased by x% during Ist year,
increased by y% during II nd and again increased by z% during IIIrd. The
population after 3 years will be,
P * [ (100+x)/100 ] * [ (100+y)/y ] * [ (100+z)/100 ]
3. % of effect
(i) Inc of x% Dec of x% xy[(x*y)/100]
(ii) Inc of x% Inc of y% (x+y)+[(x*y)/100]
(iii) Dec of x% Inc of y% [x2/100]
(iv) Dec of x% Dec of y% (xy)+[(x*y)/100]
(v) Inc of x% Dec of x% [x2/100]
(vi) Inc of x% Inc of x% 2*x+[x2/100]4. (i) If the sides of the triangle, rectangle, square, circle, rhombus etc
is increased by x%. Its area is increased by
2x+(x2/100)
(ii)If decreased x%.Its ares is decreased by,
2x+(x2/100)
5. In an examination x% failed in Hindi and y% failed in Science, if z%
of the candidates failed in both of the subjects. The percentage of
students who passed in both of the subjects is,
100(x+yz)
6. If A's income is r% more than B's income, the B's income is less
than A's income by
(r/100+r) * 100%
7. If A's income is r% less than B's income, then B's income is more
than A's income by
(r/100r) * 100
8. (i) If the price of commodity increases by r% then reduction in
consumption so as not to increase the expenditure is
(r/100+r) * 100
(ii)If the price of commodity decreases by r% then,
(r/100r)*100
9. If the population of town (or) length of a tree is 'p' and its annual
increase is r% then,
(i)population (or) length of a tree after 'n' years is,
p[1+(r/100)]
(ii)population (or) length of a tree 'n' years ago is,
p/[1+(r/100)n]
10. If the population of town (or) value of a machine is 'p' and annual
decrease is r% then,
(i)population (or) value of machine after 'n' years is,
p[1(r/100)n]
(ii)population (or) value of a machine 'n' years ago is,
p/[1(r/100)n]
11. If 'A' is x% of 'C' and 'B' is y% of 'C' then 'A' is
(x/y) * 100% of 'B'.
12. If two values are respectively x% and y% more than a third value,
then the first is
[(100+x) / (100+y)] * 100%
of second
13. Total no. of votes =
(Difference in votes/Difference in %) * 10014. Maximum marks =
[(pass marks/pass %) * 100]
15.Total marks =
(Difference in marks / Difference in %)*100
16. (i)Reduced rate =
[(Amount/Quantity more) * (Reduction % /100)]
(ii)Original rate (or) previous rate =
[(Amount/Quantity more) * (Reduction % /100reduction%)]
17. (i)Increased rate =
[(Amount/Quantity less) * (increase % /100)]
(ii)Original rate (or) previous rate =
[(Amount/Quantity less) * (Increase % /100increase%)]
18. If the numerator of fraction is increased by x% and its denominator
is diminished by y% ,the value of the fraction is A/B. Then the original
fraction is, (A/B) * [(100y) / (100+x)]
19. If A is R% more than B, then B is less than A by R / (100+R) * 100
20. If A is R% less than B, then B is more than A by R / (100R) * 100
21. If the price of a commodity increases by R%, then reduction in consumption, not to increase the expenditure is : R/(100+R)*100
22. If the price of a commodity decreases by R%, then the increase in consumption, not to decrease the expenditure is: R/(100R)*100
PERCENTAGES – FRACTIONS CONVERSIONS:
For faster calculations we can convert the percentages or decimal equivalents into
their respective fraction notations. The following is a table showing the conversions
of percentages and decimals into fractions:
Percentage Decimal Fraction
10% 0.1 1/10
12.5% 0.125 1/8
16.66% 0.1666 1/6
20% 0.2 1/5
25% 0.25 1/4
30% 0.3 3/10
33.33% 0.3333 1/3
40% 0.4 2/5
50% 0.5 1/2
60% 0.6 3/5
62.5% 0.625 5/8
66.66% 0.6666 2/3
70% 0.7 7/10
75% 0.75 3/4
80% 0.8 4/5
83.33% 0.8333 5/6
90% 0.9 9/10
100% 1.0 1
IMPORTANT POINTS TO NOTE:
When any value increases by
10%, it becomes 1.1 times of itself. (since 100+10 = 110% = 1.1)
20%, it becomes 1.2 times of itself.
36%, it becomes 1.36 times of itself.
4%, it becomes 1.04 times of itself.
Thus we can see the effects on the values due to various percentage increases.
When any value decreases by
10%, it becomes 0.9 times of itself. (Since 10010 = 90% = 0.9)
20%, it becomes 0.8 times of itself
36%, it becomes 0.64 times of itself
4%, it becomes 0.96 times of itself.
Thus we can see the effects on a value due to various percentage decreases.
Note: 1. When a value is multiplied by a decimal more than 1 it will be increased and
When multiplied by less than 1 it will be decreased.
2. The percentage increase or decrease depends on the decimal multiplied.
Example: When the actual value is x, find the value when it is 30% decreased.
Solution: 30% decrease => 0.7 x.
Example: If 600 is decrease by 20%, what is the new value?
Solution: new value = 0.8 * 600 = 480. (Since 20% decrease)
Thus depending on the decimal we can decide the % change and vice versa.
Example: When a value is increased by 20%, by what percent should it be reduced
to get the actual value?
Solution: (It is equivalent to 1.2 reduced to 1 and we can use % decrease formula)
%decrease = (1.2–1/ 1.2 )* 100 = 16.66%
When a value is subjected multiple changes, the overall effect of all the changes can
be obtained by multiplying all the individual factors of the changes.
Example: The population of a town increased by 10%, 20% and then decreased by
30%. The new population is what % of the original?
Solution: The overall effect = 1.1 * 1.2 * 0.7 (Since 10%, 20% increase and 30%
decrease)
= 0.924 = 92.4%.
Example: Two successive discounts of 10% and 20% are equal to a single discount
of ___
Solution: Discount is same as decrease of price.
So, decrease = 0.9 * 0.8 = 0.72 => 28% decrease (Since only 72% is remaining)
POINTS TO REMEMBER
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 100
For example,
20percent = 20% = 20 * (1 /100) =15
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.
By a certain percent, we mean that many hundredths. Thus x percent means x
Hundredths, written as x%.
To express x% as a fraction: We have , x% = x /100.
Thus,20% = 20/100 =15
48% = 48/100 = 12 /25
To express a/b as a percent: We have a / bx100%
Thus, 1/ 4 = ¼ X 100= 25%;
0.6 = 6/10 =3/5
3/5X 100% = 60%.
A quick reminder about percentages
We truly hope you are familiar with the following percentage formula:
% = Fraction X 100
This formula allows us to alternate between fractions and their percent form. Let's take a look at the following example:
If we decide to put 25 on the % side, we get the following equation:
25% = fraction X 100
If we divide the equation by 100, we get:
25/100 = 1/4 = Fraction
Therefore, 25% is merely another way of presenting 1/4. This formula is sometimes elaborated to include the components of a fraction (nominator and denominator), but for our purposes the formula above is satisfactory.
Percentages and decimal numbers
Many people find it easier to calculate percent changes using decimal numbers. Since a percent is actually a fraction, 1% can be written as 0.01. Therefore, increasing a number by 1% means multiplying it by 1+0.01= 1.01, and decreasing a number by 0.01 percent means multiplying it by 10.01=0.99. Thus, a percent increase means multiplying by numbers greater than one, and a percent decrease means multiplying by numbers that are smaller than one.
How to calculate % changes without the calculator's % function
To speed up the calculation process, we shall use a different format of the above formula. From now on, when asked to calculate a percent increase in value, use the following formula:
% = [(value after change/value before change)  1] X 100
When asked to calculate a decrease in value use this formula and then multiply by (1), or use the +\ sign in your calculator. Take a look at the following simplified example:
"The price of X was 30 and is now 40. What is the percent difference between the two prices..."
In this case it is clear that we are looking for an increase change, so we are looking for the ratio between 40 and 30, which constitutes the fraction in our formula:
[(40/30)  1] X 100 = 33.33%
If you feel comfortable with numbers you can always skip the multiplication by 100 to get 0.33, and thus conclude that this represents 33%. Here is another example with a decrease in value:
"The price of X was 40 and is now 30. What is the percent difference between the two prices..."
In this case, the new number has decreased in respect to the original. If we use the same formula, we get a negative number:
[(30/40) 1] X 100 = ( 25)%
This still represents the true absolute value we are looking for, so you can simply multiply by (1) or use the +\ sign in your calculator to get the correct answer. In theory, we could also switch places in the formula:
% = [1 (value before change/value after change)]X 100
However, this formula can slow down the calculation if we are using the simplest calculator, as it will require us for two steps because a subtraction precedes a multiplication.
EXERCISE
1) a is what percent of b? What % is a of b? What % of b is a?
sol: a/b x 100
To calculate percent the example problem is given below.
Ex: 50 is what percentage of 100
Sol: (50/100) x 100 = 50%
2) If a is greater than b, then in terms of % , a is more than b by (ab)/b x 100
If a is less than b then in terms of % , a is less than b by (ba)/b x 100
3) If a% and b% are two successive changes, then the overall change = a+b+(ab)/100%
Note: Use '+' for increase and '' for decrease
4) % of increase = (Increase/Initial value) x 100
% of decrease = ( Decrease/Initial value) x 100
5. What percentage of 70 is 28?
Sol: The problem means 28 is how much percent in 70 so we have to calculate percent for every hundred
Percentage = 28/70 x 100 = 40%
6. 60 is what percent of 240?
Sol: 60/240 x 100 = 25%
7. 24% of 450 = 12% of 'a' then what is value of 'a'?
Sol: 24/100 x 450 = 12/100 x a
a = (24 x 450)/12 = 900 :( both 100's are cancelled)
8. 40% of a number is 160 then 32% of it is?
Sol 1: Let’s take the number as 'a'
40/100 x a = 160
a = (160 x 100)/40 = 400
So 32% of a = 400 x 32/100 = 128
Sol 2: 40% of number = 160 then
32% of number = ?
32/40 x 160 = 128
9. A number decreased by 20% gives 72 the number is?
Sol: 100% of a number decreased by 20% then the number is 80%
Lets take the number as 'a'
80% of 'a' = 80/100 x a = 72
'a' = (72 x 100)/80
= 90
10. 30% of a number when added to 77 the result is the number itself. the number is?
Sol: Lets take the number as 'a'
30/100 x a +77 = a (30a + 7700)/100 = a 30a + 7700 = 100a 70a = 7700 a = 7700/70 = 110 11. Madhu saves 7% of his income. what is his expenditure if his income is rs.11,400? Sol: Less the savings in income the expenditure is 93% expenditure = 93/100 x 11,400 = 10602 12. In a school 8% of the students failed in exam what is the number of students passed in the exam if 1600 appeared for it? Sol: Passed students = 92/100 x 1600 = 1472 13. Madhu spends 11% on house rent, 18% on kids education, 21% for food, 16% on clothes and 5% on transport. What is his monthly savings if his income is 13000? Sol: calculate all his expenditures = 11+18+21+16+5 = 71% Less the expenditure from his income. So the savings is 29% Savings = 29/100 x 13000 = 3770 14. 'A' is 25% more than 'B' by how much percent 'B' is less than 'A'? Sol: lets the value of 'B' is 100% the value of 'A' is increased 100% to 125% Difference between 'A' and 'B' is 25% 'B' s reduced percent than 'A' = (Difference/Value of 'A') x 100 = 25/125 x 100 = 20% 15. The price of phone is less than the price of the bag by 20% what % the price of bag is more than the price of phone? Sol: Lets take price of bag is 100% then price of phone is 80% Difference between price bag and price of phone is 20% The price of bag more then price of phone = 20/80 x 100 = 25% 16. Two numbers are 25% and 40% less than the third number. By how much percentage the first number more then the second number?Sol: If you take third number as 100%, the first number is 75%, the second number is 60%. Lets find First number is more then the second number in percentage = 15/60 x 100 = 25%
17. "A car's value was reduced by 10% and is now worth £900. What was its original price. "
SoL: The car is now worth only 0.9 of its original value, as it experienced a 10% decrease. According to
[(900) / (0.9)] = 1000
18. A car's value was increased by 10% and is now worth £550. What was its original price?
[(550)/(1.1)] = 500
19. Subtracting 40% of a number: from the number, we get the result as30. The number is:
20. 12.5% of 192=50% of?.
21. What percent is 3% of 5% ?
22. 45% of ?+30% of 90 = 30% of 210
23. If 75% of a number is added to 75, the result is the number itself. Then, the number is
24. One fourth of one third of two fifth of a number is 15. What will be40% of that number
25.13937.869 ÷ 199.54 + 15% of 201 = ?
26. The population of a town has increased from 133575 to 138918. The percent increase in population is
27.218% of 1674=? x 1800
28.If 37% of a number is 990.86, what will be approximately 19% of that number
Sol: Let 37% of x = 990.86 then 37/100 * x = 990.86 so x = 2678
Now 19% of 2678 = 19/100 * 2678 = 508.82 Sol: Total money = Rs. 300. Saving of the lady = 10% of 300 = 30/ With 30/ she bought 5 kg sugar => each kg costs Rs. 6/ 30. From a 20lt solution of Salt and water with 20% salt, 2lt of water is evaporated. Find the new % concentration of salt Sol: In 20lt, salt = 20% => 4 lt. New volume = 18 lt (2 lt evaporated) So, new % = 4/18 X 100 = 22.22%
31. In a list of weights of candidates appearing for police selections, the weight of A is marked as 58 kg instead of 46.4 kg. Find the percentage of correction required.
Sol: % correction = (5846.4)/58 X 100 = 20%
32. A person spends 20% of his income on rent, 20% of the rest on food, 10% of the remaining on clothes and 10% on groceries. If he is left with Rs. 9520/ find his income.
Sol: Three successive decreases of 20%, 20% and 10% => 0.8 X 0.8 X 0.9 = 0.576Again 10% decrease => 0.576 – 0.1 = 0.476. So, 0.476 x = 9520 => x = 20000
33. A shopkeeper offers three successive discounts of 10%, 20% and 30% to a customer. If the actual price of the item is Rs. 10000, find the price the customer has to pay to the shopkeeper.
Sol: Total discount = 0.9 X 0.8 X 0.7 = 0.504 of actual price.So, price = 0.504 X 10000 = 5040 34. 1100 boys and 700 girls are examined in a test; 42% of the boys and 30% of the girls pass. The percentage of the total who failed is:
35. 218% of 1674 = ? x 1800
36.10% of the votes did not cast their vote in an election between two candidates. 10% of the votes polled were found invalid. The successful candidate got 54% of the valid votes and won by a majority of 1620 votes. The number of votes enrolled on the voters list was:
37. The difference to two numbers is 20% of the larger number. If the smaller number is 20, then the larger number is:
38. While purchasing one item costing Rs. 400, I had to pay the sales tax at 7% and on another costing Rs. 6400, the sales tax was 9%. What percent of the sales tax I had to pay, taking the two items together on an average?
39. Which of the following multipliers will cause a number to be increased by 29.7%?
40. 65% of ? = 20% of 422.50
41. What percentage of 7.2 kg is 18 gms?
65.In an election between two candidates, one got 55% of the total valid votes, 20% of the votes were invalid. If the total number of votes was 7500, the number of valid votes that the other candidate got, was
 
 
66. Gauri went to the stationers and bought things worth Rs. 25, out of which 30 paisa went on sales tax on taxable purchases. If the tax rate was 6%. then what was the cost of the tax free items ?
 
 
67. In a competitive examination in State A, 6% candidates got selected from the total appeared candidates. State B had an equal number of candidates appeared and 7% candidates got selected with 80 more candidates got selected than A. What was the number of candidates appeared from each State?
 
 
68.A housewife saved Rs. 2.50 in buying an item on sale. If she spent Rs. 25 for the item, approximately how much percent she saved in the transaction?
 
 
69.Two students appeared at an examination. One of them secured 9 marks more than the other and his marks was 56% of the sum of their marks. The marks obtained by them are
 
 
70. The price of a car is Rs. 3, 25,000. It was insured to 85% of its price. The car was damaged completely in an accident and the insurance company paid 90% of the insurance. What was the difference between the price of the car and the amount received?
 
Sol:
 

71. The population of a town increased from 1, 75,000 to 2, 62,500 in a decade. The average percent increase of population per year is
 
 
72.

A student has to obtain 33% of the total marks to pass. He got 125 marks and failed by 40 marks. The maximum marks are
 
 
73

A fruit seller had some apples. He sells 40% apples and still has 420 apples. Originally, he had
 
 
74.

270 students appeared for an examination, of which 252 passed. The pass percentage is
 
 
75.

Raman's salary was decreased by 50% and subsequently increased by 50%. How much percent does he loss?
 
 
76.

How many litres of pure acid are there in 8 litres of a 20% solution?
 
 
77. Gauri went to the stationers and bought things worth Rs. 25, out of which 30 paise went on sales tax on taxable purchases. If the tax rate was 6%, then what was the cost of the tax free items?
 
Let the amount taxable purchases be Rs. x.
Then, 6% of x =

30

100

30

x

100
 
100

6

78. Two tailors X and Y are paid a total of Rs. 550 per week by their employer. If X is paid 120 percent of the sum paid to Y, how much is Y paid per week?

Let the sum paid to Y per week be Rs. z.
Then, z + 120% of z = 550.
120

z = 550
 
100

11

z = 550
 
5

550 x 5
 
11
 
79.Three candidates contested an election and received 1136, 7636 and 11628 votes respectively. What percentage of the total votes did the winning candidate get?

Total number of votes polled = (1136 + 7636 + 11628) = 20400.
11628

x 100
 
20400

80.If 20% of a = b, then b% of 20 is the same as:
 
20% of a = b

20

a = b.

100

b

x 20

=

20

a x

1

x 20

=

4

a = 4% of a.
 
100

100

100

100

81. In an election between two candidates, one got 55% of the total valid votes, 20% of the votes were invalid. If the total number of votes was 7500, the number of valid votes that the other candidate got, was:

Number of valid votes = 80% of 7500 = 6000.
=

45

x 6000

= 2700.
 
100

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