RATIO AND PROPORTION Important Facts: 1.Ratio : The ratio of two qualities a and b in the same units, is the fraction a/b and we write it as a:b. In the ratio, a:b, we call â ãâ as the first term of antecedent and b, the second term consequent. Ex: The ratio 5:9 represents 5/9 with antecedent=5,consequent=9 3Rule: The multiplication or division of each term of 9 ratio by the same non-zero number does not affect the ratio. 4.Proportion: The equality of two ratios is called proportion. If a:b=c:d, we write a:b::c:d and we say that a,b,c and d are in proportion. Here a and b are called extremes, while b and c are called mean terms. Product of means=product of extremes Thus, a:b::c:d => (b*c)=(a*d) 5.Fourth proportional: If a:b::c:d, then d is called the fourth proportional to a,b and c. 6.Third proportional: If a:b::b:c, then c is called third proportional to a and b. 7.Mean proportional: Mean proportional between a and b is SQRT(a*b). COMPARISION OF RATIOS: We say that (a:b)>(c:d) => (a/b)>(c/d) 8.Compounded ratio: The compounded ratio of the ratios (a:b), (c:d),(e:f) is (ace:bdf). 9.Duplicate Ratio: If (a:b) is (a2: b2 ) 10.Sub-duplicate ratio of (a:b) is (SQRT(a):SQRT(b)) 11.Triplicate ratio of (a:b) is (a3: b3 )
12.Sub-triplicate ratio of (a:b) is (a1/3: b1/3 ). 13.If a/b=c/d, then (a+b)/(a-b)=(c+d)/(c-d) (componend o and dividend o) VARIATION: 14.we say that x is directly proportional to y, if x=ky for some constant k and we write. 15.We say that x is inversely proportional to y, if xy=k for some constant and we write. 16. X is inversely proportional to y. If a/b=c/d=e/f=g/h=k then k=(a+c+e+g)/(b+d+f+h) If a1/b1,a2/b2, a3/b3...an/bn are unequal fractions then the ratio. SIMPLE PROBLEMS 1.If a:b =5:9 and b:c=4:7 Find a:b:c? Sol: a:b=5:9 and b:c=4:7=4*9/4:9*4/9=9:63/9 a:b:c=5:9:63/9=20:36:63 2.Find the fourth proportion to 4,9,12 Sol: d is the fourth proportion to a,b,c a:b=c:d 4:9=12:x 4x=9*12=>x=27 3.Find third proportion to 16,36 Sol: if a:b=b:c then c is the third proportion to a,b 16:36=36:x 16x=36*36 x=81 4.Find mean proportion between 0.08 and 0.18 Sol: mean proportion between a and b=square root of ab
mean proportion =square root of 0.08*0.18=0.12 5.If a:b=2:3 b:c=4:5, c:d=6:7 then a:b:c:d is Sol: a:b=2:3 and b:c=4:5=4*3/4:5*3/4=3:15/4 c:d=6:7=6*15/24:7*15/24=15/4:35/8 a:b:c:d=2:3:15/4:35/8=16:24:30:35 6.2A=3B=4C then A:B:C? Sol: let 2A=3B=4C=k then A=k/2, B=k/3, C=k/4 A:B:C=k/2:k/3:k/4=6:4:3 7.15% of x=20% of y then x:y is Sol: (15/100)*x=(20/100)*y 3x=4y x:y=4:3 8.a/3=b/4=c/7 then (a+b+c)/c= Sol: let a/3=b/4=c/7=k (a+b+c)/c=(3k+4k+7k)/7k=2 9.Rs 3650 is divided among 4 engineers, 3 MBAâ s and 5 CAâ s such that 3 CAâ s get as much as 2 MBAâ s and 3 Engâ s as much as 2 CAâ s.find the share of an MBA. Sol: 4E+3M+5C=3650 3C=2M, that is M=1.5C 3E=2C that is E=.66 C Then, (4*0.66C)+(3*1.5C)+5C=3650 C=3650/12. 166 C=450. C=300 M=1.5 and DIFFICULT PROBLEMS 1.Three containers A,B and C are having mixtures of milk and water in the ratio of 1:5 and 3:5 and 5:7 respectively. If the capacities of the containers are in the ratio of all the three containers are in the
ratio 5:4:5, find the ratio of milk to water, if the mixtures of all the three containers are mixed together. Sol: Assume that there are 500,400 and 500 liters respectively in the 3 containers. Then,we have, 83.33, 150 and 208.33 liters of milk in each of the three containers. Thus, the total milk is 441.66 liters. Hence, the amount of water in the mixture is 1400-441.66=958.33liters. Hence, the ratio of milk to water is 441.66:958.33 => 53:115(using division by.3333) The calculation thought process should be (441*2+2):(958*3+1)=1325:2875 Dividing by 25 => 53:115. 2.A certain number of one rupee,fifty parse and twenty five paise coins are in the ratio of 2:5:3:4, add up to Rs 210. How many 50 paise coins were there? Sol: the ratio of 2.5:3:4 can be written as 5:6:8 let us assume that there are 5 one rupee coins,6 fifty paise coins and 8 twenty-five paise coins in all. their value=(5*1)+(6*.50)+(8*.25)=5+3+2=rs 10 If the total is Rs 10,number of 50 paise coins are 6. if the total is Rs 210, number of 50 paise coins would be 210*6/10=126. 3.The incomes of A and B are in the ratio of 4:3 and their expenditure are in the ratio of 2:1. if each one saves Rs 1000,what are their incomes? Sol: Ratio of incomes of A and B=4:3 Ratio of expenditures of A and B=2:1 Amount of money saved by A=Amount of money saved by B=Rs 1000 let the incomes of A and B be 4x and 3x respectively let the expense of A and B be 2y and 1yrespectively Amount of money saved by A=(incomeexpenditure)=4x-2y=Rs 1000 Amount of money saved by B=3x-y=Rs 1000 this can be even written as 6x-2y=Rs 2000 now solve 1 and 3 to get
x=rs 500 therefore income of A=4x=4*500=Rs 2000 income of B=3x=3*500=Rs 1500 4.A sum of Rs 1162 is divided among A,B and C. Such that 4 times A's share share is equal to 5 times B's share and 7 times C's share. What is the share of C? Sol: 4 times of A's share =5 times of B's share=7 times of C's share=1 therefore, the ratio of their share =1/4:1/5:1/7 LCM of 4,5,7=140 therefore, ¼:1/5:1/7=35:28:20 the ratio now can be written as 35:28:20 therefore C's share=(20/83)*1162=20*14=rs 280. 5.The ratio of the present ages of saritha and her mother is 2:9, mother's age at the time of saritha's birth was 28 years, what is saritha's present age? Sol: ratio of ages of saritha and her mother =2:9 let the present age of saritha be 2x years. then the mother's present age would be 9x years Difference in their ages =28 years 9x-2x=28 years 7x=28=>x=4 therefore saritha's age =2*4=8 years