Chapter 1, Pre-Requisites PDF

 God Created Natural Numbers, All Else is Work of Man" -L. Kronecker Chapter 1.. Pre-Requisites Contents 1.1 Natural Numbers and Integers 7.2 H.C.F. andL.C.M. 1.3 F...

 God Created Natural Numbers, All Else is Work of Man" -L. Kronecker Chapter 1.. Pre-Requisites Contents 1.1 Natural Numbers and Integers 7.2 H.C.F. andL.C.M. 1.3 Fractions 1.4 Indices i.5 Ratio 1.6 Proportion H Variation.8 Percentage Key Words : Ratio, Proportion, Direct Natural Numbers Integers, Fractions, H.C.F., L.C.M., Continued ratio variation, Joint Proportion, Continued proportion, Inverse proportion, variation, Inverse variation, Percentage. Objectives fractions, indices, ratio proportion and To understand the concept of number system, percentage and their use in commercial activities. 1.1 Natural Numbers and Integers mankind's necessity of counting The origin of Mathematics may be traced back to dates for counting the livestock objects. In the pre-historic age, people used pebbles or... made this task so simple that (sheep, coWs etc.), The discovery of numbers 1, 2, 3, mathematicians did not think todefine them formally. Cantor gave serious thought to this At the end of 19" century Peano, Dedekind and problem. Mathematician defined natural Sir Bertrand Russell, a twentieth century Philosopher, number as follows : class". "Anatural number is a class of all the classes that are similar to that (1.1)  F.Y.B.Com. Business Maths & Stats 1.2 Though the definition looks clumsy and bit humorous, it carries lot of Pre-Requisites meaning. The set N={1, 2, 3,...} is called set of natural ordered set such that every natural number is numbers. The natural numbers form a cannot be greatest natural number. followed by next one. It is clear that thers Properties of natural numbers : (1) Addition in N is commutative as well as associative. ie. if a, b, ce N, then a + b=b+aand a +(b+ c) = (a + b) +c. (2) Multiplication in N is commutative as well as associative. i.e. a xb = bxa and a x (bX c) = (ax b) Xc (3) Multiplication is distributive over addition. i.e. a X(b + c) = Integers : axb+axc The operations of addition and multiplication on the set of numbers only. In this sense, may not always give N is closed for addition and natural numbers give natural generally, if a +.X = cnatural number; and if c n) (3) (a")=a mn (4) (ab)"= a" bn m a (5) (b0) Meanings of a', Suppose we assume the first law a"xa=a for all rational yalues of mand n. Then. a"xa' = a+0 am a' = m=1 Thus the zeroth power of any non-zeronumber is 1, (0° = 0) Similarly., we can deduce a n a" and a9=a Note : a =, aetc. 2 a =a, 64 =64) -(64) -(4)' =16 etc. F.Y.B.Com. Business Maths && Stats 1,8 Pre-Requisites Illustrative Examples 1/n Example 1.6:Evaluate : Solution: Given expression (3²a+14 3x(3,2 (: Va=a 33n 12 x (qntl,/2qlh 3x3-n2 32n + 1/2 x3n2 +i/2n 3 [32n + 1/2 +n/2 + 1/21/n 3!-n2 - z(Bn +n2 +1)- (1 -n2),1n = [32h +n2 +1-1*n,lh = (35n,n=3n x1n =3 =27 Example 1.7:If(2.381)' = (0.2381Y' = 10, prove that 1 y Solution : Let (2.381 =(0.2381' =10=k (2.381' = k (2.381* = k 2.381 = kx Similarly, 0.2381 = ky and 10 = 2 Now, 2.381 0.2381 = 10 X (if base is same + indices must be equal) X yz EVB.Com. Business Maths & Stats 1.9 Pre-Requisites Example 1.8 : Evaluate : [1 -(l -(1-x'rrwhen x= 100. : Solution : {1 -(1:-*))= -i Given expression -- 3l/3 3-1/3 -l/3 =(x)=X=100 1.9 : Ifa' = b'= = d and ab =cd, showt h a t - 4 Example Solution:Let a = b'=c=d=k. a' = k (ayla a' = a=k* (:: (a = a Similarly, b = k, c =k and d= kw ab= cd (given) Now, kl. k = k. kx+ 1/y llz+ lw 1 1 11 (If base is same, indices are equal.) + + W Xy , show that 2x + 6x-3 = 0. Example 1.10:Ifx = 2-28 X = 2/3-2-/3 Solution: x = (2/3 = ( 2 -(23x 2 x2-l (2-g3) (" (a -b)=a'-b-3ab (a - b)] = 2'-2-3x2xx 1 (: 2° = 1) = 2-5-3x 2x = 4-1- 6x.:. 2x' + 6x -3 = 0  Pre-Requisite 1.10 F.Y.B.Com. Business Maths & Stats 2/3 and Example 1.11:Find greater of the two numbers and 3) is 6. Solution:The L.C.M. of denominators of indices (viz. 2 Hence, we raise both quantities to power 6. = (0.75)'=0.0750 4 2/3-76 =(0.8)* =0.4096 2/3 3/2 1.5 Ratio make In day-to-day life, we come across several situations in which we have to comparisons among quantities. For example, salaries of persons, prices of commodities, sales of different firms etc. In mathematics, the operations of subtraction and division are mainly used for comparison. When two quantities are compared by division, we use ratio. Suppose that a firm having sale ofR 40crores in a year made a profit of ? 13 crores and another firm having sale of 60crores in thae year made a profit of 16crores. Therefore. do we infer that the performance of second firm is better ? Taking the quotient profit/sales we can compare the performance of the two firms. (Note that this is one of the criteria for judging the performance of a firm). In other words, we are using ratio (of profit to sales) to compare the performance of the two companies, and then our conclusion is that the 13 16 performance of first firm is better than that of the second 40 60 From the above illustration, it is clear that, we can find the ratio of two same type. Moreover, the unit for measurement must be same. quantities of the Thus, we have a formal definition. Definition : If 'u' and 'b' are magnitudes of same kind, expressed in same units, then the quolient is called the ratio of 'a' to 'b' and it is denoted by a : b. Note : (i) Ratio is a pure number i.e. it has no units. (ü) In the ratio a : b, a is called /i f we multiply he antecedent and b is called consequent. numerator and denoninator In any ratio by the same number, the rauo renains the same. (non-zero) Ma i.e. mb (m 0) EY.B.Com. Business Maths &Stats. 1.11 Pre-Requisites From this it is clear that, the antecedent and consequent in a ratio may not be actual nantities. It also indicates that, if the ratio of two quantities is a : b, the actual quantities should be taken as xa and xb (x # 0), Continued Ratio : It is the relation between the magnitudes of three or more quantities ofthe same kind. The continued ratio of three similar quantities a, b,c is denoted by a:b:c. Illustrative Examples Example 1.12 : Two numbers are in the ratio 7,:8 and their sum is 195. Find the numbers. Solution : Let the numbers be 7x and &x. 7x+8x 195 i.e 15x = 195 X=13 Required numbers are 91 and 104. Example 1.13 : Ifa : b=4:7and b. c9: 5, find a: c. a Solution : b 7a= 4b 4b 7 b Again, C 5b = 9c 5b C= 9 4 36 ie. a:c = 36:35. 5 35 Example 1.14 : The sum of presentages of 3 persons is 66 years. Five years ago, their ages were in the ratio 4 :6:7. Find their present ages. (April 2010) Solution: Let the ages of three persons, five years ago be 4x, 6x and 7x years respectively. Their present ages are 4x + 5, 6x +5 and 7x + 5. From the information given, (4x + 5)+ (6x + 5) + (7x + 5) = 66 17x + 15 = 66  F.Y.B.Com. Business Maths & Stats 1,12 Pre-Requisites 17x = 51 X= 3 Present ages are 4x + 5, 6x + 5 and 7x +5. i.e. 17, 23 and 26 years respectively. Example 1.15 : The monthly salaries of. two persons are in the ratio 3 : 5. 1f each receives an increase of ?200 in monthly salary, the new ratio is 13 : 21. Find their salaries. original (Oct. 2009) Solution: Let the original salaries be 3x and 5x. Due to increase in salaries, the revised salaries are (3x + 200) and given that (5x + 200). It is 3x + 200 13 5x + 200 21 63x + 4200 = 65x + 2600 2x = 1600 X = 800: Original salaries were 2,400 and4,000. Example 1.16 : The ratio of price of first had risen by 10% andprices that of two houses was 4: 5. Two of the second by 6,000, the years later, when the Find the new prices of the houses. ratio became 11 : 152 Solution: Let the original prices be 4x and ? 5x. house increased to 4x + 4x Two years later, the price of first and that of the second to ? (5x + 6,000). Ratio of new prices, 4x 4x + 10 L1 5x + 6000 = 15 (given) 154x +4x 10, = 11 (5x + 6000) 60x + 6x = 55x + 66000 11x = 66000 X = 6000 New prices of the houses are 4x + 4x 10 and 5x + 6000 i.e. 24000 + 2400 and 30000 + 6000 i.e. 26400 and 36000.  be in litre EY.B.Com. 5y, the Due Solution:Example Thus, But Solution Example less i.e. 7y, ratio (which In to In in Business J0 it Their 9y is Original 19% Ratio 8Savings 5:7:9. 100, givenrespectively. 100, I00. 1.17 Y savings Let : 1.18 is increase Let of 8 19 rate 11900- 10,000 less one one the Find : Maths savings of that the X 100 y wil IfP : Incomes getsoriginal Price & P, P incomes of 119x than gets original and Q, saves in Stats 10x saves be milk 10,000 119x 119x/100 price, X100 is R 2x the of 10: 28 wil - ofP, is =119x 1900 100 milk 25y - 2x (1/5)th earlier). litres. rate rate = X 25y8x 5y 1/5th ofP, X= neW respectively. 19:28. y be -5y, is = 16/litrë = of of g50 = = of Q, 0, rate increased of 1900 1 milk milk. 25 2x 5 2x 3x Rare 119 y- his R his 10,000 1.13 -7y, be 119x is income income, in 15.9664 16 be 5y, = 100 119x by 2x, the x/litre. g75 4x litres. 19% i.e. 9y 3x, y-7y, find ratio 4x as respectively. respectively the 2: a ratio 3:4 result and ofand of 100 which their and their respectively, y9y savings. a their expenditures person Pre-Requisites expenditures (Oct. 2009) gets are I  Pre-Requisites 1.14 F.Y.B.Com. Business Maths & Stats 1.6 Proportio ratios are equal, then the four quantities given by them are said to be in If two b. c. d are said to be in proportion proportion. i.e. if the ratios a :b and c:dare equal, then a, called extremes called means while a and dare and we write a : b::c:d. Here, b and care further d is called 4th proportional to a, b and c. Note : If a, b, c, d are in proportion, then, a b ad = bc. ie. Product of extremes = Product of means Continued Proportion (April 2010):f a, b;'c are three quantities of the same kind and if ab = b/c, then a, b, care said to be in continued proportion. In thiscase, b is called mean proportional to a and c. Note that b² = ac. The concept of continued proportion can alsobe extended to more th¡n three quantities of the same kind. Direct Proportion (Oct. 2009) : Petrol costs ? 71 per litre. If aperson buys 3 litres of petrol, clearly he has to pay ? 213, Thus, as the consumption of petrol increases, expeniture on it also increases. Similarly, if the consumption is less,,expenditure is also less. Thus, we have a relation between two variables viz., Consumption of petrol and expenditure on it. They are said to be in directproportion: Definition : When two variables are so related that an increase (or reduction) in one causes an increase (or reduction) in the other in same ratio then the proportion is called direct proportion. Inverse Proportion (Oct. 2008, April 2011) : Suppose that a man 15days working 4 hours per day. Then we know that if the job is to completes a job in he will have to work 6 hours per day. Thus, if the be completed in 10days, job is to be completed in lesser days, the man has to work more everyday. In this illustration, number of days and number of hours is increased, number of days isworking hours are two variables such that if of bours is decreased, the decreased in the same ratio, Also if number number of days is increased in the same ratio. This type of variation is called inverse variation and two variables since in this case, one ratio is reciprocal of the other as are said to be in inverse shown below : proportion, No. of days 15 Working4 hours 10 6 15 10 = -) EY.B.Com. Business Maths & Stats 115 Pre-Requisites Definition : If two variables are so related that, an increase (or reduction) in one aUses a reduction (or increase) in the same ratio in the other, then they are said to be in inverse proportion. 1.7 Variation If two variables X and y are in direct proportion, we write it as x ox y, then, X = ky, where, k is called constant of proportionality. If avalue of x and corresponding value of y are known, then this constant can be obtained at once. For a circle, circumference « radius is an illustration of direct variation. 1 Inverse Variation : If xand y are two variables such that x varies directly as then we y saythat x varies inversely as y and write, X k then, X where, k is constant of proportionality. constant can be If a value of x and corresponding value of y are khown, then this obtained at once. Joint Variation : y and z, if it varies 1. Avariable x is said to vary jointly with respect to the variables as their product i.e. if X « yz then, X = kyz base and altitude. For example,We know that area of a triangle varies jointly as its 2. A variable x is said to vary directly as y and inversely as z, if it varies as ie. X ky then, X = Illustrative Examples| Type 1: Example 1.19 : Find x, if (i) 6:15::2: x, (ii) 15 : 27 :: x:45. Solution: (i) 6: 15 ::2:x 6 2 i.e. 15 X i.e. 6x = 30 X = 5 F.Y.B.Com. Business Maths && Stats 1.16 Pre-Requisites (ii) 15:27:: x: 45 1.e. 15 X 27 45 5 X i.e.. 45 x 5 = X X = 25 Example 1.20 : Find fourth proportional to6, 8, 10. Solution: Let xbe the fourth proportional. 6:8:: 10: x 1.e. 6 10 X i.e. 6x = 80 40 X = 3 = 13.3333 Example 121 : Ages of IfMadhav is 4 years old and DilipMadhav, Ajit, and Dilip are is 9 years old, what is the age ofin continued Ajit ? proportion. Solution :Let Ajit be x years old. 4:x:: x:9 (since they are in continued proportion) 1.e. 4 = X 1.¬. X? = 36 X =6 Hence. the age of Ajit is 6 years. Type 2: Example 1.22 48.000 ? : If sugar costs 32 per kg, how many tonnes can be Solution: The price of sugar and quantity bought for If x kg sugar can be purchased are in direct proportion. bought for ? 48,000. 32 48000 X 1500 kg = 1.5 Type 3: tonnes Example 1.23: Astudent finishes a book by many pagesreuding 30 pages per day in 16 wants to finish the book in 12 duys, hoW Solution: We know that the should be read number of pages read days. Ifhe in inverse proportion. and the ? everyday number of days required are EY.B.Com. Business Maths & Stats 1,17 Pre-Requisites Let x be the number of pages that he has to read everyday to finish the book in 12 days. No. of pages No. of days Original data 30 16 New data X 12 Because of inverse proportion, X 1 30 12/16 X = 30 3 X = 40 He has toread 40 pages per day. Type 4: Example 1.24 : If 75 persons can perform a piece of work in 12 days of 10 hours each, how many persons could perform apiece of work twice as large in half the number of days, working 8 hours daily ? Solution:In this type of problems, we prepare a table as follows: Variables Original Set New Set Type of Variation Persons 75 X Work Direct 12 6 Inverse Days Hours 0 Inverse Note that the proportions are decided with respect to the unknown x i.e. we know that when the work is more, number of persons required is more, so that proportion is direct. When number of days is reduced, number of persons required will be more-so that the proportion is inverse and so on. Then we have a concise formula as follows : 2 12 10 Original quantity 1íx 6 X 8 X i.e. = 5 75 X = 375 Note that on the R.H.S., we take the product of ratios : new value/original value and original value/new value depending on the type of variation. The ratio new value/original value is taken for direct variation and original value/new vaiue for inverse variation. F.Y.B.Com. Business Maths & Stats 1.18 Pre-Requisites Example 1.25 : 20 men require 25 days to dig atrench 50 mlong, 20 mn broad and I m deep. How many days will be required for 60 men to dig a trench 90 mlong, 60 mbroad, and 1/3 m deep ? Solution : Variables Original Set New Set Proportion Men 20 60 Inverse Days 25 X Length 50 90 Direct Breadth 20 60 Direct Depth 1 1/3 Direct 20 90 60 1/3 X 25 60 50 X 20o X 1 X 1 25 X = 15 Hence 15 davs will be required. Type 5 : Exampie 1.26 : f A B and A= 4 when B=6, find thevalue ofA when B = 27. Solution : A « B;... (1) A = kB When A = 4. and B = 6. 4 = 6k 2 k =a B from (1) A B when B = 27 A = 2 X 27 A = 18 Example 1.27 : lfx wuries direcly as y and inversely as z and x= i= l0, find ywhen s = 9 and z = 24. 12whenv=9 and SolutivD: and X 8x NN X = ky... ( ) Whcn A= 12, y =9, Z= 16, 9k 12 = 16 k = 64 from l , X  EY.B.Com. Business Maths & Stats 1,19 Pre-Requisites When x=9, z= 24, 81 9 = 64 yx 24 y = 8 3 Example 1.28 : A diamond worth 25,600 is accidentally broken into two pieces whose weights are in the ratio3 : 3. The value of the diamond varies as the square of the weight. Calculate the loss due to the breakage. Solution: Let c denote the cost and w denote the weight of diamond. Then, C c = kw2..(1) where, k is constant of proportionality. Let weight of the diamond be 8 gm. 25,600 i k (64) k 400 :: (1) gives C = 400 w². (2) Due to breakage, the two pieces now weigh 3gm and 5gm respectively. The cost of the diamond weighing 3gm is c= 400 (9) =3,600 Similarly, the cost of the diamond weighing 5gmis C = 400 (25). = 10,000 The total cost of the two pieces = 3,600+ 10,000 = 13,600 Loss= 25,600- 13,600 1.e. Loss =12,000 Example 1.29: Find fourth proportional to 2, 4, 6, x. Solution : If 2, 4, 6, x are in proportion. 2 6 = 4 X 2x = 24 X = 12 Fourth proportional to 2, 4, 6 is 12. 1.8 Percentage It is aspecial type of ratio, in which the consequent (denominator) is l00. When a ratio X 1S expressed in this form, the numerator is said to express the percentage. Thus denotes 100 X%. Consider the illustration studied in earlier section, in which profits of two firms were 13 32.5 and the ratio for other Compared. The ratio of profit to sales of the first firm is 40 I00  F.Y.B.Com. Business Maths & Stats 1.20 Pre-Requisites 80 80 16 3 3 firm is 60 100 In other words, the percentage profits of the two firms are 32.5 and 100 Hence, according to this criterion, the performance of first firm is better than that of the second. Note that with the help of percentage, we could compare the performance of the two companies very easily. Remarks : 1. As percentage is a ratio, it has no units. 2. Percentages are very useful in profit and loss, commission and brokerage, simple interest, compound interest etc. X 3. If we want to find x% of a quantity, we should multiply the quantity by 100 in X other words, x% of Y = Y X 100 3 For example, 3% of 58 = 58 x 100. = 1.74. 4. If we want to write a given fraction in percentages, we should multiply the fraction by 100. 3 3 For example, 4 = 4 X 100% = 75%. Ilustrative Examples Example 1.30 : The population.of a city according to rose to 1,11,200 in 1981. Find the percentage increase in the 1971 census was 84,500 and it Solution : The increase in population = 1,11,200 -84,500= population. 26,700. This growth is w.r.t. original population 84,500. on 84,500, increase is 26,700 100 on 100, the increase is 31.6 (approx.)84500 X 26700= 31.6 The percentage increase in population over the decade is 31.6 (approx ) Example 1.31 :A salesnnan gets 5%% commission in his 7870in amonth, find his sales in that month. sales. Ifhe gets a commission of Solution : For a commission of? 5, sales are 100. For a commission of 870, sales are 17,400 870 5X 100 = 17,400. His sales in that mnonth are R 17,400. Example 1.32 : The price ? 50 of an article was inCreased by 12%. As a resul, consumption decreased by 10%. FFind the percentage increase or decrease in the income. original  Stats EY.B.Com. Business Maths & 1.21 Pre-Requisites Solution : Suppose that the original consumption is 100 articles. Due to 12% rise, the article now costs 56 Original price 100, new price 112 80 = 89.6 100 Original price 50, new price 56 Due to 10% reduction in number of articles, now the consumption is 90 articles. New income = 90 x56 = 5,040 There is an increase of (5,040 - 5,000) = 40. For original income of 5,000, increase 40. 8 For original income of 100, increase 10 Thus, there is an increase of0.8% in the original income. Example 1.33 : In a school, there are 12% girls. If 5 boys and 15 girls are newly admitted to the school, the percentage of girls becomes 15. What is the total strength of the school ? Solution : Let the original strength of the school be x. 12 x Number of girls 100 12 x After new admissions, number of girls = 100,+ 15 Revised strength of the school = x+ 20 12 x + 15 100 Revised percentage of girls = X+ 20 X 100 12 x + 1500 X + 20 But the percentage is given to be 15. 12 x + 1500 = 15 (X + 20) 12x + 1500 = 15 x + 300 3x = 1200 X = 400 Original strength of the _chool is 400. Example 1.34 : A sold acar to B ut 15% profit. B sold the car to Cat 5% profit for 488,300. Finddthe price at which Ahas purchased the car. Solution : Suppose that A purchased the car for x. Since his profit is I5% i.e. 15 x he 100 has sold the car to B for 15 x 115 x. x+ 100, 100 F.Y.B.Com. Business Maths & Stats 1.22 Pre-Requisites B takes a profit of 5% on this. For C.P. 100, S.P. is 105 115 x 105 483 x X 100 = 400 For C.P. 115 x 483 x 100 100 ,S.P. is 400 483 Thus. 400 X = 48,300 X 100 400 X = 40,000 Ahad purchased the car for 40,000. Example 1.35 : Rates of electricity charges increased by 25%. In order to keep expenses on elertricity at the same level, by what per cent a family should reduce its consumption of eleciricity ? Solution: Let the original consumption of electricity be 100 units and rate be Re. 1 per unit. Original bill = 100 ×1 =7100 Due to the increase in the rates, new rate is ? 1.25 per unit. Let the new consumption be x units. Revised expenditure = 1.25 × x = 4 X As the family wants to maintain expenditure at original level, 5 4 X = 100 X = 80 Thus, the family should reduce its consumption by 20o. Example 1.36 Price of sugar increased by 10% as a : kg. less in 88. Find the original rate. result of which aperson gets l Solution: Let the original rate be ?x per kg. In ?88, the person would get 88 Due to increase, the new rate is X 11 x X kg. X + 10 10.. In ? 88, the person would get 88 80 11x 10 88 80 t`4 5:%en information, X X X = 8 Thus, the original rate of sugar is 8 per kg.  Stats Pre-Requisites EY.B.Com. Business Maths & 1.23 Exercise 1.1 Numerical Problems : numbers : 1 Exercise : Find the L.C.M. of H.C.F. of following (i) 240and 924, (i) 96, 72, (iii) 72, 240, 196. their H.C.F. is 33. Find their LCM. 2. The product of two numbers is 84942 and 3. Fill in the blanks 2005. His salary increased to (i) A person was getting a salary of 37500 in salary in 2005 is 41,500 in 2007.. Ratio of salary in 2007 to the (i) Arvind has T8 while Sameer has 40 paise... Ratio of amount with Arvind to that of Sameer is.....(ii) If x: y = 5:7and x= 30, then y (iv) Ahas3.200 and Bhas2.600.:. B has..... times the amount with A. (v) Manish has 3 bananas and Hari has 7.dozen bananas. Ratio of Manish's bananas to that of Hari is.. Ratio of two numbers is 4:7. The bigger number is 147. Hence the smaller (vi) number is Ratio of two numbers is 3 5 aDnd the sum of the numbers is 232. Hence, the (vii) bigger number is.... the is 8:3. If the perimeter of (vi) The ratio of length to breadth of a rectangle rectangle is 352 cm, then the sides of the rectangle are of Band C are in the ratio ratio 9:11 and those 4. Monthly incomes of Aand B are in the : 10. If monthly income of C is? 1,430, then find the incomes of AandB. 13 x. x and 50 are in continued proportion, th¹n findincreased 5. If8, by same non-zero constant, are each 6. f numerator and denominator in a ratio about the ratio? the ratio remains the same. What can you say 7. alloy of gold and copper weighing 100 gm contains 97 gm of gold. How much gold An of gold to 98 ? should be added to the alloy to increase the percentage 8. If an article is sold at 25% profit, find the ratio of cost price to selling price. 9. Sand and cement are in the ratio 6: 5 in a mixture weighing 671 kg. How much sand the ratio 8:5? must be added to the mixture so as to make and B are two alloys of gold and copper prepared by mixing them in the ratio 3: 2 10. A alloys are melted to form a third alloy, and 5: 8 respectively. If equal quantities of the alloy. then find the ratio of the gold to copper in the new 11. illin the blanks : () of an amount = % of that amount. (ii) A person having income of ? 1,640 spends 1,230, therefore, his expenses are % of his income. Astudent scores 42 marks out of 60... His percentage score is (ii) A clerk writes 9 in place of 90... He committees an error of..... % (iv)  F.Y.B.Com. 20. 19. 18. T7. 16. I3. 14. 13. 12. 28. 97 26. 25. 24. 23. 22. 21. TheMonthlywater book 7:9.IfA ln The Find (Assume Which cO..iferemaining overAwhat expenditure items, A Aorganisation sales. percentage byTheWhat another The student each twpassing. o In A 2:3:4. month. TheIncomesmuchincomes An ratio Theyears. 135 ratio 3416 person man customer 10% a student man total ornament monthly ages is theBusiness stock sales is market candidates, constituency 7:5. ages. their Find should litres and what should A of incomes 4:5.If value investment was spends the company, secured Find If of pure of of the at and declared and salary man of eams 30% gave on in of A A, If tw o be of cost Maths is paid share 2 it sharevotes saves gold A, incomes face 75,donated miscellaneous be in shopesreduced the each earns B of 10%B added milk the of 20% his 30% of bill gold and of x, subjectto telephone the ? B in the of 24 maximum andmustbe friends 36000, of cost a if & value gives the salary 330 by it of cast 80500 "Passed"marks Woman 20%% one-third C. of mixed pencil to of percentageon th e increased an winner of weighing C of and so A (i) Stats remainder his 8o. were A's of x for year by a are A are find of charity a a ?What house organisation in company more? added income that B a :3 Lotal favour and note to share30%0better items sanu ? 220 The in 10% votersmarks in and inthe are the with rent, telephone the 2008 obtained hence he than if of th e that =(x 1ncomela anmount iS reduction difference by his 28 the Bincomes in book to ratio water, to return:rust. to theirom ratio as in40% of 60% at securesemploying income, ratio it, gmnexceeds are ratio the of + be eaen 30% the each the a 2):5, compared in scored woman, so in ratio of ispen a 1:2:3contains amount 1.24 Tolal to his 4:7. 2007 in examination. 2600 cast the the 100.) O1 on calls. t candidatehe as 16: of milk 15, is 40 or nree sSalary.fo0d, in in the their 40% find to 16% ratio (i) After year 4 6:7 ratio B amount Son.spent the new year votes on 25 make 27and to find 2:3 20 The to year marks of thewhile gold andwater of number th e 2006 votes anmen of 9:7 and stockdaughters, at 12%% advertising, sales 2008 the the : Or he under lf telephone 2007. ? more average. ratio.of the and C's C. milk while x those and the year and more total and theirratio income and the becomes cost =4:5. spends on is as was copper share 90. remainder, each clothing, of 79 2006 Due than in difference to that 35 of frontax 2 ench The calls articles. compared 50%. a marks. than expenditures What their of those water of of charges sales women B a heading ? to straight his savings. gold in by and pencil. 5 cost 1350 competitionAfter that ? of :2. is remaining 10% in In is the o he in opponent. to 120. B in ol order increased to are monthly an is copperratio are C 7 gave ? on on Find shop the contest are and their : a miscellaneous the examination 7,80,000per necessary Find in 2. pell Pre-Requisite to the A yeartakeover C the How amount S0% fuel in 13: are age keep by increased 15:1? to and original by between What salary th e monthly 1. ratio to 20017. other in is much a 10% ratio How SS note his the the is for the