RATIO AND PROPORTION, INDICES, LOGARITHMS

CHAPTER RATIO AND PROPORTION, INDICES, LOGARITHMS UNIT I: RATIO LEARNING OBJECTIVES After reading this unit a student will learn How to compute and compare two ratios; Effect of increase or decrease of a quantity on the ratio; The concept and application of inverse ratio. UNIT OVERVIEW Overview of Ratios Inverse Ratio Type of Ratios Ratio Triplicate Ratio Sub-triplicate Ratio Duplicate Ratio Sub-duplicate Ratio We use ratio in many ways in practical fields. For example, it is given that a certain sum of money is divided into three parts in the given ratio. If first part is given then we can find out total amount and the other two parts. In the case when ratio of boys and girls in a school is given and the total number of student is also given, then if we know the number of boys in the school, we can find out the number of girls of that school by using ratios.

. BUSINESS MATHEMATICS. RATIO A ratio is a comparison of the sizes of two or more quantities of the same kind by division. If a and b are two quantities of the same kind (in same units), then the fraction a/b is called the ratio of a to b. It is written as a : b. Thus, the ratio of a to b = a/b or a : b. The quantities a and b are called the terms of the ratio, a is called the first term or antecedent and b is called the second term or consequent. For example, in the ratio 5 : 6, 5 & 6 are called terms of the ratio. 5 is called first term and 6 is called second term... Remarks Both terms of a ratio can be multiplied or divided by the same (non zero) number. Usually a ratio is expressed in lowest terms (or simplest form). Illustration I: : 6 = /6 = ( 4)/(4 4) = /4 = : 4 The order of the terms in a ratio is important. Illustration II: : 4 is not same as 4 :. Ratio exists only between quantities of the same kind. Illustration III: (i) There is no ratio between number of students in a class and the salary of a teacher. (ii) There is no ratio between the weight of one child and the age of another child. Quantities to be compared (by division) must be in the same units. Illustration IV: (i) Ratio between 50 gm and kg = Ratio between 50 gm and 000 gm = 50/000 = /40 = : 40 (ii) Ratio between 5 minutes and 45 seconds = Ratio between (5 60) sec. and 45 sec. = 500/45 = 00/ = 00 : Illustration V: (i) Ratio between kg & 5 kg = /5 To compare two ratios, convert them into equivalent like fractions. Illustration VI: To find which ratio is greater : ;.6 : 4.8

RATIO AND PROPORTION, INDICES, LOGARITHMS. Solution: : = 7/ : 0/ = 7 : 0 = 7/0.6 : 4.8 =.6/4.8 = 6/48 = /4 L.C.M of 0 and 4 is 0. So, 7/0 = (7 )/(0 ) = 4/0 And /4 = ( 5)/(4 5) = 5/0 As 5 > 4 so, 5/0 > 4/0 i. e. /4 > 7/0 Hence,.6 : 4.8 is greater ratio. If a quantity increases or decreases in the ratio a : b then new quantity = b of the original quantity/a The fraction by which the original quantity is multiplied to get a new quantity is called the factor multiplying ratio. Illustration VII: Rounaq weighs 56.7 kg. If he reduces his weight in the ratio 7 : 6, find his new weight. Solution: Original weight of Rounaq = 56.7 kg Applications: He reduces his weight in the ratio 7 : 6 His new weight = (6 56.7)/7 = 6 8. = 48.6 kg Example : Simplify the ratio / : /8 : /6 Solution: L.C.M. of, 8 and 6 is 4. / : /8 : /6 = 4/ : 4/8 : 4/6 = 8 : : 4 Example : The ratio of the number of boys to the number of girls in a school of 70 students is : 5. If 8 new girls are admitted in the school, find how many new boys may be admitted so that the ratio of the number of boys to the number of girls may change to :. Solution: The ratio of the number of boys to the number of girls = : 5 Sum of the ratios = + 5 = 8 So, the number of boys in the school = ( 70)/8 = 70 And the number of girls in the school = (5 70)/8 = 450 Let the number of new boys admitted be x, then the number of boys become (70 + x). After admitting 8 new girls, the number of girls become 450 + 8 = 468 According to given description of the problem, (70 + x)/468 = / or, (70 + x) = x 468 or, 80 + x = 96 or, x = 6 or, x = 4. Hence the number of new boys admitted = 4.

.4 BUSINESS MATHEMATICS.. Inverse Ratio One ratio is the inverse of another if their product is. Thus a : b is the inverse of b : a and viceversa.. A ratio a : b is said to be of greater inequality if a>b and of less inequality if a<b.. The ratio compounded of the two ratios a : b and c : d is ac : bd. For example compound ratio of : 4 and 5 : 7 is 5 : 8. Compound ratio of :, 5 : 7 and 4 : 9 is 40 : 89.. A ratio compounded of itself is called its duplicate ratio. Thus a : b is the duplicate ratio of a : b. Similarly, the triplicate ratio of a : b is a : b. For example, duplicate ratio of : is 4 : 9. Triplicate ratio of : is 8 : 7. 4. The sub-duplicate ratio of a : b is a : b and the sub-triplicate ratio of a : b is a : b. For example sub-duplicate ratio of 4 : 9 is 4 : 9 = : And sub-triplicate ratio of 8 : 7 is 8 : 7 = :. 5. If the ratio of two similar quantities can be expressed as a ratio of two integers, the quantities are said to be commensurable; otherwise, they are said to be incommensurable. : cannot be expressed as the ratio of two integers and therefore, and are incommensurable quantities. 6. Continued Ratio is the relation (or compassion) between the magnitudes of three or more quantities of the same kind. The continued ratio of three similar quantities a, b, c is written as a : b : c. Applications: Illustration I: The continued ratio of ` 00, ` 400 and ` 600 is ` 00 : ` 400 : ` 600 = : :. Example : The monthly incomes of two persons are in the ratio 4 : 5 and their monthly expenditures are in the ratio 7 : 9. If each saves ` 50 per month, find their monthly incomes. Solution: Let the monthly incomes of two persons be ` 4x and ` 5x so that the ratio is ` 4x : ` 5x = 4 : 5. If each saves ` 50 per month, then the expenditures of two persons are ` (4x 50) and ` (5x 50). 4x 50 7 = or 6x 450 = 5x 50 5x 50 9 or, 6x 5x = 450 50, or, x = 00 Hence, the monthly incomes of the two persons are ` 4 00 and ` 5 00 i.e. ` 400 and ` 500.

RATIO AND PROPORTION, INDICES, LOGARITHMS.5 Example : The ratio of the prices of two houses was 6 :. Two years later when the price of the first has increased by 0% and that of the second by ` 477, the ratio of the prices becomes : 0. Find the original prices of the two houses. Solution: Let the original prices of two houses be ` 6x and ` x respectively. Then by the given conditions, 6x +0% of 6x = x + 477 0 6x +.6x or, =, or, 0x + x = 5x + 547 x + 477 0 or, 5x 5x = 547, or, 99x = 547; x = 5 Hence, the original prices of two houses are ` 6 5 and ` 5 i.e. ` 848 and `,9. Example : Find in what ratio will the total wages of the workers of a factory be increased or decreased if there be a reduction in the number of workers in the ratio 5 : and an increment in their wages in the ratio : 5. Solution: Let x be the original number of workers and ` y the (average) wages per workers. Then the total wages before changes = ` xy. After reduction, the number of workers = (x)/5 After increment, the (average) wages per workers = ` (5y)/ The total wages after changes = 5 5xy ( x) ( ` y) = ` 5 6 Thus, the total wages of workers get decreased from ` xy to ` 5xy/6 Hence, the required ratio in which the total wages decrease is EXERCISE (A) 5xy xy: =6:5. 6 Choose the most appropriate option (a) (b) (c) or (d).. The inverse ratio of : 5 is (a) 5 : (b) : 5 (c) : 5 (d) none of these. The ratio of two quantities is : 4. If the antecedent is 5, the consequent is (a) 6 (b) 60 (c) (d) 0. The ratio of the quantities is 5 : 7. If the consequent of its inverse ratio is 5, the antecedent is (a) 5 (b) 5 (c) 7 (d) none of these

.6 BUSINESS MATHEMATICS 4. The ratio compounded of :, 9 : 4, 5 : 6 and 8 : 0 is (a) : (b) : 5 (c) : 8 (d) none of these 5. The duplicate ratio of : 4 is (a) : (b) 4 : (c) 9 : 6 (d) none of these 6. The sub-duplicate ratio of 5 : 6 is (a) 6 : 5 (b) 6 : 5 (c) 50 : 7 (d) 5 : 6 7. The triplicate ratio of : is (a) 8 : 7 (b) 6 : 9 (c) : (d) none of these 8. The sub-triplicate ratio of 8 : 7 is (a) 7 : 8 (b) 4 : 8 (c) : (d) none of these 9. The ratio compounded of 4 : 9 and the duplicate ratio of : 4 is (a) : 4 (b) : (c) : (d) none of these 0. The ratio compounded of 4 : 9, the duplicate ratio of : 4, the triplicate ratio of : and 9 : 7 is (a) : 7 (b) 7 : (c) : (d) none of these. The ratio compounded of duplicate ratio of 4 : 5, triplicate ratio of :, sub duplicate ratio of 8 : 56 and sub-triplicate ratio of 5 : 5 is (a) 4 : 5 (b) : (c) : (d) none of these. If a : b = : 4, the value of (a+b) : (a+4b) is (a) 54 : 5 (b) 8 : 5 (c) 7 : 4 (d) 8 : 5. Two numbers are in the ratio :. If 4 be subtracted from each, they are in the ratio : 5. The numbers are (a) (6, 4) (b) (4, 6) (c) (, ) (d) none of these 4. The angles of a triangle are in ratio : 7 :. The angles are (a) (0, 70, 90 ) (b) (0, 70, 80 ) (c) (8, 6, 99 ) (d) none of these 5. Division of ` 4 between X and Y is in the ratio : 7. X & Y would get Rupees (a) (04, 0) (b) (00, 4) (c) (80, 44) (d) none of these 6. Anand earns ` 80 in 7 hours and Promode ` 90 in hours. The ratio of their earnings is (a) : (b) : (c) 8 : 9 (d) none of these 7. The ratio of two numbers is 7 : 0 and their difference is 05. The numbers are (a) (00, 05) (b) (85, 90) (c) (45, 50) (d) none of these 8. P, Q and R are three cities. The ratio of average temperature between P and Q is : and that between P and R is 9 : 8. The ratio between the average temperature of Q and R is (a) : 7 (b) 7 : (c) : (d) none of these 9. If x : y = : 4, the value of x y + xy : x + y is (a) : (b) : (c) : (d) none of these

RATIO AND PROPORTION, INDICES, LOGARITHMS.7 0. If p : q is the sub-duplicate ratio of p x : q x then x is p q pq (a) (b) (c) (d) none of these p+q p+q p+q. If s : t is the duplicate ratio of s p : t p then (a) p = 6st (b) p = 6st (c) p = st (d) none of these. If p : q = : and x : y = 4 : 5, then the value of 5px + qy : 0px + 4qy is (a) 7 : 8 (b) 7 : 8 (c) 7 : 8 (d) none of these. The number which when subtracted from each of the terms of the ratio 9 : reducing it to : 4 is (a) 5 (b) 5 (c) (d) none of these 4. Daily earnings of two persons are in the ratio 4:5 and their daily expenses are in the ratio 7 : 9. If each saves ` 50 per day, their daily earnings in ` are (a) (40, 50) (b) (50, 40) (c) (400, 500) (d) none of these 5. The ratio between the speeds of two trains is 7 : 8. If the second train runs 400 kms. in 5 hours, the speed of the first train is (a) 0 Km/hr (b) 50 Km/hr (c) 70 Km/hr (d) none of these SUMMARY A ratio is a comparison of the sizes of two or more quantities of the same kind by division. If a and b are two quantities of the same kind (in same units), then the fraction a/b is called the ratio of a to b. It is written as a : b. Thus, the ratio of a to b = a/b or a : b. The quantities a and b are called the terms of the ratio, a is called the first term or antecedent and b is called the second term or consequent. The ratio compounded of the two ratios a : b and c : d is ac : bd. A ratio compounded of itself is called its duplicate ratio. a : b is the duplicate ratio of a b. Similarly, the triplicate ratio of a : b is a : b. For any ratio a : b, the inverse ratio is b : a The sub-duplicate ratio of a : b is a : b and the sub-triplicate ratio of a : b is a / : b /. Continued Ratio is the relation (or compassion) between the magnitudes of three or more Quantities of the same kind. The continued ratio of three similar quantities a, b, c is written as a : b : c.

.8 BUSINESS MATHEMATICS UNIT II: PROPORTIONS LEARNING OBJECTIVES After reading this unit a student will learn What is proportion? Properties of proportion and how to use them. UNIT OVERVIEW Overview of Proportions Properties of Proportions Proportions Crossproducet rule Invertendo Altertendo Componendo Dividendo Compodendo and Dividendo Addendo Cross Product Rule Continued Propritons Mean Proportonal. PROPORTIONS If the income of a man is increased in the given ratio and if the increase in his income is given then to find out his new income, Proportion problem is used. Again if the ages of two men are in the given ratio and if the age of one man is given, we can find out the age of the another man by Proportion. An equality of two ratios is called a proportion. Four quantities a, b, c, d are said to be in proportion if a : b = c : d (also written as a : b :: c : d) i.e. if a/b = c/d i.e. if ad = bc. The quantities a, b, c, d are called terms of the proportion; a, b, c and d are called its first, second, third and fourth terms respectively. First and fourth terms are called extremes (or extreme terms). Second and third terms are called means (or middle terms). If a : b = c : d then d is called fourth proportional. If a : b = c : d are in proportion then a/b = c/d i.e. ad = bc i.e. product of extremes = product of means.

RATIO AND PROPORTION, INDICES, LOGARITHMS.9 This is called cross product rule. Three quantities a, b, c of the same kind (in same units) are said to be in continuous proportion if a : b = b : c i.e. a/b = b/c i.e. b = ac If a, b, c are in continuous proportion, then the middle term b is called the mean proportional between a and c, a is the first proportional and c is the third proportional. Thus, if b is mean proportional between a and c, then b = ac i.e. b = ac. When three or more numbers are so related that the ratio of the first to the second, the ratio of the second to the third, third to the fourth etc. are all equal, the numbers are said to be in continued proportion. We write it as x/y = y/z = z/w = w/p = p/q =... when x, y, z, w, p and q are in continued proportion. If a ratio is equal to the reciprocal of the other, then either of them is in inverse (or reciprocal) proportion of the other. For example 5/4 is in inverse proportion of 4/5 and vice-versa. Note: In a ratio a : b, both quantities must be of the same kind while in a proportion a : b = c : d, all the four quantities need not be of the same type. The first two quantities should be of the same kind and last two quantities should be of the same kind. Applications : Illustration I: ` 6 : ` 8 = toffees : 6 toffees are in a proportion. Here st two quantities are of same kind and last two are of same kind. Example : The numbers.4,.,.5, are in proportion because these numbers satisfy the property the product of extremes = product of means. Here.4 = 4.8 and..5 = 4.8 Example : Find the value of x if 0/ : x : : 5/ : 5/4. Solution: 0/ : x = 5/ : 5/4 Using cross product rule, x 5/ = (0/) 5/4 Or, x = (0/) (5/4) (/5) = 5/ Example : Find the fourth proportional to /, /7, 4. Solution: If the fourth proportional be x, then /, /7, 4, x are in proportion. Using cross product rule, (/) x = ( 4)/7 Example 4: Find the third proportion to.4 kg, 9.6 kg. Solution: or, x = ( 4 )/(7 ) = 8/7. Let the third proportion to.4 kg, 9.6 kg be x kg. Then.4 kg, 9.6 kg and x kg are in continued proportion since b = ac So,.4/9.6 = 9.6/x or, x = (9.6 9.6)/.4 = 8.4

.0 BUSINESS MATHEMATICS Hence the third proportional is 8.4 kg. Example 5: Find the mean proportion between.5 and.8. Solution: Mean proportion between.5 and.8 is.5.8 =.5 =.5... Properties of Proportion. If a : b = c : d, then ad = bc a c Proof. = ; ad = bc (By cross - multiplication) b d. If a : b = c : d, then b : a = d : c (Invertendo) a c a c b d Proof. = or =, or, = b d b d a c Hence, b : a = d : c.. If a : b = c : d, then a : c = b : d (Alternendo) a c Proof. = or, ad = bc b d Dividing both sides by cd, we get ad bc a b =, or =, i.e. a : c = b : d. cd cd c d 4. If a : b = c : d, then a + b : b = c + d : d (Componendo) a c a c Proof. =, or, += + b d b d a + b c + d or, =, i.e. a + b : b = c + d : d. b d 5. If a : b = c : d, then a b : b = c d : d (Dividendo) a c a c Proof. =, = b d b d a b c d =, i.e. a b : b = c d : d. b d 6. If a : b = c : d, then a + b : a b = c + d : c d (Componendo and Dividendo) a c a c a + b c + d Proof. =, or = +, or = b d b d b d... Again a,= c, or a b = c d b d b d...

RATIO AND PROPORTION, INDICES, LOGARITHMS. Dividing () by () we get a + b c + d =, i.e. a + b: a b = c +d : c d a b c d 7. If a : b = c : d = e : f =...., then each of these ratios (Addendo) is equal (a + c + e +..) : (b + d + f +.) a c e Proof. =,= =...(say) k, b d f a = bk, c = dk, e = fk,... Now a + c + e... a + c + e... =k(b + d + f)... or =k b + d + f... Hence, (a + c + e +..) : (b + d + f +.) is equal to each ratio Example : If a : b = c : d =.5 :.5, what are the values of ad : bc and a + c : b + d? Solution: We have a = c, =.5 b d.5...() From () ad = bc, or, Again from () a + c.5 5 5 = = =, b + d.5 5 ad =, i.e. ad : bc = : bc a c a + c = = b d b + d i.e. a + c : b + d = 5 : Hence, the values of ad : bc and a + c : b + d are : and 5 : respectively. Example : If a = b = c 4 7, then prove that a + b +c = c Solution: We have a = b = c 4 7 = a + b + c = a + b + c + 4 + 7 4 a + b + c c a + b + c 4 = or = = 4 7 c 7 Example : A dealer mixes tea costing ` 6.9 per kg. with tea costing ` 7.77 per kg and sells the mixture at ` 8.80 per kg and earns a profit of he mix them? 7 % on his sale price. In what proportion does Solution: Let us first find the cost price (C.P.) of the mixture. If S.P. is ` 00, profit is 65 7 Therefore C.P. = ` (00-7 ) = ` 8 = `

. BUSINESS MATHEMATICS If S.P. is ` 8.80, C.P. is (65 8.80)/( 00) = ` 7.6 C.P. of the mixture per kg = ` 7.6 nd difference = Profit by selling kg. of nd kind @ ` 7.6 = ` 7.77 ` 7.6 = 5 Paise st difference = ` 7.6 ` 6.9 = 4 Paise We have to mix the two kinds in such a ratio that the amount of profit in the first case must balance the amount of loss in the second case. Hence, the required ratio = (nd diff) : (st diff.) = 5 : 4 = :. EXERCISE (B) Choose the most appropriate option (a) (b) (c) or (d).. The fourth proportional to 4, 6, 8 is (a) (b) (c) 48 (d) none of these. The third proportional to, 8 is (a) 4 (b) 7 (c) 6 (d) none of these. The mean proportional between 5, 8 is (a) 40 (b) 50 (c) 45 (d) none of these 4. The number which has the same ratio to 6 that 6 has to is (a) (b) 0 (c) (d) none of these 5. The fourth proportional to a, a, c is (a) ac/ (b) ac (c) /ac (d) none of these 6. If four numbers /, /, /5, /x are proportional then x is (a) 6/5 (b) 5/6 (c) 5/ (d) none of these 7. The mean proportional between x and 7y is (a) 8xy (b) 8xy (c) 8xy (d) none of these (Hint: Let z be the mean proportional and z = (x x 7y ) 8. If A = B/ = C/5, then A : B : C is (a) : 5 : (b) : 5 : (c) : : 5 (d) none of these 9. If a/ = b/4 = c/7, then a + b + c/c is (a) (b) (c) (d) none of these 0. If p/q = r/s =.5/.5, the value of ps : qr is (a) /5 (b) : (c) 5/ (d) none of these. If x : y = z : w =.5 :.5, the value of (x + z)/(y + w) is (a) (b) /5 (c) 5/ (d) none of these. If (5x y)/(5y x) = /4, the value of x : y is (a) : 9 (b) 7 : (c) 7 : 9 (d) none of these

RATIO AND PROPORTION, INDICES, LOGARITHMS.. If A : B = : and B : C = : 5, then A : B : C is (a) 9 : 6 : 0 (b) 6 : 9 : 0 (c) 0 : 9 : 6 (d) none of these 4. If x/ = y/ = z/7, then the value of (x 5y + 4z)/y is (a) 6/ (b) /6 (c) / (d) 7/6 5. If x : y = :, y : z = 4 : then x : y : z is (a) : : 4 (b) 4 : : (c) : : 4 (d) none of these 6. Division of ` 750 into parts in the ratio 4 : 5 : 6 is (a) (00, 50, 00) (b) (50, 50, 50) (c) (50, 50, 50) (d) 8 : : 9 7. The sum of the ages of persons is 50 years. 0 years ago their ages were in the ratio 7 : 8 : 9. Their present ages are (a) (45, 50, 55) (b) (40, 60, 50) (c) (5, 45, 70) (d) none of these 8. The numbers 4, 6, 5, 4 are not in proportion. The fourth term for which they will be in proportion is (a) 45 (b) 40 (c) (d) none of these 9. If x/y = z/w, implies y/x = w/z, then the process is called (a) Dividendo (b) Componendo (c) Alternendo (d) none of these 0. If p/q = r/s = p r/q s, the process is called (a) Subtrahendo (b) Addendo (c) Invertendo (d) none of these. If a/b = c/d, implies (a + b)/(a b) = (c + d)/(c d), the process is called (a) Componendo (b) Dividendo (c) Componendo (d) none of these and Dividendo. If u/v = w/p, then (u v)/(u + v) = (w p)/(w + p). The process is called (a) Invertendo (b) Alternendo (c) Addendo (d) none of these., 6, *, 0 are in proportion. Then * is (a) 5 (b) 4 (c) 5 (d) none of these 4. 4, *, 9, ½ are in proportion. Then * is (a) 6 (b) 8 (c) 9 (d) none of these 5. The mean proportional between.4 gms and 5.6 gms is (a) 8 gms (b).8 gms (c). gms (d) none of these 6. If a b c a+b+c = = then is 4 5 9 c (a) 4 (b) (c) 7 (d) none of these. 7. Two numbers are in the ratio : 4; if 6 be added to each terms of the ratio, then the new ratio will be 4 : 5, then the numbers are (a) 4, 0 (b) 7, 9 (c) 8 and 4 (d) none of these

.4 BUSINESS MATHEMATICS 8. If a = b 4 5 then (a) a + 4 b - 5 = a - 4 b + 5 (b) a + 4 b + 5 = a - 4 b - 5 (c) a - 4 b + 5 = a + 4 b - 5 (d) none of these 9. If a b a : b = 4 : then + is b a (a) 5/ (b) 4 (c) 5 (d) none of these 0. x y z If = = then b+ca c+ab a+bc (b c)x + (c a)y + (a b)z is (a) (b) 0 (c) 5 (d) none of these SUMMARY p : q = r : s => q : p = s : r (Invertendo) (p/q = r/s) => (q/p = s/r) a : b = c : d => a : c = b : d (Alternendo) (a/b = c/d) => (a/c = b/d) a : b = c : d => a + b : b = c + d : d (Componendo) (a/b = c/d) => (a + b)/b = (c + d)/d a : b = c : d => a b : b = c d : d (Dividendo) (a/b = c/d) => (a b)/b = (c d)/d a : b = c : d => a + b : a b = c + d : c d (Componendo & Dividendo) (a + b)/(a b) = (c + d)/(c d) a : b = c : d = a + c : b + d (Addendo) (a/b = c/d = a + c/b + d) a : b = c : d = a c : b d (Subtrahendo) (a/b = c/d = a c/b d) If a : b = c : d = e : f =... then each of these ratios = (a c e...) : (b d f...) The quantities a, b, c, d are called terms of the proportion; a, b, c and d are called its first, second, third and fourth terms respectively. First and fourth terms are called extremes (or extreme terms). Second and third terms are called means (or middle terms).

RATIO AND PROPORTION, INDICES, LOGARITHMS.5 If a : b = c : d are in proportion then a/b = c/d i.e. ad = bc i.e. product of extremes = product of means. This is called cross product rule. Three quantities a, b, c of the same kind (in same units) are said to be in continuous proportion if a : b = b : c i.e. a/b = b/c i.e. b = ac If a, b, c are in continuous proportion, then the middle term b is called the mean proportional between a and c, a is the first proportional and c is the third proportional. Thus, if b is mean proportional between a and c, then b = ac i.e. b = ac. UNIT III: INDICES LEARNING OBJECTIVES After reading this unit a student will learn A meaning of indices and their applications. Laws of indices which facilitates their easy applications. UNIT OVERVIEW Overview of Indices Meaning of Indices Laws of Indices Law Law Law. INDICES: We are aware of certain operations of addition and multiplication and now we take up certain higher order operations with powers and roots under the respective heads of indices. We know that the result of a repeated addition can be held by multiplication e.g. 4 + 4 + 4 + 4 + 4 = 5(4) = 0 a + a + a + a + a = 5(a) = 5a Now, 4 4 4 4 4 = 4 5 ; a a a a a = a 5.

.6 BUSINESS MATHEMATICS It may be noticed that in the first case 4 is multiplied 5 times and in the second case a is multiplied 5 times. In all such cases a factor which multiplies is called the base and the number of times it is multiplied is called the power or the index. Therefore, 4 and a are the bases and 5 is the index for both. Any base raised to the power zero is defined to be ; i.e. a o =. We also define r r a =a. If n is a positive integer, and a is a real number, i.e. n N and a R (where N is the set of positive integers and R is the set of real numbers), a is used to denote the continued product of n factors each equal to a as shown below: a n = a a a.. to n factors. Here a n is a power of a whose base is a and the index or power is n. For example, in = 4, is base and 4 is index or power. Law a m a n = a m+n, when m and n are positive integers; by the above definition, a m = a a.. to m factors and a n = a a.. to n factors. a m a n = (a a.. to m factors) (a a.. to n factors) = a a.. to (m + n) factors = a m+n Now, we extend this logic to negative integers and fractions. First let us consider this for negative integer, that is m will be replaced by n. By the definition of a m a n = a m+n, we get a n a n = a n+n = a 0 = For example 4 5 = ( ) ( ) = 4 + 5 = 9 Again, 5 = / 5 = /( ) = /4 Example : Simplify / - x x if x = 4 Solution: We have / - x x = / - /- 6x x =6x = / 6x 6 6 6 6 = = = = = / / / x 4 ( ) Example : Simplify 6ab c 4b c d. Solution: 6ab c 4b c d = 4 a b b c c d = 4 a b +( ) c +( ) d

RATIO AND PROPORTION, INDICES, LOGARITHMS.7 = 4 a b c d = 4a b 0 c 0 d = 4ad Law a m /a n = a m n, when m and n are positive integers and m > n. By definition, a m = a a.. to m factors m m n a aa...to m factors Therefore, a a = = n a aa...to n factors = a a to (m n) factors = a m n Now we take a numerical value for a and check the validity of this Law 7 7 4...to7 factors = = 4...to4factors =.. to (7 4) factors. =.. to factors = = 8 or 7 7 4 = = 4 = = ++ = = 8 Example : Find the value of 4 x x - -/ 4 x Solution: / x = - - (-/) 4x = - + / 4x 4 -/ = 4x or x /

.8 BUSINESS MATHEMATICS Example 4: Simplify 7 a a 6a 9a - 5 a if a= 4 Solution: 7 a a 6a 9a - 5 a if a= 4 =...a.a 7 5 - = 4 a a ( + 4-4) / 6 (-0 + 9) / 6 4 a =. a - 7/6 -/6 = 7 4 a 6 6 = 4 a - = 4. a = 4. 4 Law (a m ) n = a mn. where m and n are positive integers By definition (a m ) n = a m a m a m.. to n factors = (a a.. to m factors) a a.. to n factors..to n times = a a.. to mn factors = a mn Following above, (a m ) n = (a m ) p/q (We will keep m as it is and replace n by p/q, where p and q are positive integers) Now the qth power of (a m ) is p/q m (a ) = (a m ) (p/q)x q = a mp p/q q If we take the qth root of the above we obtain mp /q q a = a mp Now with the help of a numerical value for a let us verify this law. ( 4 ) = 4 4 4 = 4+4+4 = = 4096

RATIO AND PROPORTION, INDICES, LOGARITHMS.9 Law 4 (ab) n = a n. b n when n can take all of the values. For example 6 = ( ) = = First, we look at n when it is a positive integer. Then by the definition, we have (ab) n = ab ab. to n factors = (a a... to n factors) (b b. n factors) = a n b n When n is a positive fraction, we will replace n by p/q. Then we will have (ab) n = (ab) p/q The qth power of (ab) p/q = {(ab) (p/q) } q = (ab) p Example 5: Simplify (x a.y b ). (x y ) a Solution: (x a.y b ). (x y ) a = (x a ). (y b ). (x ) a. (y ) a = x a a. y b a. = x 0. y b a. = b+a y 4b 6 / - -b Example 6: 6 a x.(a x ) 4b 6 / - -b Solution: 6 a x.(a x ) 4b 6 -b - -b = 6 (a x ).( a ).(x ) = = = 4b 6 6 6 b --b (a ).(x ). a.x b b a.x.a.x b b a.x +b b = a 0. x +b = x +b Example 7: Find x, if / x x/ Solution: x (x) x.x x x x = (x x )

.0 BUSINESS MATHEMATICS or, / x x/ x x or, / x/ x x [If base is equal, then power is also equal] x i.e. or, x x = Example 8: Find the value of k from ( 9 ) 7 ( ) 5 = k Solution: ( 9 ) 7 ( ) 5 = k or, ( / ) 7 ( ½ ) 5 = k or, 7 5/ = k or, 9/ = k or, k = 9/ SUMMARY a m a n = a m + n (base must be same) Ex. = + = 5 a m a n = a m n Ex. 5 = 5 = (a m ) n = a mn Ex. ( 5 ) = 5 = 0 a o = Ex. 0 =, 0 = a m = /a m and /a m = a m Ex. = / and / 5 = 5 If a x = a y, then x=y If x a = y a, then x=y m a = a /m, x = x ½, 4 = / / = = Ex. 8 = 8 / = ( ) / = / =

RATIO AND PROPORTION, INDICES, LOGARITHMS. EXERCISE (C) Choose the most appropriate option (a) (b) (c) or (d).. 4x /4 is expressed as (a) 4x /4 (b) x (c) 4/x /4 (d) none of these. The value of 8 / is (a) (b) 4 (c) (d) none of these. The value of () /5 is (a) (b) 0 (c) 4 (d) none of these 4. The value of 4/() /5 is (a) 8 (b) (c) 4 (d) none of these 5. The value of (8/7) / is (a) / (b) / (c) /9 (d) none of these 6. The value of (56) /8 is (a) (b) (c) / (d) none of these 7. ½. 4 ¾ is equal to (a) a fraction (b) a positive integer (c) a negative integer (d) none of these 8. 8x -8 y 4 4 has simplified value equal to (a) xy (b) x y (c) 9xy (d) none of these 9. x a b x b c x c a is equal to (a) x (b) (c) 0 (d) none of these p q 0. The value of xy 0 where p, q, x, y 0 is equal to (a) 0 (b) / (c) (d) none of these. {( ) (4 ) (5 ) } / {( ) (4 ) (5 ) } is (a) /4 (b) 4/5 (c) 4/7 (d). Which is True? (a) 0 > (/) 0 (b) 0 < (/) 0 (c) 0 = (/) 0 (d) none of these. If x /p = y /q = z /r and xyz =, then the value of p + q + r is (a) (b) 0 (c) / (d) none of these 4. The value of y a b y b c y c a y a b is (a) y a+b (b) y (c) (d) /y a+b

. BUSINESS MATHEMATICS 5. The True option is (a) x / = x (b) x / = x (c) x / > x (d) x / < x 6. The simplified value of 6x y 8 x y is (a) xy (b) xy/ (c) (d) none of these 7. The value of (8/7) / (/4) /5 is (a) 9/4 (b) 4/9 (c) / (d) none of these 8. The value of {(x + y) / (x y) / /x+y (x y) } 6 is (a) (x + y) (b) (x y) (c) x + y (d) none of these 9. Simplified value of (5) / 5 5 5 / is (a) 5 (b) /5 (c) (d) none of these 0. [{() /. (4) /4. (8) 5/6. (6) 7/8. () 9/0 } 4 ] /5 is (a) A fraction (b) an integer (c) (d) none of these. [ { ( x ) } ] / is equal to (a) x (b) /x (c) (d) none of these n - n+ x n is equal to n. (a) x n (b) x n+ (c) x n (d) none of these. If a b = (a b) (a + ab + b ), then the simplified form of l l l m n x x x m n l x x x + m+m m +mn+n l + ln+n (a) 0 (b) (c) x (d) none of these 4. Using (a b) = a b ab(a b) tick the correct of these when x = p / p / (a) x +x = p + /p (b) x + x = p /p (c) x + x = p + (d) none of these 5. On simplification, /(+a m n +a m p ) + /(+a n m +a n p ) + /(+a p m +a p n ) is equal to (a) 0 (b) a (c) (d) /a 6. The value of 7. If a+b b+c c+a a b c x x x b c a x x x (a) (b) 0 (c) (d) none of these x = + -, then x -9x is (a) 5 (b) 0 (c) (d) none of these

RATIO AND PROPORTION, INDICES, LOGARITHMS. 8. If a x = b, b y = c, c z = a, then xyz is (a) (b) (c) (d) none of these (a +ab+b ) (b +bc+c ) (c +ca+a ) a b c x x x 9. The value of b c a x x x (a) (b) 0 (c) (d) none of these 0. If x = y = 6 -z, + + is x y z (a) (b) 0 (c) (d) none of these UNIT IV: LOGARITHM LEARNING OBJECTIVES After reading this unit a student will learn After reading this unit, a student will get fundamental knowledge of logarithm and its application for solving business problems. UNIT OVERVIEW Overview of Logarithms Change of Base Fundamental Laws of Logarithms Definition of Logarithms Logarithms of the numbers same base Quotient of two numbers Product of two numbers Anti Logarithms.4 LOGONITHMS: The logarithm of a number to a given base is the index or the power to which the base must be raised to produce the number, i.e. to make it equal to the given number. If there are three quantities indicated by say a, x and n, they are related as follows: If a x = n, where n > 0, a > 0 and a then x is said to be the logarithm of the number n to the base a symbolically it can be expressed as follows:

.4 BUSINESS MATHEMATICS log a n = x i.e. the logarithm of n to the base a is x. We give some illustrations below: (i) 4 = 6 log 6 = 4 i.e. the logarithm of 6 to the base is equal to 4 (ii) 0 = 000 log 0 000 = (iii) i.e. the logarithm of 000 to the base 0 is 5 = log 5 - i.e. the logarithm of (iv) = 8 log 8 = 5 = - 5 5 to the base 5 is i.e. the logarithm of 8 to the base is. The two equations a x = n and x = log a n are only transformations of each other and should be remembered to change one form of the relation into the other.. The logarithm of to any base is zero. This is because any number raised to the power zero is one. Since a 0 =, log a = 0. The logarithm of any quantity to the same base is unity. This is because any quantity raised to the power is that quantity only. Since a = a, log a a = ILLUSTRATIONS:. If log a = find the value of a., 6 We have a /6 6 = a = ( ) = = 8. Find the logarithm of 58 to the base. Let us take log 58 = x x 6 6 6 6 We may write, ( ) =58=8 79= =( ) () =( ) Hence, x = 6 Logarithms of numbers to the base 0 are known as common logarithm..4. Fundamental Laws of Logarithm. Logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers to the same base, i.e. log a mn = log a m + log a n

RATIO AND PROPORTION, INDICES, LOGARITHMS.5 Proof: Let log a m = x so that a x = m (I) Log a n = y so that a y = n (II) Multiplying (I) and (II), we get m n = a x a y = a x+y log a mn = x + y (by definition) log a mn = log a m + log a n. The logarithm of the quotient of two numbers is equal to the difference of their logarithms to the same base, i.e. Proof: log a m n = log a m log a n Let log a m = x so that a x = m (I) log a n = y so that a y = n (II) Dividing (I) by (II) we get x m a = =a y n a x-y Then by the definition of logarithm, we get log a m n = x y = log a m log a n Similarly, log a = log a - logan = 0 - logan = log n a n [ log a = 0] Illustration I: log ½ = log log = log. Logarithm of the number raised to the power is equal to the index of the power multiplied by the logarithm of the number to the same base i.e. log a m n = n log a m Proof: i.e. Let log a m = x so that a x = m Raising the power n on both sides we get (a x ) n = (m) n a xn = m n log a m n = nx log a m n = n log a m (by definition)

.6 BUSINESS MATHEMATICS Illustration II: (a) Find the logarithm of 78 to the base. Solution: We have 78 = 6 = 6 () 6 = () 6 ; and so, we may write log 78 = 6 (b) Solve log 0 5 - log 0 + log 08 Solution: The given expression.4. Change of Base = log 5 - log +log 8 0 0 0 = log 5 - log 0 0 9 + log 0 8 5x8 = log 0 0 =log 0 = 9 If the logarithm of a number to any base is given, then the logarithm of the same number to any other base can be determined from the following relation. log a m logam =log bm log a b log bm = log b Proof: Let log a m = x, log b m = y and log a b = z Then by definition, a x = m, b y = m and a z = b Also a x = b y = (a z ) y = a yz Therefore, x = yz log a m = log b m log a b log m a log bm = log a b Putting m = a, we have log a a = log b a log a b log b a log a b =, since log a a =. Example : Change the base of log 5 into the common logarithmic base. a Solution: log x b Since log a x = log b a

RATIO AND PROPORTION, INDICES, LOGARITHMS.7 log 0 log 5 = log 0 5 Example : log 8 Prove that = log 0 log 9 6 log 40 Solution: Change all the logarithms on L.H.S. to the base 0 by using the formula. log Logarithm Tables: log x a bx=, we may write loga b log 0 8 log 0 log 0 log 8 log log log 0 0 0 log 06 log 0 4log 0 log 96 log 9 log log 0 0 0 log 0 0 log 40 log00 = log 4 log log 0 0 0 4 0 0 0 L.H.S.= log 0 = = log 0 = R.H.S. log log log log 4log 0 0 The logarithm of a number consists of two parts, the whole part or the integral part is called the characteristic and the decimal part is called the mantissa where the former can be known by mere inspection, the latter has to be obtained from the logarithm tables. Characteristic: The characteristic of the logarithm of any number greater than is positive and is one less than the number of digits to the left of the decimal point in the given number. The characteristic of the logarithm of any number less than one () is negative and numerically one more than the number of zeros to the right of the decimal point. If there is no zero then obviously it will be. The following table will illustrate it. Number Characteristic 7 One less than the number of digits to 4 6 the left of the decimal point 6. 0 Number [ ] Characteristic.8 One more than the number of zeros on.07 the right immediately after the decimal point. 0 [ ]

.8 BUSINESS MATHEMATICS.00507.000670 4 Zero on positive characteristic when the number under consideration is greater than unity: Since 0 0 =, log = 0 0 = 0, log 0 = 0 = 00, log 00 = 0 = 000, log 000 = All numbers lying between and 0 i.e. numbers with digit in the integral part have their logarithms lying between 0 and. Therefore, their integral parts are zero only. All numbers lying between 0 and 00 have two digits in their integral parts. Their logarithms lie between and. Therefore, numbers with two digits have integral parts with as characteristic. In general, the logarithm of a number containing n digits only in its integral parts is (n ) + a decimal. For example, the characteristics of log 75, log 796, log.76 are, 4 and 0 respectively. Negative characteristics Since 0 = =0. log 0.=- 0-0 = =0.0 log 0.0=- 00 - All numbers lying between and 0. have logarithms lying between 0 and, i.e. greater than and less than 0. Since the decimal part is always written positive, the characteristic is. All numbers lying between 0. and 0.0 have their logarithms lying between and as characteristic of their logarithms. In general, the logarithm of a number having n zeros just after the decimal point is (n + ) + a decimal. Hence, we deduce that the characteristic of the logarithm of a number less than unity is one more than the number of zeros just after the decimal point and is negative. Mantissa The mantissa is the fractional part of the logarithm of a given number. Number Mantissa Logarithm Log 4594 = ( 66) =.66 Log 459.4 = ( 66) =.66 Log 45.94 = ( 66) =.66 Log 4.594 = ( 66) = 0.66 Log.4594 = ( 66) =.66

RATIO AND PROPORTION, INDICES, LOGARITHMS.9 Thus with the same figures there will be difference in the characteristic only. It should be remembered, that the mantissa is always a positive quantity. The other way to indicate this is Log.004594 = +.66 =.66. Negative mantissa must be converted into a positive mantissa before reference to a logarithm table. For example.687 = 4 + (.687) = 4 + 0.8 = 4.8 It may be noted that 4.8 is different from 4.8 as 4.8 is a negative number whereas, in 4.8, 4 is negative while.8 is positive. Illustration I: Add 4.7468 and.467 4 +.7468 Antilogarithms +.467 +.6995 = 0.6995 If x is the logarithm of a given number n with a given base then n is called the antilogarithm (antilog) of x to that base. This can be expressed as follows: If log a n = x then n = antilog x For example, if log 670 = 4.7904 then 670 = antilog 4.7904 Number Logarithm 06.9 0.6.9.06 0.9.06.9.006.9 Example : Find the value of log 5 if log is equal to.00. Solution: 0 log5 = log =log 0 - log =.00 =.6990 Example : Find the number whose logarithm is.4678. Solution: From the antilog table, for mantissa.467, the number = 9 for mean difference 8, the number = 5 for mantissa.4678, the number = 96

.0 BUSINESS MATHEMATICS The characteristic is, therefore, the number must have digits in the integral part. Hence, Antilog.4678 = 9.6 Example : Find the number whose logarithm is.4678. Solution: -.4678 = - + -.4678 = - +.5 =.5 For mantissa.5, the number = 404 For mean difference, the number = for mantissa.5, the number = 406 The characteristic is, therefore, the number is less than one and there must be two zeros just after the decimal point. Thus, Antilog (.4678) = 0.00406 Relation between Indices and Logarithm Let x = log a m and y = log a n so a x = m and a y = n a x. a y = mn or a x+y = mn or x +y = log a mn or log a m + log a n = log a mn [ log a a = ] or Also, or or (m/n) = a x /a y log a mn = log a m + log a n (m/n) = a x y log a (m/n) = (x y) or log a (m/n) = log a m log a n [ log a a = ] Again m n = m.m.m. to n times so log a m n = log a (m.m.m to n times) or log a m n or = log a m + log a m + log a m + + log a m log a m n = n log a m Now a 0 = 0 = log a Let log b a = x and log a b = y a = b x and b=a y so a = (a y ) x or a xy = a or xy =

RATIO AND PROPORTION, INDICES, LOGARITHMS. or log b a log a b = let log b c = x & log c b = y c = b x & b = c y so c = c xy or xy = l log b c log c b = l Example : Find the logarithm of 64 to the base Solution: log 64=log 8 = log 8 =log () = 4 log = 4x= 4 Example : Solution: If log a bc = x, log b ca = y, log c ab = z, prove that x + y + z + x+ = log a bc + log a a = log a abc y+ = log b ca + log b b = log b abc z+ = log c ab + log c c = log c abc Therefore x + y + z + log abc log abc log abc Example : a b c = log abc a +log abc b + log abc c = log abc = (proved) abc If a=log 4, b=log 6 4, and c=log 48 6 then prove that +abc = bc Solution: +abc = + log 4 log 6 4 log 48 6 = + log 6 log 48 6 = + log 48 = log 48 48 + log 48 = log 48 48 = log 48 ( ) = log 48 4 = log 6 4 x log 48 6 = bc

. BUSINESS MATHEMATICS SUMMARY log a mn = log a m + log a n Ex. log ( ) = log + log log a (m/n) = log a m log a n Ex. log (/) = log log log a m n = n log a m Ex. log = log log a a =, a = Ex. log 0 0 =, log =, log = etc. log a = 0 Ex. log = 0, log 0 = 0 etc. log b a log a b = Ex. log log = log b a log c b = log c a Ex. log log 5 = log 5 log b a = log a/log b Ex. log = log/log log b a = /log a b a log ax = x (Inverse logarithm Property) Notes: The two equations ax= n and x = logan are only transformations of each other and should be remembered to change one form of the relation into the other. Since a = a, log a a = (A) If base is understood, base is taken as 0 (B) Thus log 0 =, log = 0 (C) Logarithm using base 0 is called Common logarithm and logarithm using base e is called Natural logarithm {e =. (approx.) called exponential number}. EXERCISE (D) Choose the most appropriate option. (a) (b) (c) or (d).. log 6 + log 5 is expressed as (a) log (b) log 0 (c) log 5/6 (d) none of these

RATIO AND PROPORTION, INDICES, LOGARITHMS.. log 8 is equal to (a) (b) 8 (c) (d) none of these. log /4 is equal to (a) log /log 4 (b) log log 4 (c) (d) none of these 4. log ( ) is equal to (a) log + log + log (b) log (c) log (d) none of these 5. The value of log 0.000 to the base 0. is (a) 4 (b) 4 (c) /4 (d) none of these 6. If log x = 4 log, the x is equal to (a) (b) 9 (c) (d) none of these 7. log 64 is equal to (a) (b) 6 (c) (d) none of these 8. log 78 is equal to (a) (b) (c) 6 (d) none of these 9. log (/8) to the base 9 is equal to (a) (b) ½ (c) (d) none of these 0. log 0.065 to the base is equal to (a) 4 (b) 5 (c) (d) none of these. Given log = 0.00 and log = 0.477 the value of log 6 is (a) 0.900 (b) 0.954 (c) 0.778 (d) none of these. The value of log log log 6 (a) 0 (b) (c) (d) none of these. The value of log to the base 9 is (a) ½ (b) ½ (c) (d) none of these 4. If log x + log y = log (x+y), y can be expressed as (a) x (b) x (c) x/x (d) none of these 5. The value of log [log {log (log 7 )}] is equal to (a) (b) (c) 0 (d) none of these 6. If log x + log 4 x + log 6 x = /4, these x is equal to (a) 8 (b) 4 (c) 6 (d) none of these 7. Given that log 0 = x and log 0 = y, the value of log 0 60 is expressed as (a) x y + (b) x + y + (c) x y (d) none of these 8. Given that log 0 = x, log 0 = y, then log 0. is expressed in terms of x and y as (a) x + y (b) x + y (c) x + y (d) none of these 9. Given that log x = m + n and log y = m n, the value of log 0x/y is expressed in terms of m and n as (a) m + n (b) m + n (c) m + n + (d) none of these

.4 BUSINESS MATHEMATICS 0. The simplified value of log 0 5 + log 0 8 ½ log 0 4 is (a) / (b) 4 (c) (d) none of these. log [ { ( x ) } ] / can be written as (a) log x (b) log x (c) log /x (d) none of these - -4/. The simplified value of log 4 79 9.7 is (a) log (b) log (c) log ½ (d) none of these. The value of (log b a log c b log a c) is equal to (a) (b) 0 (c) (d) none of these 4. The logarithm of 64 to the base is (a) (b) (c) ½ (d) none of these 5. The value of log 8 5 given log = 0.00 is (a) (b) (c).548 (d) none of these ANSWERS Exercise (A). (a). (d). (c) 4. (a) 5. (c) 6. (d) 7. (a) 8. (c) 9. (a) 0. (c). (d). (d). (a) 4. (c) 5. (d) 6. (a) 7. (c) 8. (b) 9. (b) 0. (c). (a). (c). (a) 4. (c) 5. (c) Exercise (B). (a). (b). (c) 4. (d) 5. (a) 6. (c) 7. (a) 8. (c) 9. (c) 0. (b). (c). (d). (a) 4. (d) 5. (d) 6. (a) 7. (a) 8. (b) 9. (d) 0. (a). (c). (d). (c) 4. (a) 5. (b) 6. (b) 7. (c) 8. (b) 9. (a) 0. (b) Exercise (C). (c). (c). (c) 4. (b) 5. (a) 6. (a) 7. (b) 8. (d) 9. (b) 0. (c). (d). (c). (b) 4. (d) 5. (a) 6. (c) 7. (a) 8. (c) 9. (d) 0. (b). (a). (c). (b) 4. (b) 5. (c) 6. (a) 7. (b) 8. (a) 9. (a) 0. (b) Exercise (D). (b). (c). (b) 4. (a) 5. (b) 6. (b) 7. (a) 8. (c) 9. (c) 0. (d). (c). (c). (a) 4. (c) 5. (c) 6. (a) 7. (b) 8. (c) 9. (a) 0. (c). (b). (a). (c) 4. (d) 5. (c)

RATIO AND PROPORTION, INDICES, LOGARITHMS.5 ADDITIONAL QUESTION BANK - - 6 7 6 7. The value of -4-5 is 6 7 6 7 7-5 (a) 0 (b) 5 (c) 50 (d) 48 /7 /5-9/7 5/6. The value of x x x z -/ / / /5 is z z z x (a) (b) (c) 0 (d) None x+ x- y x+ y+ y+ 5 6. On simplification reduces to x+ y+ x 6 0 5 (a) (b) 0 (c) (d) 0 y y y 9-7 = then x y is given by x. 7 (a) (b) (c) 0 (d) None -..( ) - 4. If 5. Show that a-c b-a c-b a-b b-c c-a x x x is given by (a) (b) (c) (d) 0 x x x x 6. 4 5 5 6. Show that is given by x x x 5 6 5 (a) (b) (c) 4 (d) 0 7. Show that a+b b+c c+a a b c x x x x x x b c a is given by (a) 0 (b) (c) (d) a b c 8. Show that a+b x b+c x c+a x x x x b c a reduces to (a) (b) 0 (c) (d) None 9. Show that b+c a-b c+a b-c a+b c-a c-a a-b b-c x x x reduces to (a) (b) (c) (d) None

.6 BUSINESS MATHEMATICS 0. Show that a b c b c a x x x x x x c a b reduces to (a) (b) (c) 0 (d) bc ca ab b c a x x x. Show that c a b x x x reduces to (a) (b) 0 (c) (d) None. Show that a +ab+b b +bc+c c +ca+a a b c x x x x x x b c a is given by (a) (b) (c) 0 (d). Show that a-c b-a c-b a-b b-c c-a x x x is given by (a) 0 (b) (c) (d) None 4. Show that b+c-a c+a-b a+b-c b c a x x x x x x c a b is given by (a) (b) 0 (c) (d) None 5. Show that 6. a -ab+b b -bc+c c -ca+a a b c x x x x x x -b -c -a is reduces to - (a) (b) a +b +c x (c) a +b +c x (d) - a +b +c x a b - c - b c - a - c a - b - x. x. x x³ would reduce to zero if a b c is given by (a) (b) (c) 0 (d) None 7. The value of z is given by the following if (a) (b) 8. + + b -c c -a a -b x + x + x + x + x + x + (c) - (d) 9 4 would reduce to one if a b c is given by (a) (b) 0 (c) (d) None 9. On simplification would reduces to +z a-b +z a-c +z b-c +z b-a +z c-a +z c-b

RATIO AND PROPORTION, INDICES, LOGARITHMS.7 (a) a+b+c z (b) x y z 0. If5.678 = 0.5678 =0 then (a) - + = x y z (b) z a+b+c - - =0 x y z (c) (d) 0 (c) - + =- x y z (d) None -. If x=4 +4 prove that 4x -x is given by (a) (b) (c) 5 (d) 7 -. If x=5 +5 prove that 5x -5x is given by (a) 5 (b) 6 (c) 7 (d) 0. If ax +bx +c=0 then the value of a x +b x+c is given by (a) abcx (b) abcx (c) abc (d) abc p q r 4. Ifa =b, b =c, c =a the value of pqr is given by (a) 0 (b) (c) (d) None p q r 5. Ifa =b =c and b =ac the value of q (p r)/ pr is given by (a) (b) (c) (d) None x 6. On simplification x a a-b a a+b x x b b-a b b+a a+b reduces to (a) (b) (c) 0 (d) None ab b +c ca x x x 7. On simplification a +b bc c +a x x x (a) -a x a+b (b) b+c c+a reduces to a x (c) - a +b +c x (d) a +b +c x 8. On simplification (a) -a x ab ab bc bc ca ca x x x a b b c c a x x x (b) x y m m y z m m reduces to a x (c) - a +b +c x z 9. On simplification m m x+y y+z x-z x (d) a +b +c x mreduces to

.8 BUSINESS MATHEMATICS (a) (b) (c) - (d) 0. The value of + y-x x-y is given by +a +a (a) (b) 0 (c) (d) N one. If xyz then the value of x y y z z x is (a) (b) 0 (c) (d) None c, then reduces to c b a (a) (b) 0 (c) (d) None a b. If. If a b -c = =6 then the value of + + reduce to a b c (a) 0 (b) (c) (d) ab-c a+b reduces to a b 4. If =5 = 75 c, then the value of (a) (b) 0 (c) (d) 5 a b 5. If = = c, then the value of 6. If ab-c a+b reduces to (a) 0 (b) (c) (d) a b c =4 =8 andabc=88 then the value (a) 8 (b) a 4b 8c - (c) 8 96 is given by p q r s 7. If a =b =c =d and ab = cd then the value of reduces to p q r s (d) - 96 (a) a (b) b (c) 0 (d) b a 8. If a =b, then the value of a b a b -a a - b reduces to (a) a (b) b (c) 0 (d) None 9. If x m=b, y y x n=b and ( m n ) =b the value of xy is given by (a) (b) 0 (c) (d) None

RATIO AND PROPORTION, INDICES, LOGARITHMS.9 m- n- p- 40. If a= xy b=xy, c= x y then the value of a n p b p m c m n reduces to (a) (b) (c) 0 (d) None n+p m p+m n m+n p 4. Ifa= x y, b= x y, c = x y then the value of n-p p-m m-n a b c reduces to (a) 0 (b) (c) (d) None 4. If a= +- - then the value of a +a- is (a) (b) 0 (c) (d) - 4. If a = x + x then a -a is (a) - x + x (b) - x - x (c) x (d) 0-44. If 4 4 a = + and - 4 4 b = - then the value of ( ) a +b is (a) 67 (b) 65 (c) 64 (d) 6 45. If x = + and y = then x² y² is (a) 5 (b) (c) (d) 4 46. If a = 47. If 4 6 + a+ a+ then the value of + a- a- is given by (a) (b) (c) (d) P + Q + 5 R + 5 S = + + 5 then the value of P is (a) 7/ (b) / (c) -/ (d) -/ 48. If a = + then the value of - a + a is (a) (b) - (c) (d) - 49. If a = + then the value of - a - a is (a) (b) (c) (d) - 50. If a = ( 5- ) 5. If - - - then the value of a + a - 5a - 5a + a + a is (a) 0 (b) (c) 5 (d) a = 7+4 7-4 then the value of a a-4 is (a) 4 (b) 7 (c) (d) 5. If a = - 5 then the value of a 4 a 0a 6a + 4 is (a) 0 (b) 4 (c) 0 (d) 5

.40 BUSINESS MATHEMATICS + 5. If a = - then the value of 4 a - a + a -a + is (a) (b) (c) (d) None 54. The square root of 5 is (a) 5 + (b) 5 + (c) Both the above (d) None 55. If x = - - the value of x is given by (a) (b) (c) (d) 0 + 56. If a = -, - b = then the value of a + b is + (a) 0 (b) 00 (c) 98 (d) 99 + 57. If a = -, - b = + then the value of a + b is (a) 0 (b) 00 (c) 98 (d) 99 + 58. If a = -, - b = + then the value of + a b is (a) 0 (b) 00 (c) 98 (d) 99 59. The square root of x + x -y is given by (a) +y + -y x x (b) x+y - x -y (c) x+y + x -y (d) x+y - x-y 60. The square root of 0 is given by (a) 6 5 (b) 6 5 (c) (d) 6. log ( + + ) is exactly equal to (a) log + log + log (b) log (c) Both the above (d) None 6. The logarithm of 95 to the base of 7 and 968 to the base of are (a) Equal (b) Not equal (c) Have a difference of 69 (d) None 8 6 6. The value of is 4log log log5 is 5 5 (a) 0 (b) (c) (d) 64. logb-logc logc-loga loga-logb a b c has a value of (a) (b) 0 (c) (d) None

RATIO AND PROPORTION, INDICES, LOGARITHMS.4 + + 65. log ( abc) log ( abc) log ( abc ) ab bc ca is equal to (a) 0 (b) (c) (d) + + 66. +log ( bc) +log ( ca) +log ( ab ) a b c is equal to (a) 0 (b) (c) (d) + + log x log x log x is equal to 67. ( ) ( ) ( ) a b c b c a (a) 0 (b) (c) (d) 68. log ( a ).log b c ( b ).loga ( c ) is equal to (a) 0 (b) (c) (d) None 69. log b a. log c (b ). log a (c ) is equal to (a) 0 (b) (c) (d) None b c a log log log 70. The value of is c a b a.b.c (a) 0 (b) (c) (d) None log b log c log a c a b 7. The value of ( ) ( ) ( ) bc. ca. ab is (a) 0 (b) (c) (d) None 7. The value of a b c log + log + log n n n n b n c n a is (a) 0 (b) (c) (d) None a b c 7. The value of log + log + log bc ca ab is (a) 0 (b) (c) (d) None 74. log (a 9 ) log a 0 if the value of a is given by 75. If (a) 0 (b) 0 (c) (d) None loga logb logc = = y-z z-x x-y the value of abc is (a) 0 (b) (c) (d) None