RATIO AND PROPORTION

Introduction

Ratio means two quantities in the same units like Hari’s height is 100 cm and Ram’s height is 200 cm so the ratio of their heights is 100/200 or 1:2. the ratio has no units.

Proportion is equality of two ratio’s a:b = c:d we write as a:b::c:d and a,d are extremes and b,c are means. The product of means = product of extremes.
                                                a * d = b * c

When we have to divide a quantity ‘x’ into a:b we need to break it into two parts one is a/a+b and second is b/b+a.

Componendo – Dividendo rule:

 

if a:b = c:d then (a+b)/(a-b) = (c+d)/(c-d)


Proportionality:

If x ∝ y i.e. x is directly proportional to y which means when x increases y too increases and when x decreases y to decreases.

If x ∝ 1/y i.e. x is inversely proportional to y which means if x increases y decreases and when x decreases y increases.

    • If we multiply the numerator and the denominator of a ratio by the same number, the ratio remains unchanged. ab=mamb

    • If we divide the numerator and the denominator of a ratio by the same number, the ratio remains unchanged. Thus a/db/d=ab

    • The magnitudes of two ratios can be compared by equating the denominators of the two ratios and then checking for the value of the numerator. Thus, if we have to check for 83=114. We convert 8/3 into 81.3331.33=10.664Now denominators are equal and so we can compare the numerators and find which are greater and lesser.

    • If a/b = c/d = e/f = g/h = k then k=a+e+c+gb+d+f+h

    • If a1 / b1 , a2/b2, a3/b3… an/bn are unequal fractions then we have a1+a2+a3+..+anb1+b2+b3+..+bn lies between the lowest and the highest of these fractions.

    • If we have two equations containing three unknowns as a1x + b1y + c1z = 0 and a2x + b2y + c2z = 0 then we can find the proportion x : y : z. This will be given by b1c2b2c1:a2c1c2a1:a1b2a2b1

    • If the ratio ab>1 (called a ratio of greater inequality) and if k is a positive number: a+kb+k<ab and akbk>ab Similarly if ab<1 then a+kb+k>ab and akbk<ab

    • Maintenance of equality when numbers are added in both the numerator and the denominators. This if best illustrated through an example: 2030=20+230+3 i.e. ab=a+cb+d if and only if cd=ab

    • Consequently, if cd>ab then a+cb+d>ab and if cd<ab then a+cb+d<ab

To get the consolidated ratio A:B:C:D:E when individual ratios A:B, B:C, C:D, D:E are given

    • A:B = 2:3, B:C = 4:5, C:D = 6:11, D:E = 12:17

    • In order to create one consolidated ratio for this situation using the LCM process becomes too long. The short cut goes as follows:

    • A will correspond to the product of all numerators (2 × 4 × 6 × 12) while B will take the first denominator and the last 3 numerators (3 × 4 × 6 × 12). C on the other hand takes the first two denominators and the last 2 numerators (3 × 5 × 6 × 12), D takes the first 3 denominators and the last numerator (3 × 5 × 11 × 12) and E takes all the four denominators (3 × 5 × 11 × 17).

    • In mathematical terms this can be written as:

    • ab=N1D1,bc=N2D2,cd=N3D3,de=N4D4 then a:b:c:d:e=N1N2N3N4:D1N2N3N4:D1D2N3N4:D1D2D3N4:D1D2D3D4

Cross multiplication method : Compare two ratios

    • Two ratios can be compared using the cross multiplication method as follows. Suppose you have to compare 12/17 with 15/19. Then, to test which ratio is higher cross multiply and compare 12 × 19 and 15 × 17.

    • If 12 × 19 is bigger the Ratio 12/17 will be bigger. If 15 × 17 is higher, the ratio 15/19 will be higher. In this case, 15 × 17 being higher, the Ratio 15/19 is higher

Method for calculating the value of a percentage change in the ratio:

    • if 20/40 becomes 22/50 then Effect of numerator = 20 -> 22(10% increase) Effect of denominator = 50 -> 40(25% decrease) (reverse fashion)

    • Overall effect on the ratio:

    • 100 increased by 10% gives 110 which when reduced by 25% gives 82.5.

    • Hence the overall change in the ratio is 17.5%

    • When two ratios are equal, the four quantities composing them are said to be proportionals. Thus if a/b = c/d, then a, b, c, d are proportionals. This is expressed by saying that a is to b as c is to d, and the proportion is written as a : b : : c : d OR a : b = c : d

    • The terms a and d are called the extremes while the terms b and c are called the means.

    • If four quantities are in proportion, the product of the extremes is equal to the product of the means. Let a, b, c, d be the proportionals. Then by definition a/b = c/d we get ad = bc Hence if any three terms of proportion are given, the fourth may be found. Thus if a, c, d are given, then b = ad/c

    • If three quantities a, b and c are in continued proportion, then a : b = b : c then ac=b2 also a : c = a2:b2

    • If four quantities a, b, c and d form a proportion, many other proportions may be deduced by the properties of fractions. The results of these operations are very useful. These operations are

      1. Invertendo: If a/b = c/d then b/a = d/c

      1. Alternando: If a/b = c/d, then a/c = b/d

      1. Componendo: If a/b = c/d, then a+bb=c+dd

      1. Dividendo: If a/b = c/d, then abb=cdd

    1. Componendo and Dividendo: If a/b = c/d, then a+bab=c+dcd

Solved Question Papers


Q.
5783 is divided among Sherry, Berry, and Cherry in such a way that if t` 28, t` 37 and t` 18 be deducted from their respective shares, they have money in the ratio 4 : 6 : 9. Find Sherry’s shar

1256
1228
1456
1084

Ans .1228

Explanation :
The problem clearly states that when we reduce 28, 37 and 18 rupees respectively from
Sherry’s, Berry’s and Cherry’s shares, the resultant ratio is: 4 : 6 : 9.
Thus, if we assume the reduced values as
4x, 6x and 9x, we will have Æ
Sherry’s share is 4x + 28, Berry’s share is 6x + 37 and Cherry’s share is 9x + 18 and thus we have
(4x + 28) + (6x + 37) + (9x + 18) = 5783
so 19x = 5783 – 83 = 5700
Hence, x = 300.
Hence, Sherry’s share is 1228.

 

Q.
Divide 1870 into three parts in such a way that half of the first part, one-third of the second part and one-sixth of the third part are equal.

241, 343, 245
400, 800, 670
470, 640, 1160
None of these

Ans .None of these

Explanation :
Solve this question using options. 1/2 of the first part should equal 1/3
rd of the second part and of the third part. This means that the first part should be divisible by 2, the second one by 3, and
the third one by 6. Looking at the options, none of the first 3 options has its third number divisible
by 6. Thus, option (d) is correct.

 

Q.
Divide 500 among A, B, C and D so that A and B together get thrice as much as C and D together, B gets four times of what C gets and C gets 1.5 times as much as D. Now the value of what B gets is

300
75
125
150

Ans .300

Explanation :
(A + B) = 3 (C + D) so A + B = 375 and C + D = 125.
Also, since C gets 1.5 times D we have
C = 75 and D = 50, and B = 4 * C = 300.

 

Q.
If ab+c=ba+c=cb+a, then each fraction is equal to

(a + b + c)2
1/2
1/4
0

Ans .1/2

Explanation :
The given condition has a, b and c symmetrically placed. Thus, if we use a = b = c = 2 (say) we get each fraction as 1/2

 

Q.
If 6×2 + 6y2 = 13xy, what is the ratio of x to y?

1 : 4
3 : 2
4 : 5
1 : 2

Ans .3 : 2

Explanation :
Solve using options. Since x > y > 0 it is clear that a ratio of x:y as 3:2 fits the equation

 

Q.
If a : b = c : d then the value of a2+b2c2+d2

1/2
a+b / c+d
a-b / c-d
ab/cd

Ans .ab/cd

Explanation :
1 : 2 = 3 : 6 So, (a2 + b2)/(c2 + d2) = 5/45 = 1/9; From the given options, only ab/cd gives us this value.

 

Q.
If 4 examiners can examine a certain number of answer books in 8 days by working 5 hours a day, for how many hours a day would 2 examiners have to work in order to examine twice the number of answer books in 20 days.

6
7.5
8
9

Ans .8

Explanation :
4 × 8 × 5 = 160 man-hours are required for ‘x’ no. of answer sheets. So, for ‘2x’ answer sheets we would require 320 man-hours = 2 × 20 × n Æ n = 8 Hours a day.

 

Q.
In a mixture of 40 litres, the ratio of milk and water is 4 : 1. How much water must be added to this mixture so that the ratio of milk and water becomes 2 : 3.

20
32
40
30

Ans .40

Explanation :
In 40 litres, milk = 32 and water = 8. We want to create 2 : 3 milk to water mixture, for this we would need: 32 milk and 48 water. (Since milk is not increasing). Thus, we need to add 40 litres
of water

 

Q.
If three numbers are in the ratio of 1 : 2 : 3 and half the sum is 18, then the ratio of squares of the numbers is:

6 : 12 : 13
1 : 2 : 4
36 : 144 : 324
3 : 5 : 7

Ans .6 : 144 : 324

Explanation :
1 : 2 : 3 then x, 2x and 3x add up to 36.
So the numbers are: 6, 12 and 18.
Ratio of squares = 36 : 144 : 324

 

Q.
The ratio between two numbers is 3 : 4 and their LCM is 180. The first number is

6
45
1
20

Ans .45

Explanation :
The numbers would be 3x and 4x and their LCM would be 12x. This gives us the values as 45 and 60. The first number is 45.

 

Q.
If a, b, c, d are proportional, then (a – b) (a – c)/a =

a + c + d
a + d – b – c
a + b + c + d
a + c – b – d

Ans .a + d – b – c

Explanation :
Assume a set of values for a, b, c, d such that they are proportional i.e. a/b = c/d. Suppose we take
a:b as 1:4 and c:d as 3:12 we get the given expression:
(a – b)(a – c)/a = –3 × –2/1 = 6. This value is also given by a + d – b – c and hence option (b) is
correct

 

Q.
A and B are two alloys of argentum and brass prepared by mixing metals in proportions 7 : 2 and 7 : 11 respectively. If equal quantities of the two alloys are melted to form a third alloy C, the proportion of argentum and brass in C will be:

5 : 9
5 : 7
7 : 5
9 : 5

Ans .7 : 5

Explanation :
Since equal quantities are being mixed, assume that both alloys have 18 kgs (18 being a number which is the LCM of 9 and 18).
The third alloy will get, 14 kg of argentum from the first alloy and 7 kg of argentum from the
second alloy. Hence, the required ratio: 21:15 = 7:5

 

Q.
If 30 men working 7 hours a day can do a piece of work in 18 days, in how many days will 21 men working 8 hours a day do the same work?

24
22.5
30
45

Ans .22.5

Explanation :
The total number of manhours required = 30 × 7 × 18 = 3780
21 × 8 × no. of days Æ 3780/168 = 22.5 days
Note, you could have done this directly by: (30 × 7 × 18)/(21 × 8)

Quiz

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Q1: If a:b = 3:4 and b:c is 5:4 then find a:b:c
15:20:16
15:20:25
15:20:10
15:20:30

ANS.15:20:16

Q2: Divide Rs. 600 in the ratio 4:2
400,200
450,150
500,100
350,250

ANS.400,200

Q3: A bag contains blue balls , red balls and yellow balls in ratio of 3:4:5. If total balls are 96 how many are of blue type
24
30
36
42

ANS.24

Q4: A mixture contains alcohol and water in the ratio 4 : 3. If 5 litres of water is added to the mixture, the ratio becomes 4: 5. Find the quantity of alcohol in the given mixture.
10
15
20
25

ANS.10

Q5: x is directly proportional to y which means
when x increases y too increases
when x increases y decreases
when x decreases y increases
when x increases y remains constant

ANS.when x increases y too increases