business mathematics

Ratio and Proportions concepts

Ratio and Proportions:


Ratio is denoted by ":"
The form of ratio is a:b. Here a,b are called terms.

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 ‘a’ 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

2.Rule:
The multiplication or division of each term of 9 ratio by the same non-zero number does not affect the ratio.


3.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)


4.Fourth proportional:
If a:b::c:d, then d is called the fourth
proportional to a,b and c.

5.Third proportional:
If a:b::b:c, then c is called third
proportional to a and b.

6.Mean proportional:
Mean proportional between a and b is SQRT(a*b).


7.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 )

Sub-duplicate ratio of (a:b) is (SQRT(a):SQRT(b))

Triplicate ratio of (a:b) is (a3: b3 )

Sub-triplicate ratio of (a:b) is (a1/3: b1/3 ).
If a/b=c/d, then (a+b)/(a-b)=(c+d)/(c-d) (componend o and dividend o)

RATIO AND PROPORTION practice problems:

For RATIO AND PROPORTION concept click: http://business-maths.blogspot.com/2008/11/ratio-and-proportions.html

For RATIO AND PROPORTION solved problems click: http://business-maths.blogspot.com/2008/11/ratio-and-proportions-problems.html

Practice problems:
1.If a:b=3:7,find the value of (5a+b):(4a+5b)
a)15:44 b)22:35 c)15:49 d)22:47

2.The ratio between two number is 3:5 and their is 40.Find the larger of two numbers
a)15 b)20 c)25 d)40

3.A bag contains one rupee,50 paise and 25 paise coins in the ratio 1:2:4.If the
total amount is Rs 75,then find the number of 50 paise coins in the bag
a)25 b)50 c)75 d)100

4.If (x+4):(3x+15) is the triplicate of 2:3.Find the value of x
a)1 b)3 c)4 d)none of these

5.If a is 75% of b,b is 150% of c and d is 25% of c,then find a:d
a)9:1 b)9:2 c)8:3 d)8:1

6.The present ages of two persons are in the ratio 7:8.Twenty years ago the ratio of
their ages was 9:11.Find the present age of the older son
a)64 years b)72 years c)56 years d)40 years

7.Find x:y:z,if 2x+y-5z = 0 and 3x-2y-4z=0
a)1:2:1 b)1:1:1 c)1:1:2 d)2:1:1

8.A certain sum is divided among A,B and C in a manner that for every rupee that A
gets,B gets 75 paise and for every rupee that B gets,C gets 50 paisee.If B's share
in the total sum is Rs 840.Find the share of A
a)Rs 2380 b)Rs 2240 c)Rs 1750 d)Rs 1120

9.Seventy eight is divided into two parts such that five times the first part and four times the second part are in the ratio 15:14.Find the first part
a)32 b)36 c)42 d)46

10.A certain amount of money is divided among A,B and C such that A gets half of what B and C gets together.B gets one third of what A and C together get.If A got Rs 500 more than B,then how much money was divided?
a)Rs4500 b)Rs 6000 c)Rs 8000 d)none of these

11.In a school there are 650 students.The ratio of the boys to that of the girls is 8:5. How many more girls should join the school so that the ratio becomes 4:3?
a)25 b)50 c)100 d)200

12.A variable x varies directly as the cube of another variable y.if x=4,y=2,then find y when x = 32
a)4 b)8 c)16 d)32

13.What should be subtracted from both the numbers which are in the ratio 3:4 so that the ratio becomes 2:3
a)4 b)6 c)10 d)cannot be determined

14.A varies directly as the sum of the quantities B and C.B in turn varies directly as x and C varies inversely as x.When x=2,A = 6 and when x=4,A=9.Find the value of A when the value of x=16
a)2 1/2 b)1 c)8 1/2 d)32 1/4

15.The volume of a sphere varies directly as the cube of its radius.If three cubes of radii 3 cm,4 cm and 5 cm are melted and recast into one sphere,then find the radius of the sphere
a)5.5 cm b)6 cm c)7 cm d)7.5 cm

16. 15 liters of mixture contains 20% alcohol and the rest water.If 3 liters of water be mixed with it,the percentage of alcohol in the new mixture would be
a)15 b)16 2/3 c)17 d)none of these

17.20 liters of mixture contains milk and water in the ratio 5:3.If 4 liters of this mixture be replaced by 4 liters of milk,the ratio of milk to water in the new mixture would be
a)2:1 b)7:3 c)8:3 d)4:3

18.85 kg of a mixture contains milk and water in the ratio 27:7.How much more water is to be added to get a new mixture containing milk and water in the ratio 3:1?
a)5kg b)6.5kg c)7.25kg d)8kg

19.The ages of A and B are in the ratio 3:1.Fifteen years hence,the ratio will be 2:1.Their present ages are
a)30,10 b)45,15 c)21,7 d)60,20

20.The average age of three boys is 25 years and their ages are in the proportion 3:5:7.The age of the youngest boy is
a)21 b)18 c)15 d)9

Ratio and Proportions problems

Ratio and Proportions problems

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 MBA’s and 5 CA’s
such that 3 CA’s get as much as 2 MBA’s and 3 Eng’s as
much as 2 CA’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=300
M=1.5 and C=450.

Difficule 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=(income-expenditure)=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