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What is 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 every100 of the reference value
Why Percentage?
Percentage is a concept evolved so that there can be a uniform platform for comparisonof various things. (Since each value is taken to a common platform of 100.)
Eg: To compare three different students depending on the marks they scored we cannot directlycompare their marks until we know the maximum marks for which they took the test. But bycalculating percentages they can directly be compared with one another.
Before going deeper into the concept of percentage, let u have a look at some basics and tips forfaster calculations:
Prerequesties(Important facts and formulae):
1. Concept of Percentage: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 =1/5; 48% =48/100 =12/25, etc.To express a/b as a percent: We have, a/b =((a/b)*100)%.Thus, ¼ =[(1/4)*100] = 25%; 0.6 =6/10 =3/5 =[(3/5)*100]% =60%.
2. If the price of a commodity increases by R%, then the reduction in consumption so asnot toincrease the expenditure is[R/(100+R))*100]%.If the price of the commodity decreases by R%, then the increase in consumption so as todecrease the expenditure is[(R/(100-R)*100]%.
3. Results on Population: Let the population of the town be P now and suppose it increases at therate of R% per annum, then :1. Population after nyeras = P [1+(R/100)]^n.2. Population n years ago = P /[1+(R/100)]^n.
4. Results on Depreciation: Let the present value of a machine be P. Suppose it depreciates at therate
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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.
5. If A is R% more than B, then B is less than A by[(R/(100+R))*100]%.If A is R% less than B , then B is more than A by[(R/(100-R))*100]%.
Calculation of Percentage:
Percentage = (Value / Total value) X 100
Eg: 50 is what % of 200?Soln: Percentage = (50/200) X 100 = 25%.Calculation of Value:Value = (Percentage/100) X total value
Eg: What is 20% of 200?Soln: Value = (20/100) X 200Note: Percentage is denoted by “%”, which means “/100”.
Eg: What is the decimal notation for 35%?Soln: 35% = 35/100 = 0.35.For faster calculations we can convert the percentages or decimal equivalents into theirrespective fraction notations.
Percentages – Fractions Conversions:The following is a table showing the conversions of percentages and decimals into
fractions:
Percentage Decimal Fraction10% 0.1 1/1012.5% 0.125 1/816.66% 0.1666 1/620% 0.2 1/525% 0.25 1/430% 0.3 3/1033.33% 0.3333 1/340% 0.4 2/550% 0.5 1/260% 0.6 3/562.5% 0.625 5/866.66% 0.6666 2/370% 0.7 7/10
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75% 0.75 3/480% 0.8 4/583.33% 0.8333 5/690% 0.9 9/10100% 1.0 1
Similarly we can go for converting decimals more than 1 from the knowledge of the abovecited conversions as follows:
We know that 12.5% = 0.125 = 1/8Then, 1.125 = [8(1)+1]/8 = 9/8 (i.e., the denominator will add to numerator once, denominatorremaining the same.
Also, 2.125 = [8(2)+1]/8 = 17/8 (here the denominator is added to numerator twice)3.125 = [8(3)+1]/8 = 25/8 and so on.Thus we can derive the fractions for decimals more than 1 by using those les than 1.We will see how use of fractions will reduce the time for calculations:
Eg: What is 62.5% of 320?Soln: Value = (5/8) X 320 (since 62.5% = 5/8)= 200.
Percent change:
A change can be of two types – an increase or a decrease.When a value is changed from initial value to a final value,% change = (Difference between initial and final value/initial value) X 100Eg: If 20 changes to 40, what is the % increase?Soln: % increase = (40-20)/20 X 100 = 100%.
Note:If a value is doubled the percentage increase is 100.If a value is tripled, the percentage change is 200 and so on.
Percentage Difference:% Difference = (Difference between values/value compared with) X 100.
Eg: By what percent is 40 more than 30?Soln: % difference = (40-30)/30 X 100 = 33.33%(Here 40 is compared with 30. So 30 is taken as denominator)
Eg: By what % is 60 more than 30?Soln: % difference = (60-30)/30 X 100 = 100%.(Here is 60 is compared with 30.)Hint: To calculate percentage difference the value that occurs after the word “than” in thequestion can directly be used as the denominator in the formula.
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Important Points to Note:
When any value increases by10%, 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 100-10 = 90% = 0.9)20%, it becomes 0.8 times of itself36%, it becomes 0.64 times of itself4%, 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 whenmultiplied by less than 1 it will be decreased.
2. The percentage increase or decrease depends on the decimal multiplied.
Eg: When the actual value is x, find the value when it is 30% decreased.Soln: 30% decrease => 0.7 x.
Eg: A value after an increase of 20% became 600. What is the value?Soln: 1.2x = 600 (since 20% increase)ð x = 500.
Eg: If 600 is decrease by 20%, what is the new value?Soln: new value = 0.8 X 600 = 480. (Since 20% decrease)Thus depending on the decimal we can decide the % change and vice versa.
Eg: When a value is increased by 20%, by what percent should it be reduced to get the actualvalue?Soln: (It is equivalent to 1.2 reduced to 1 and we can use % decrease formula)% decrease = (1.2 – 1)/1.2 X 100 = 16.66%.
When a value is subjected multiple changes, the overall effect of all the changes can be obtainedby multiplying all the individual factors of the changes.
Eg: The population of a town increased by 10%, 20% and then decreased by 30%. The newpopulation is what % of the original?Soln: The overall effect = 1.1 X 1.2 X 0.7 (Since 10%, 20% increase and 30% decrease)= 0.924 = 92.4%.Eg: Two successive discounts of 10% and 20% are equal to a single discount of ___Soln: Discount is same as decrease of price.So, decrease = 0.9 X 0.8 = 0.72 => 28% decrease (Since only 72% is remaining)
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SOLVED EXAMPLES:
1. A batsman scored 110 runs which included 3 boundaries and 8 sixes. What percent of his totalscore did he make by running between the wickets?
A.45% B.455
%11
C.546
%11
D.55%
Explanation:Number of runs made by running = 110 - (3 x 4 + 8 x 6)= 110 - (60)= 50.
Required percentage =50
x 100% = 45
5%
110 11
2. Two students appeared at an examination. One of them secured 9 marks more than the otherand his marks was 56% of the sum of their marks. The marks obtained by them are:A.39, 30 B.41, 32C.42, 33 D.43, 34Answer: Option CExplanation:Let their marks be (x + 9) and x.
Then, x + 9 =56
(x + 9 + x)100
25(x + 9) = 14(2x + 9)3x = 99x = 33
So, their marks are 42 and 33.
3. A fruit seller had some apples. He sells 40% apples and still has 420 apples. Originally, hehad:A.588 apples B.600 applesC.672 apples D.700 applesAnswer: Option DExplanation:Suppose originally he had x apples.Then, (100 - 40)% of x = 420.
60x x = 420
100
x =420 x 100
= 700.60
4. What percentage of numbers from 1 to 70 has 1 or 9 in the unit's digit?Answer: Option CExplanation:
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Clearly, the numbers which have 1 or 9 in the unit's digit, have squares that end in the digit 1.Such numbers from 1 to 70 are 1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69.Number of such number =14
Required percentage =14
x 100% = 20%.70
5. If A = x% of y and B = y% of x, then which of the following is true?A.A is smaller than B. B.A is greater than B
C.Relationship between A and B cannot bedetermined.
D.If x is smaller than y, then A is greater thanB.
E.None of theseAnswer: Option EExplanation:
x% of y =x
x y =y
x x = y% of x100 100
A = B.6. If 20% of a = b, then b% of 20 is the same as:
A.4% of a B.5% of aC.20% of a D.None of theseAnswer & ExplanationAnswer: Option AExplanation:
20% of a = b20
a = b.100
b% of 20 =b
x 20 =20
a x1
x 20 =4
a = 4% of a.100 100 100 100
7. In a certain school, 20% of students are below 8 years of age. The number of students
above 8 years of age is of the number of students of 8 years of age which is 48. What isthe total number of students in the school?A.72 B.80C.120 D.150E.100Answer & ExplanationAnswer: Option EExplanation:Let the number of students be x. Then,Number of students above 8 years of age = (100 - 20)% of x = 80% of x.
80% of x = 48 +2of 48
380
x = 80100x = 100.
8. Two numbers A and B are such that the sum of 5% of A and 4% of B is two-third of thesum of 6% of A and 8% of B. Find the ratio of A : B.A.2 : 3 B.1 : 1
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C.3 : 4 D.4 : 3Answer & ExplanationAnswer: Option DExplanation:
5% of A + 4% of B =2
(6% of A + 8% of B)3
5A +
4B=
2 6A +
8B
100 100 3 100 1001
A +1
B=1
A +4
B20 25 25 75
1-
1
A =
4-1
B20 25 75 251
A =1
B100 75
A=
100=
4.
B 75 3Required ratio = 4 : 3
9.A student multiplied a number by
3instead of
5.
5 3What is the percentage error in the calculation?A.34% B.44%C.54% D.64%Answer & ExplanationAnswer: Option DExplanation:Let the number be x.
Then, error =5x -
3x =
16x.
3 5 15
Error% =16x
x3
x 100% = 64%.15 5x
10. In an election between two candidates, one got 55% of the total valid votes, 20% of thevotes were invalid. If the total number of votes was 7500, the number of valid votes thatthe other candidate got, was:A.2700 B.2900C.3000 D.3100Answer & ExplanationAnswer: Option AExplanation:Number of valid votes = 80% of 7500 = 6000.
Valid votes polled by other candidate = 45% of 6000
=45
x 6000 = 2700.100
11. Three candidates contested an election and received 1136, 7636 and 11628 votes respectively.What percentage of the total votes did the winning candidate get?A.57% B.60%C.65% D.90%
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Answer & ExplanationAnswer: Option AExplanation:Total number of votes polled = (1136 + 7636 + 11628) = 20400.
Required percentage =11628
x 100% = 57%20400
12. Two tailers X and Y are paid a total of Rs. 550 per week by their employer. If X is paid 120percent of the sum paid to Y, how much is Y paid per week?A.Rs. 200 B.Rs. 250C.Rs. 300 D.None of theseAnswer: Option BExplanation:Let the sum paid to Y per week be Rs. z.Then, z + 120% of z = 550.
z +120
z = 550100
11z = 550
5
z =550 x 5
= 25011
13. Gauri went to the stationers and bought things worth Rs. 25, out of which 30 paise went onsales tax on taxable purchases. If the tax rate was 6%, then what was the cost of the tax freeitems?A.Rs. 15 B.Rs. 15.70C.Rs. 19.70 D.Rs. 20Answer: Option CExplanation:Let the amount taxable purchases be Rs. x.
Then, 6% of x =
30
100
= 5.
x =30
x100
= 5.100 6
Cost of tax free items = Rs. [25 - (5 + 0.30)] = Rs. 19.70
14. Rajeev buys good worth Rs. 6650. He gets a rebate of 6% on it. After getting the rebate, hepays sales tax @ 10%. Find the amount he will have to pay for the goods.A.Rs. 6876.10 B.Rs. 6999.20C.Rs. 6654 D.Rs. 7000
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Answer: Option AExplanation:
Rebate = 6% of Rs. 6650 = Rs.6
x 6650 = Rs. 399.100
Sales tax = 10% of Rs. (6650 - 399) = Rs.10
x 6251 = Rs. 625.10100
Final amount = Rs. (6251 + 625.10) = Rs. 6876.10
15. The population of a town increased from 1,75,000 to 2,62,500 in a decade. The averagepercent increase of population per year is:A.4.37% B.5%C.6% D.8.75%Answer & ExplanationAnswer: Option BExplanation:Increase in 10 years = (262500 - 175000) = 87500.
Increase% =87500
x 100% = 50%.175000
Required average =50
% = 5%.10
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EXERCISE PROBLEMS:
1.The original price of a shirt was $20. It was decreased to $15 . What is the percent decrease ofthe price of this shirt.
Solution to Problem 1:The absolute decrease is20 - 15 = $5The percent decrease is the absolute decrease divided by the the original price(part/whole).percent decease = 5 / 20 = 0.25Multiply and divide 0.25 to obtain percent.percent decease = 0.25 = 0.25 * 100 / 100 = 25 / 100 = 25%
2.Mary has a monthly salary of $1200. She spends $280 per month on food. What percent of hermonthly salary does she spend on food?
Solution to Problem 2:The part of her salary that is spent on food is $280 out of her monthly salary of$1200percent = part / whole = 280 / 1200 = 0.23 (rounded to 2 decimal places)Multiply and divide 0.23 by 100 to convert in percentpercent = 0.23 * 100 / 100 = 23 / 100 = 23%
3.The price of a pair of trousers was decreased by 22% to $30. What was the original price ofthe trousers?
Solution to Problem 3:Let x be the original price and y be the absolute decrease. If the price wasdecreased to $30, thenx - y = 30y is given byy = 22% of x = (22 / 100) * x = 0.22 xSubstitute y by 0.22 x in the equation x - y = 30 and solve for x which the originalprice.x - 0.22 x = 300.78 x = 30x = $38.5Check the solution to this problem by reducing the original price found $38.5 by22% and see if it gives $30.
4.The price of an item changed from $120 to $100. Then later the price decreased again from$100 to $80. Which of the two decreases was larger in percentage term?
Solution to Problem 4:First decrease in percentpart / whole = (120 - 100) / 120 = 0.17 = 17%Second decrease in percentpart / whole = (100 - 80) / 100 = 0.20 = 20%The second decrease was larger in percent term. The part were the same in bothcases but the whole was smaller in the second decrease.
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5.The price of an item decreased by 20% to $200. Then later the price decreased again from$200 to $150. What is the percent of decrease from the original price to the final price of $150?
Solution to Problem 5:We first need to find the original price x. The first decrease givesx - 20% x = 2000.8 x = 200x = 200 / 0.8 = 250The percentage decrease fro the original price 250 to 150 is given bypart / whole = (250 - 150) / 250 = 0.4 = 40%
6.A number increases from 30 to 40 and then decreases from 40 to 30. Compare the percent ofincrease from 30 to 40 and that of the decrease from 40 to 30.
Solution to Problem 6:Percent increase from 30 to 40 is given by(40 - 30) / 30 = 10 / 30 = 0.33 = 33% (2 significant digits)Percent decrease from 40 to 30 is given by(40 - 30) / 40 = 0.25 = 25%In absolute term, the percent decrease is less than the percent increase.
7.A family had dinner in a restaurant and paid $30 for food. They also had to pay 9.5% sale taxand 10% for the tip. How much did they pay for the dinner?
Solution to Problem 7:They paid for food, sales tax and tip, hencetotal paid = $30 + 9.5% * 30 + 10% * 30 = $35.85
8.A shop is offering discounts on shirts costing $20 each. If someone buys 2 shirts, he will beoffered a discount of 15% on the first shirt and another 10% discount on the reduced price for thesecond shirt. How much would one pay for two shirts at this shop?
Solution to Problem 8:The reduced price for the first shirt20 - 15% * 20 = $17The reduced price for the second shirt. The 10% discount will be on the alreadyreduced price, hence the price of the second shirt is given by17 - 10% * 17 = $15.3 The total cost for the two shirts is17 + 15.3 = $32.3
9.Smith invested $5000 for two years. For the first year, the rate of interest was 7% and thesecond year it was 8.5%. How much interest did he earn at the end of the two year period?
Solution to Problem 9:Interest at the end of the first year7% * 5000 = $350Interest at the end of the second year8.5% * (5000 + 350) = $454.75Total interest at the end of the two year period is$350 + $454.75 = $804.75
10.Janette invested $2000 at 5% compounded annually for 5 years. How much interest did sheearn at the end of the 5 year period?
Solution to Problem 10:At the of the first year, she has the principal plus the interest on the principalP1 = 2000 + 5% * 2000 = 2000(1 + 5%)
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At the of the second year, she has the principal P1 plus the interest on P1P2 = P1 + 5% * P1 = P1(1 + 5%)Substitute P1 by 2000(1 + 5%) found above to findP2 = 2000 * (1 + 5%) 2
Continuing with this process, it can easily be shown that a the end of the 5th year,the principal is given byP5 = 2000 * (1 + 5%) 5
= 2000 * (1 + 0.05) = $2552.56The interest earned at the end of 5 years is$2552.56 - $2000 = $552.56
11.Tom borrowed $600 at 10% per year, simple interest, for 3 years. How much did he have torepay (principal + interest) at the end of the 3 year period?
Solution to Problem 11:The interest to pay is given byInterest = 600 * 10% * 3 = $180Total to repay600 + 180 = $780
12.Out of a world population of approximately 6.6 billion, 1.2 billion people live in the richercountries of Europe, North America, Japan and Oceania and is growing at the rate of 0.25% peryear, while the other 5.4 billion people live in the lees developed countries and is growing at therate of 1.5%. What will be the world population in 5 years if we assume that these rates ofincrease will stay constant for the next 5 years. (round answer to 3 significant digits)
Solution to Problem 12:Let us first calculate the population PR in 5 years in the richer countriesPR = (1.2 + 0.25% * 1.2) = 1.2(1 + 0.25%) after one yearPR = 1.2(1 + 0.25%) + 0.25% * 1.2(1 + 0.25%)= 1.2(1 + 0.25%) 2after two yearsContinue with the above and after 5 years, PR will bePR = 1.2(1 + 0.25%) 5 after 5 yearsSimilar calculations can be used to find the population PL in less developedcountries after 5 years.PL = 5.4(1 + 1.5%) 5 after 5 yearsThe world population P after 5 years will beP = PR + PL = 1.2(1 + 0.25%) 5 + 5.4(1 + 1.5%) 5 = 7.03 billion.
13.Cassandra invested one part of her $10,000 at 7.5% per year and the other part at 8.5% peryear. Her income from the two investment was $820. How much did she invest at each rate?
Solution to Problem 13:Let x and y be the amount invested at 7.5% and 8.5% respectivelyIncome = $820 = 7.5% * x + 8.5% * yThe total amount invested is also known10,000 = x + ySolve the system of the equations to find x and y.x = $3000 and y = $7000As a practice check that 7.5% of $3000 and 8.5% of $7000 gives $820.
14.The monthly salary S of a shop assistant is the sum of a fixed salary of $500 plus 5% of allmonthly sales. What should the monthly sales be so that her monthly salary reaches $1500?
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Solution to Problem 14:Let S be the total monthly salary and x be the monthly sales, henceS = 500 + 5% * xFind sales x so that S = 1500, hence1500 = 500 + 5% * x = 500 + 0.05 xSolve for xx = (1500 - 500) / 0.05 = $20000
15.A chemist has a 20% and a 40% acid solutions. What amount of each solution should be usedin order to make 300 ml of a 28% acid solution?
Solution to Problem 15:Let x be the solution at 20% and y be the solution at 40%, hencex + y = 300 mlWe now write an equation that expresses that the total acid in the final 300 ml isequal to the sum of the amounts of acid in x and y28% * 300 = 20% * x + 40% * ySolve the above system of equations to findx = 180 and y = 120
16.What percent of the total area of the circular disk is colored red?
Solution to Problem 16:Total area of diskAd = pi * r 2
Angle t in radians of central angle of red sectort = (360-120)* pi / 180 = (4/3) piArea of red sectorAs = (1/2) t * r 2
Percentage of total area in redP = [ (1/2) t * r 2 ] / [ pi * r 2 ]= 4 / 6 = 66.7% (3 significant digits)THINK: compare 66.7% to 240 / 360, why are they equal?
17.What percent of the total area of the rectangle is colored red?
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Solution to Problem 17:Total area of rectangleAr = L * WArea of triangleAt = (1/2) base * height = (1/2) [ L * (1/2) W ]Percentage of area in redP = (1/2) [ L * (1/2) W ] / [L*W] = 1/4= 25%
18. In Chicago in the year 2000, there were approximately 1.053 million African Americans,907 thousand whites (non-Hispanic), and 754 thousand Hispanics, and 181 thousand others(other races or two or more races). What percent of Chicagoans in 2000 were of Hispanicorigin?
Solution:
% of Hispanics = # of Hispanics / Total Population% of Hispanics = 754,000 / (1,053,000 + 907,000 + 754,000 + 181,000)% of Hispanics = 0.26 * 100%% of Hispanics = 26%
19. 54% of DePaul’s student body of approximately 21,000 is female. Approximate howmany females attend DePaul?
Solution:
% of female students = # of female students / 21,000# of female students = % of female students * 21,000# of female students = 54% of female students * 21,000# of female students = 0.54 * 21,000# of female students = 11,340
Percentage Change (Note: second problem is over 100%)
20. Chicago’s population grew from 2.78 million in 1990 to 2.90 million in 2000. By howmany percent did it grow?
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Solution:
Percentage Change = (new – old ) / old
Percentage Change = (2.90 – 2.78) / 2.78Percentage Change = 0.0432Percentage Change = 0.0432 * 100%Percentage Change = 4.32%
21. In 1996, 94,007 tons of waste was recycled in Chicago’s blue bag recycling program. Inthe year 2000, 296,363 tons were recycled. By how many percent did it increase from 1996 to2000?
Solution:
Percentage Change = (296,363 tons – 94,007 tons) / 94,007 tonsPercentage Change = 202,356 / 94,007Percentage Change = 2.15 * 100%Percentage Change = 215%
Percent More Than; Times More Than22. The life expectancy in Canada is 79.1 years; the life expectancy in the US is 76.0 years.By how many percent is the life expectancy of Canada higher than the life expectancy in the US?
Solution:
The key word for this kind of problems is "than". Whatever is after "than" should go in thedenominator (bottom part).
How many percent more = (Canada – US) / USHow many percent more = (79.1 – 76) / 76How many percent more = 3.1 / 76How many percent more = 0.0407 * 100%How many percent more = 4.07%
23. By how many percent is the life expectancy of people in the US lower than the lifeexpectancy in Canada?
Solution:
How many percent less = (US – Canada) / CanadaHow many percent less = (76 – 79.1) / 79.1How many percent less = –3.1 / 79.1How many percent less = –0.0392 * 100%How many percent less = –3.92%
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24. How many times more is the life expectancy of Canada higher than the life expectancy inthe US?
Solution:
We expect a number here!How many times more = Canada / USHow many times more = 79.1 / 76How many times more = 1.041
Successive Percentage Change
25. Spot prices for crude oil are rather volatile. From 1998 to 1999, spot prices for crude oildecreased by 28%. From 1999 to 2000, they increased by 106%. What was the percentagechange over the two year period from 1998 to 2000?
Solution:A short way to do this kind of problem isPercentage change over two year period = (1 ± %)(1 ± %) – 1Percentage change over two year period = (1 – 28%)(1 + 106%) – 1Percentage change over two year period = (1 – 0.28)(1 + 1.06) – 1Percentage change over two year period = (0.72)(2.06%) – 1Percentage change over two year period = 1.4832 – 1Percentage change over two year period = 0.4832 * 100%Percentage change over two year period = 48.32%
An increase of 48.32% over the original price!
26. Suppose you are earning a salary of $1,000 per week. Your company experiences aslowdown in earnings, and asks all workers to take a 20% pay cut. What is your salary after thecut? Six months later, the company recovers and it offers you a 20% increase on your currentsalary. How much will you be earning after the increase?
Solution:
New salary = (1 – 20%)(1 + 20%) – 1New salary = (1 – 0.20)(1 + 0.20) – 1New salary = (0.80)(1.20) – 1New salary = 0.96 – 1New salary = –0.04 * 100%New salary will have a decrease of 4%, or ($1000 – $1000 * 0.04) = $960
Reverse Percentage Change
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27. According to the official 2000 census, the Hispanic population of the US in 2000 was35,305,818. It rose by an astounding 57.9% from 1990 to 2000. What was the Hispanicpopulation in 1990?
Solution:1990 → 2000old newold 35,305,818
If we use the percentage change formula, then% change = (new – old) / old57.9% = (35,305,818 – old) / olddoing some algebra0.579 * old = 35,305,818 – old0.579 * old + old = 35,305,8181.579 * old = 35,305,818old = 35,305,818 / 1.579old = 22,359,606
28. From a news article: "In the first six months of 1996, the number of deaths [from AIDS] fell12% to 22,000 [compared to the same period a year earlier]." How many AIDS deaths were thereduring the first six months of 1995?
Solution:1995 → 1996old newold 22,000
Because there was a 12% decrease, we have to use –12%
–12% = (22,000 – old) / old–0.12 * old = 22,000 – old–0.12 * old + old = 22,0000.88 * old = 22,000old = 22,000 / 0.88old = 25,000
29. If 75% of a number is added to 75, the result is the number itself. Then the number isAns.300
Basic formula:
X% = x / 100a/ b as percent : (a/b x 100) %
Answer with explanation
Take the number as x
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75% of x + 75 = x
75/100 + 75 = x
75 = x – 75/100 x
75 = x – ¾ x
75 = x/4 x = 300
30. Subtracting 40% of a number from the number, we get the result as 30, the number is
Ans: 50
Answer with explanation:
Consider the number as x
x – 40/100 x = 30x - 2/5 x = 303/5 x = 303x = 150x = 150/3x = 50
1. If three fifth of a number is 40 more than 40% of the same number2. If three fifth of a number is 40 more than 40% of the same number. What
is the number Ans: 200Answer with explanationConsider the number as x
3/5 x = 40 + 40 % of x= 40 + 40/100x
3/5 x – 40/100 x = 40
3/5x - 2/5 x = 40
1/5 x = 40
x = 200
31.A number on subtracting 15 from it, reduces to its 80% what is 40% of that number?Answer with explanation:
Let the number be x
x-15 = 80%x
x – 15 – 80/100 x
x – 15 = 4/5 x
x – 4/5 x = 15
x / 5 = 15
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75
Now,
40% of 75= 40 /100 x 75
=2/5 x 15
=30
Hint:
30% (75) = 80/100 x 75 = 60 ---(1)= 75- 15 = 60 ---(2)(1)=(2)
32. Calculation shows that an angle is 37 ½ %. The size obtained by drawing and measurement is36%. The error percent is
Let the number be x
X – 15 = 80% x
X – 15 = 80% x
X-15 = 4/5 x
X – 4/5 = 15
X / 5 = 15
X = 75
Now,40% of 75=40 / 100 x 75
= 2 / 5 x 15
= 30
Hint:
30% (75) = 80/100 x 15 = 60 --- >(1) 75-15 = 60 --(2)1=2percentage, BU
33. Calculation shows that an angle is 37 ½ % The size obtained by drawing and measurement is36%. The error percent is
Ans: 4Error for 37 ½
37 ½ - 36 =75 / 2 = 36= 75- 72 / 2 = 3/2 (=1 ½)
error for 100= 3/2 x 2/75 x 100 = 4%
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34. If x is 90% of Y, what percent of x is y?ans : 111.1
Answer with explanation:X = 90 % of yX = 90/ 100 yX = 9/10 y y = 10/9 x y / x = 10/9
required percentage = y / x x 100= 10/9 x 100= 1.111 x 100= 111.1 %
35. If x % of y is the 4/5 of 80, then the value of xy isans : 6400
Answer with explanation:X % of y = 4/5 x 80X / 100 x y = 4/5 x 80xy = 4 x 16 x 100xy = 6400
36. Subtracting 6% of % from x is equivalent to multiplying % by how much?Ans : 94
Answer with explanation:Consider the number as xX – 6/100 = xy100 / 100 x – 6 / 100 x = xy100-6 / 100 x = xy94 /100 x = xy94% of x = xy
= y = 94%
37. If 8% of x = 4% of y, then 20% of x isans: 10
Answer with explanation:
8% of x = 4% of y8x /100 = 4 /100 y 8x = 4y
2x = y x = ½ y
20% of x = 20/100 x= 20/100 x ½ y= 10/100 y= 10% of y
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38. (X% of Y + Y % of X) = ?Ans: 2% of xy
Answer with explanation:X % of y + y % of X
= xy / 100 + yx / 100= 2 x y / 100= 2% xy
39. If 90% of A=30% of B and b=x% of A, then x is equal toans: 300
Answer with explanation:90% of A = 30 % of B
90 / 100 A = 30 / 100 B90A = 30B9A = 3B3A = BB = x/100 A3A = x / 100 A x = 300
40.. If 70% of students in a school are boys and the number of girls is 504, the number of boys isans: 1176Answer:
70% of students in a school are boys:(100-70)% = 30% girls percentage30% of x = 50430/100x = 5043/10 x = 504x = 504 x 10 / 3
= 1680No of boys = 70% of 1680
= 70/100 x 1680= 1176
41. A number increased 37 ½ % gives 33. The number isAns: 24
(100 + 37 ½ ) % of x = 33(100+ 75/2 ) % of x = 33(200+75/2) % of x = 33
= 275 / 2 x 1 / 100 x = 33x = (33 x 100 x 2 ) / 275
= 3 x 4 x 2x = 24
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42. The number which when decreased by 27 ½ % given 87 isAns: 120
Answer(100 – 27 ½ ) % of x = 87200 - 55 / 2 x 1/100x = 87145 / 2 x 100 = 87x = 87 x 2 x 20 / 145
= 3 x 2 x20= 120
43. 40 Quintal is what percent of 2 metric tones?
Ans: 200%1 metric tonee= 10 quintals2 metric tonee = 20 quintals
= (40 / 20 x 100) %= (2 x 100) %= 200%
44. It is known that 20% of the mangoes are rotten. If the number of rotten mangoes is 35, thenthe total number of mangoes isAns: 175
20% x = 3520 / 100 = 351/5 = 35
x = 175
45.A student has to secure 40% marks topass. He gets 178 marks and failed by 22 marks. Themaximum marks are :
40% x = 178 +2240/100 x = 2002/5 x = 200x = 100 x 5/2x = 500
46. A house owner was having his house painted. He was advised that he would require 25kg ofpaint. Allowing for 15% wastage and assuming that the paint is available in 2kg cans, whatwould be the cost of pain purchased, if one can cost Rs.16?
Ans: 240Rs.Answer:Required no of kgs 25kgs
10% 2.55% / 15% 1.25 / 28.75kg
1 can contains 2kg paint
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= 28.7514 can 1 can= 15 cans= 15 x 16 = 240
1 can = 16 Rs.
47. A reduction of 12.5% in the price of a dining table brought down its price to Rs.4375, theoriginal price (in Rs.)of the table was:Ans: Rs.5000
Answers:The original price to be x
X – 12.5% of x =4375X – 12.5/ 100 x = 4375X – 125 / 1000 x =43751000-125 / 1000 x = 4375875/ 1000 x = 4375
x = 4375 x 1000 / 875=Rs.5000
48. Of the total amount received by Kiran, 20% was spent on purchases and 5% of theremaining on transportation. If he is left with Rs.1520, the initial amount wasAns: 2000
Answer:For purchase 20% of x => 20/100 x = 1/5 xRemaining x – 1/5 = 4/5 xFor transportation == 5 % 4/5 x
= 5/100 x 4/5 x= x/25
Balance = 4x /5 – x / 25 = 20x / 25 – x/25 = 19/25 x19x / 25 = 1520x = 1520 x 25 / 19x =2000
49. In a library 20% books are in Hindi, 50% of the remaining are in English and the remaining9000 are in various other languages. What is the total number of books in English.Answers:
Let the no of books be x20% of x = 20/100 x /5Remaining x – x /5 = 4x / 5For English 56/ 100 x 4x / 5
2x/ 5Remaining 4 x / 5 – 2x / 5 = 2 x / 5For other languages
2x / 5 = 90002x = 45000x = 22500
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Another methodLet it be 100 %Hindi 20% (-20)/80
Eng 50% of remaining ( -40) / 40 = 9000= 100
9000 x 100/ 40 = 2250050. Avinash spends 30% of his income on scooter petrol, ¼ o the remaining on house rent andthe balance on food. If he spends Rs.300 on petrol, then what is the expenditure on house rent?Ans : 175
Answer :From the given information,Method: 1For scooter petrol30x / 100 = 3x / 10remaining x – 3x / 10 = 7x / 10for house ¼ x 7x / 10 = 7x / 40remaining for food -- > 7x / 10 – 7 x / 40
=> 28 x – 7x / 40 = 21x /40Amount for petrol = 30/100 x = 30
= 1000House rent,7x / 40 = 7 x 1000 / 40 = 175Method :2
Total income xX - 30/100 x = x – 300(in %) = (in Rs.)30 / 100 x = 300x = 1000
rem¼ x 100 = 175Rs.
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QUESTION BANK:
1. If 20% of 40% of a = 25% of a% of b, then what is b?a. 8/5 b. 16/25 c. 8/25 d. None2. By what % is 200 more than 50?a. 100 b. 200 c. 300 d. None3. A value changes from 30 to 80. What is the percentage change?a. 125 b. 166.66 c. 156 d. None4. The population of a city is increased by 30% and thus became 78000. What is the originalpopulation?a. 76000 b. 64200 c. 60000 d. None5. In a theatre, the number of seats is increased by 20% and the price per ticket is increased by10% but the public response decreased by 30%. What is the net effect on the economy of thetheatre?a.10% rise b. 7% fall c. 7% rise d. None6. A saves 20% of his income. His income is increased by 20% and so he increased hisexpenditure by 30%. What is the percentage change in his savings?a. 20% fall b. 4% fall c. 20% rise d. 4% rise7. The price of petrol is increased by 25%. By what percent the consumption be reduced to makethe expenditure remain the same?a. 25% b. 33.33% c. 20% d. None8. The side of a square is increased by 20%. The percentage change in its area is ___a. 20% b. 44% c. 36% d. None9. If the length of a rectangle is increased by 33.33%, by what percentage should the breadth bereduced to make the area same?a. 20% b. 33.33% c. 25% d. None10. In an election between two candidates, A and B, A secured 56% of the votes and won by48000 votes. Find the total number of votes polled if 20% of the votes were declared invalid.a. 500000 b. 400000 c. 600000 d. None11. A reduction of 10% in price of sugar enables a housewife to buy 5 kg more for Rs. 300/-.Find the reduced price per kg of sugar.a. 5/- b. 4.5/- c. 6/- d. None12. From a 20lt solution of alt and water with 20% salt, 2lt of water is evaporated. Find the new% concentration of salt.a. 20% b. 23% c. 25% d. None13. In a list of weights of candidates appearing for police selections, the weight of A is marked as58 kg instead of 46.4 kg. Find the percentage of correction required.a. 30 b. 20 c. 24 d. None14. A person spends 20% of his income on rent, 20% of the rest on food, 10% of the remainingon clothes and 10% on groceries. If he is left with Rs. 9520/- find his income.a. 10000/- b. 15000/- c. 20000/- d. None15. A shopkeeper offers three successive discounts of 10%, 20% and 30% to a customer. If theactual price of the item is Rs. 10000, find the price the custome has to pay to the shopkeeper.a. 5040/- b. 4000/- c. 6000/- d. None16. If 10lt solution of water and alcohol containing 10% alcohol is to be made 20% alcoholsolution, find the volume of alcohol to be added.
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a. 1 lt b. 1.25 lt c. 1.5 lt d. 2 lt17. A is twice B and B is 200% more than C. By what percent is A more than C?a. 400 b. 600 c. 500 d. 20018. In an examination, a student secures 40% and fails by 10 marks. If he scored 50%, he wouldpass by 15 marks. Find the minimum marks required to pass the exam.a. 250 b. 100 c. 110 d. 12519. If A is 20% taller than B, by what percent is B shorter than A?a. 20% b. 25% c. 16.66% d. None20. The population of a town increases at a rate of 10% for every year. If the present populationis 12100, find the population two years ago.a. 11000 b. 9800 c. 10000 d. 1012021. A solution of salt and water contains 15% salt. If 30 lt water is evaporated from the solutionthe concentration becomes 20% salt. Find the original volume of the liquid before waterevaporated.a. 100 lt b. 120 lt c. 200 lt d. None22. If 240 lt of oil is poured into a tank, it is still 20% empty. How much more oil is to be pouredto fill the tank?a. 300 lt b. 60 lt c. 120 lt d. None23. A and B were hired for the same salary. A got two 40% hikes whereas B got a 90% hike.What is the percentage difference in the hikes thay got?a. 16% b. 6% c. 10% d. 8%24. The population of a town doubled every 5 years from 1960 to 1975. What is the percentageincrease in population in this period?a. 800 b. 400 c. 700 d. 60025. In a test of 80 questions, Jyothsna answered 75% of the first 60 questions correctly. What %of the remaining questions she has to answer correctly so that she can secure an overallpercentage of 80 in the test?a. 80% b. 90% c. 85% D. 95%26.A person allows discount of 20% on the already discounted MP. What is the initial discountgiven if he totally gains 20% and MP is double the CP?a. 25% b. 20% c. Data inadequate d. none27.If a group of 24 men working 8 hrs a day can complete building a wall in 12 days then in howmany days 16 men working 6 hrs a day can complete 3 such walls?a. 24 b. 48 c. 72 d. none(28-31)Read the following information and answer the questions given below it :(i) ‘A + B’ means ‘A is the father of B’(ii) ‘A – B’ means ‘A is the wife of B’(iii) ‘A ´ B’ means ‘ A is the brother of B’(iv) ‘A ¸ B’ means ‘A is the daughter of B’28. If P ¸ R + S + Q, which of the following is true?a. P is the daughter of Q b. P is the mother of Q
c. P is the aunt of Q d. Q is the ant of P29. If P – R + Q, which of the following statement is true?a.P is the mother of Q b. P is the sister of Qc. Q is the daughter of P d. P is the aunt of Q30. If ‘P ´ R ¸Q’, which of the following is true?
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a.P is the uncle of Q b. P is the father of Qc. P is the son of Q d. P is the brother of Q31. If ‘P ´ R – Q’, which of the following is true?a.P is the brother-in-law of Q b. P is the father of Qc. P is the brother of Q d. P is the uncle of Q32. The average of 13 papers is 40. The average of the first 7 papers is 42 and of the last sevenpapers is 35. Find the marks obtained in the 7th paper?a. 23 b. 38 c.19 d. None of these33. In a mixture of 40 liters, the ratio of milk and water is 4:1. How much water much beadded to this mixture so that the ratio of milk and water becomes 2:3a. 20 liters b. 32 liters c. 40 liters d. 30 liters34. A shop owner sells two puppies at the same price making a profit of 20% on one and a lossof 20% on the other. Find his loss or gain percent on the whole transaction.a. Gain of 4% b. No profit no loss c. Loss of 10% d. Loss of 4%35. N, at a speed of 20 km/h reaches his office 10 min late. Next time he increases his speed by 5km/h, but is still late by 4 min. What is the distance of office from his house?20 km b. 6 km c. 12 km d. None of these36. 4 men and 3 women finish a job in6 days, and 5 men and 7 women can do the same job in 4days. How long will 1 man and 1 woman take to do the work?a. 22(2/7) days b. 19(1/2) days c. 5(1/7) days d. 12(7/22) days37. If the code for BLUE is 240, that for TEA is 18 then the code for MATCH is _____a. 3100 b. 3120 c. 3210 d. none38. Two equal sums were borrowed at 8% simple interest per annum for 2 years and 3 yearsrespectively. The difference in the interest was Rs.56. the sum borrowed werea. 690/- b. 700/- c. 740/- d. 780/-39. If A is 25% of C and B is 30% of C, then what percentage of A is B?a)83 1/3% of B b)83 9/3% of Bc)81 1/3% of B d)83 2/3% of B39 .Two numbers are respectively 25% and 50% more than a third number. What percent is thefirst of the second?a)26 2/3 % of second b)26 3/4 % of secondc)83 1/3 % of second d)83 2/3 % of second40. Two numbers are respectively 20% and 32% less than a third number. What percent is thesecond of the first?a)15% of the first b)85% of the firstc)25% of the first d)none of these41. If the price of a commodity increases by 50%, find how much percent its consumption bereduced so as not increase the expenditurea)33 1/3 % b)66 3/4 %c)75% d)none of these42. If the price of a commodity decreases by 50%, find how much percent its consumption beincreased so as not decrease the expenditurea)100% b)0%c)10% d)none of these43. If the salary of Mr. Shashi is first increased by 18% and thereafter decreased by 15%, what isthe net change in his salary?
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a)7% b)3%c)0.3% d)o.7%44. The population of a town is decreased by 20% and 40% in two successive years. Whatpercent population is decreased after two years?a)55% b)45%c)48% d)52%45. If the side of a square is increased by 10%, its area increased by k%. Find the value of ka)21 b)15c)42 d)1246. The radius of a circle is increased by 4%. Find the percentage increase in its areaa)9% b)8 4/25%c)2 2/25% d)none of these47. The population of a town increases by 4% annually. If its present population is 12500, whatwill it be in 2 years time?a)13520 b)14520c)11520 d)none of these48. In a group of 80 boys, 60% play chess, 75% play cricket and 55% play both. How many ofthem do not play any of these two games?a) 24 b) 20c) 18 d) 1649. In a bookstore 25% of books are in English, 60% of the remaining are in Hindi, % of theremaining are in Telugu and remaining 64,000 are in other languages. What is the total numberof books in that bookstore?a) 2,56,000 b) 3,20,000c) 4,50,000 d) 6,40,00050. Three numbers are in the ratio of 3:4:8 respectively. If the first is increased by 25%, thesecond is decreased by 20% and the third is unaltered, respectively their ratio will be?a) 75:64:180 b) 75:56:160c) 75:64:160 d) 125:64:160
Solutions:1.1/5 X 2/5 X a = ¼ X a X b => b = 8/252.% difference = (200-50)/50 X 100 = 300 %3.% increase = (80-30)/30 X 100 = 166.66 %4.1.3 x = 78000 => x = 60000.5.Net effect = 1.2 X 1.1 X 0.7
= 0.924 => 7.6% decrease.6.Let I be the income.
Expenditure = 0.8I Savings = 0.2I => 20%New income = 1.2I (since 20% rise)New expenditure = (0.8I) X 1.3 (Since 30% rise)= 1.04ISo, new savings = 1.2I – 1.04I = 0.16I => 16%(So income decreased form 20% to 16%)% decrease = (20-16)/20 X 100 = 20%.
7.It is equivalent to 1.25 decreased to 1.
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% decrease = (1.25-1)/1.25 X 100 = 20%8. % change in area = 1.2 X 1.2 (since area = side X side)
= 1.44 => 44%.9.It is equivalent to 1.25 decreased to 1. So 20% decrease.10. Valid Votes:
A got 56% => B got 44%Difference = 12% = 48000So, 100% = 400000. These are valid votes.But valid votes are only 80% of total votes.So, 80% of total votes = 400000 => total votes = 500000
11.Total money = Rs. 300.Saving of the lady = 10% of 300 = 30/-With 30/- she bought 5 kg sugar => each kg costs Rs. 6/-
12.In 20lt, salt = 20% => 4 lt.New volume = 18 lt (2 lt evaporated)So, new % = 4/18 X 100 = 22.22%
13.% correction = (58-46.4)/58 X 100 = 20%14.Three successive decreases of 20%, 20% and 10% => 0.8 X 0.8 X 0.9 = 0.576
Again 10% decrease => 0.576 – 0.1 = 0.476.So, 0.476 x = 9520 => x = 20000.
15.Total discount = 0.9 X 0.8 X 0.7 = 0.504 of actual price.So, price = 0.504 X 10000 = 5040.
16.In 10 lt, alcohol is 10% = 1 lt.Let x lt alcohol is added.So, (1+x)/(10+x) = 20% = 1/5 => x = 1.25 lt.
17.A = 2B and B = 3C (since 200% more)ð A = 6C => 500 % more.
18.50% of max marks – 40% of max marks = 25ð max marks = 250Pass marks = 40% of max + 10 => 100 + 10 = 110.
19.A = 1.2 B => B = A/1.2 => 0.8333A => 16.66%.(OR) Decrease from 1.2 to 1 => 16.66%.
20.1.1 X 1.1 X x = 12100 => x = 10000.21.Salt = 15% of x = 0.15x (x = volume of solution)
Now, 0.15x/(x-30) = 20% = 1/5 (since 30 lt evaporated)ð x = 120 lt
22.20% empty => 80 % full = 240 lt => 20% = 60 lt23.A => 1.4 X 1.4 = 1.96
B => 1.9 => 6% difference.24. From 1960 to 1975, in 15 years population doubled every 5 yrs => three times
So, 2 X 2 X 2 = 8 times => 700% more.25.[(75% X 60) + (x% X 20)] / 80 = 80% => x = 95. (since required is 80%)
(OR) 60 out of 80 is 3/4. So, (3/4 X 75) + (1/4 X x) = 80 => x =95.
