Comparisons of two or more numbers and changes in their sizes are the most common ways to use the ideas in this chapter. For example, you can compare ages, weights, money, savings, heights, volume, density, temperature, etc. So, this part is very helpful for figuring out how to solve problems with Data analysis.
Because of this, most competitive tests like CAT, XAT, IIFT, CMAT, GMAT, SSC CGL, and Bank PO ask about these kinds of issues.
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Ratio
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The comparison of two quantities in terms of magnitude is referred to as the ratio, which indicates how many times one quantity is that of the other.
Therefore, the ratio of any two quantities is written as \(\frac{a}{b}\) or a: b.
The numerator 'a' is the antecedent, while the denominator 'b' is the consequent.
Rule of Ratio
Comparing two quantities that are not of the same kind or in the same units (of length, volume, currency, etc.) is meaningless. We do not compare 9 lads to 4 calves, 11 litres to 7 toys, or 6 metres to 10 centimetres. To determine the ratio of two identical quantities, it is necessary to express them in the same units.
Properties of Ratios
- When multiplied by identical quantities, the numerator and denominator have no effect on the value of the ratio.
i.e. \(\frac{a}{b}= \frac{ka}{kb}= \frac{la}{lb}= \frac{1}{x}= \frac{ma}{mb}\) etc.
- When the numerator and denominator are both divided by the same numbers, the ratio's value does not change.
i.e. \(\frac{a}{b}=\frac{a/k}{b/k}=\frac{a/l}{b/l}= \frac{a/m}{b/m}\) etc.
- The ratio of two fractions can be expressed in ratio of integers
i.e. \(\frac{3/4}{5/4}\) = \(\frac{3}{4}\) x \(\frac{4}{5}\)= \(\frac{3}{5}\)
- A compounded ratio is the product of multiplying two or more ratios together.
i.e. \(\frac{2}{3}\) x \(\frac{4}{5}\) x \(\frac{6}{7}\) = \(\frac{16}{35}\) is the compounded ratio of \(\frac{2}{3}, \frac{4}{5}, \frac{6}{7}\)
- Duplicate, triple, etc. ratios are those in which the ratio is multiplied by itself.
i.e. \(\frac{6}{7}\) x \(\frac{6}{7}\)= \((\frac{6}{7})^2\) is called as duplicate ratio
\(\frac{6}{7}\) x \(\frac{6}{7}\) x \(\frac{6}{7}\)= \((\frac{6}{7})^3\) is called as triplicate ratio
- \(\frac{a}{b}= \frac{c+am}{d+bm}\) = if and only if \(\frac{c}{d}= \frac{a}{b}\). This property is useful for comparing two fractions.
- \(\frac{a+k}{b+k}< \frac{a}{b}\) if for every positive value of k, \(\frac{a}{b} > l\) and \(\frac{a-k}{b-k}> \frac{a}{b}\)
- \(\frac{a+k}{b+k}> \frac{a}{b}\) if for every positive value of k, \(\frac{a}{b}<l\) and \(\frac{a-k}{b-k} < \frac{a}{b}\)
- \(\frac{a+k}{b+k}> \frac{a}{b} \) if \(\frac{c}{d}> \frac{a}{b}\)
- \(\frac{a+k}{b+k} < \frac{a}{b}\) if \(\frac{c}{d}< \frac{a}{b}\)
- \(\frac{a}{b}= \frac{c}{d}= \frac{e}{f}= \frac{g}{h}=......= k\) then . \(\frac{a+c+e+g+.....}{b+d+f+h+.....}=k\)
- \(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}, \frac{g}{h},.....\)Let Be different ratios then value of \(\frac{a+c+e+g....}{b+d+f+h....}\) Must lie between lowest and highest ratios.
- If a:b and b:c are given then
a:b:c=(a.b) : (b.b) : (b.c)
- If the ratios between a:b, b:c, c:d, d:e are given then the combined ratio of a:b:c:d:e
i.e., a:b:c:d:e=(a.b.c.d):(b.b.c.d) :(b.c.c.d):(b.c.d.d):(b.c.d.e)
Proportion
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The equality of two ratios is referred to as a proportion, and four numbers are said to be in proportion.
If \(\frac{a}{b}= \frac{c}{d}\) or a:b=c:d then we say that a,b,c and d are in proportion.
Here a and d are called extremes (or extreme terms) and b and c are called as means (or middle terms)
Proportinality Test
If four numbers (quantities) are in proportion, then product of the extremes is equal to the product of the means and if these are not in proportion, then product of extremes is not equal to the product of the means.
a : b :: c : d, then a × d = b × c
Proportionality Theorems
- Invertendo If \(\frac{a}{b} = \frac{c}{d}\) \(\implies\) \(\frac{b}{a}= \frac{d}{c}\)
- Alternando If \(\frac{a}{b} = \frac{c}{d}\) \(\implies\) \(\frac{a}{c}= \frac{b}{d}\)
- Componendo If \(\frac{a}{b}= \frac{c}{d}\) \(\implies\) \(\frac{a+b}{b}= \frac{c+d}{d}\)
- Dividendo If \(\frac{a}{b}= \frac{c}{d}\) \(\implies\) \(\frac{a-b}{b}= \frac{c-d}{d}\)
- Componendo and Dividendo If \(\frac{a}{b}= \frac{c}{d}\) \(\implies\) \(\frac{a+b}{a-b}=\frac{c+d}{c-d}\)
Continued Proportion
If a, b, and c are three numbers such that a:b = b:c, then these three numbers are said to be in proportion or in continued proportion.
If a : b = b : c then b2 = ac
Here b is said to be the mean proportional to a and c, and c is said to be the third proportional to a and b.
For Example, 3 :9 ::9 :27
Here, 9 × 9 = 3 × 27. That means 3, 9 and 27 are in continued proportion
Continued Proportionality Theorems
If a, b and c are three numbers such that a :b = b:c, that is \(\frac{a}{b}=\frac{b}{c}\) then,
- b2 =ac
- \(\frac{b}{a}= \frac{c}{b}\)
- \(\frac{a}{c}= \frac{a^2}{b^2}\)
- \(\frac{a}{c}= \frac{b^2}{c^2}\)
- \(\frac{a}{c}= \frac{a^2+ b^2}{b^2+ c^2}\)
Unitary Method
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In this method, at first we find the value of one unit and, then we find the value of required number of units by multiplying the value of one unit with the required number of units.
Direct Proportion
When one number goes up (or down), it causes the other number to go up (or down) by the same amount, we say that the two numbers are directly proportional. e.g.,
- The price of papers depends directly on how many there are. More articles more cost, less articles less cost.
- The amount of work done depends directly on how many guys are working. When more guys work, more work gets done in the same amount of time. Less guys means less work in the same amount of time.
Inverse Proportion
If the increase (or decrease) in one quantity causes a proportional decrease (or increase) in the other quantity, the quantities are said to vary inversely. For example, the time required to complete a task varies inversely with the number of men present.
More males at work reduces the time required to complete the same task. When there are fewer males at work, it takes longer to complete the same amount of labour.
Variation
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When two or more quantities are dependent on each other, and if one is altered, the other (dependent) quantity is also altered.
For instance:
(i) When a person's income increases, so do their savings and spending.
(ii) When the number of visitors in a hotel/students in a hostel/employees varies, so do their respective costs.
There are essentially two categories of variation: direct proportion and inverse proportion variation.
(i) Variation in a (ii) Inverse variation
Direct Variation
A quantity A is said to vary directly when an increase (or decrease) in quantity B results in an increase (or decrease) in A that is not proportional. It is written as
A ∝ B \(\implies\) A = KB
K being the p roportionality constant
- K= \(\frac{A}{B}\)
Inverse Variation
A quantity A is said to vary inversely if an increase (or decrease) in B does not result in a proportional decrease (or increase) in A. It is written as
A ∝ \(\frac{1}{B}\) \(\implies\) A = \(\frac{K}{B}\)
K being the proportionality constant
- K=AB
Partnership
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When two or more people operate a business jointly by investing their money/resources, this is referred to as a joint venture or a partnership. All of these individuals who have invested capital are known as Partners.
Types of Partners
(i) Working partner A "working partner" is a business partner who is involved in the day-to-day running of the business.
(ii) Sleeping partner A sleeping partner is someone who just spends his or her money.
General Rules of Partnership
- If each partner puts in a different amount for the same amount of time, the gains are split according to how much each person put in.
- If each partner puts in the same amount of money at different times, then the earnings are split up based on how much money each person put in at each time.
- If each partner puts in a different amount for a different amount of time, their earnings are split in proportion to the results of their individual investments and time periods. So, a gain or loss is divided by the ratio of 'money-time' funds.
Previous Year CAT Questions
Ques: In a village, the ratio of number of males to females is 5: 4. The ratio of number of literate males to literate females is 2: 3. The ratio of the number of illiterate males to illiterate females is 4 : 3. If 3600 males in the village are literate, then the total number of females in the village is (CAT 2022 Slot 1)
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Ans: 43200
Based on the question,
Male literates in the village is 3600
Male literates / Female Literates= 2/3
3600/ Female Literates= 2/3
Female literates = 2/3 x 3600 = 5400
Number of Male Illiterates and Female Illiterates be 4x and 3x respectively.
Total males / Total Females= 5/4
3600 + 4x/ 5400 + 3x= 5/4
14400 + 16x=27000 + 15x
X= 12600
Total females in the village = 5400 + 3x =5400 + 3(12600) = 43200
Ques: Pinky is standing in a queue at a ticket counter. Suppose the ratio of the number of persons standing ahead of Pinky to the number of persons standing behind her in the queue is 3 : 5. If the total number of persons in the queue is less than 300, then the maximum possible number of persons standing ahead of Pinky is __(CAT 2022 Slot 1)
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Ans: Let the number of people standing in front and behind Pinky be 3x
and 5x respectively.
.. Total number of people in the line = 3x + 5x + 1
(Pinky herself) = 8x
= 8X + 1 ≤ 300
= X ≤ 299/8 = 37.375
: Maximum value of x is 37.
- Maximum number of people standing in front of Pinky = 3x = 111.
Ques: The amount Neeta and Geeta together earn in a day equals what Sita alone earns in 6 days. The amount Sita and Neeta together earn in a day equals what Geeta alone earns in 2 days. The ratio of the daily earnings of the one who earns the most to that of the one who earns the least is __
- 7:3
- 3:2
- 11:3
- 11:7
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Ans: (C)
Let n, g, and s represent Neeta's, Geeta's, and Sita's respective incomes.
Together, Neeta and Geeta earn as much in a day as Sita does in six days.
6s= n + g (1)
Sita and Neeta's combined earnings for a day are the same as Geeta's earnings for two days.
=> 2g = s+ n
=> 2(6s - n) = s + n [From (1)]
=> 11s = 3n
=> n= 11s/3…....(2)
Substituting n = 11s/3 in (1)
=> 6s = 115/3 + g
=> g=75/3=25 ............(3)
From (2) and (3)
115/3 > 75/3 > s
n>g>s
The ratio of the daily earnings of the one who earns the most to that of the one who earns the least is = 11s/3 : s = 11:3
Ques: Anil, Bobby and Chintu jointly invest in a business and agree to share the overall profit in proportion to their investments. Anil's share of investment is 70%. His share of profit decreases by & 420 if the overall profit goes down from 18% to 15%. Chintu's share of profit increases by 80 if the overall profit goes up from 15% to 17%. The amount, in INR, invested by Bobby is ___
- 2000
- 2400
- 2200
- 1800
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Ans: ( A)
Let the total investment done by all three is I.
When total profit decreases by 3% to 15% Anil's share decreases by 420.
Decrease in profit = 3% of total investment
Decrease in Anil's profit will be 70% of decrease in total profit.
=> 420 = 70% of 3% of I
=> 420 = 70/100 x 3/100 x I
=> I = 20.000
When total profit increases by 2% to 17%,Char's share increases by 80.
Increase in total profit = 2% of I.
Increase in Charu's profit = C% of I [C% = percentage on investment contribution by Charu]
=> 80 = C% of 2% of I
=> 80 = C/100 x 2/100 x 20000
=> C= 20%
. Bobby's percentage share of investment = 100 - 70 - 20 = 10%
=> Investment by Bobby = 10% of 20000 = Rs. 2000
Ques: One part of a hostel's monthly expenses is fixed, and the other part is proportional to the number of its boarders. The hostel collects & 1600 per month from each boarder. When the number of boarders is 50, the profit of the hostel is $ 200 per boarder, and when the number of boarders is 75, the profit of the hostel is 3 250 per boarder. When the number of boarders is 80, the total profit of the hostel, in INR, will be __
- 20000
- 20200
- 20800
- 20500
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Ans: (D)
Total cost = Fixed cost + Variable cost
Variable cost = k x b
where, b = number of boarders and k = variable cost per boarder.
Total cost = Fixed Cost + kb
Total collection from hotel boarders = 1600b
Case 1: Total profit = 200 x 50 = 1600 x 50 - (FC + k. 50) ...(1)
Case 2: Total profit = 250 x 75 = 1600 x 75 - (FC + k. 75) .. (2)
(2) - (1), we get
=> 18750 - 10000 = 1600 x 25 - 25k
=> k= 1250 and Fixed Cost = 7,500
Now, total profit when 80 boarders are present
(1600) ( 80) - (7500 + 1250 x 80) = 20,500
Ques: A sum of money is split among Amal, Sunil and Mita so that the ratio of the shares of Amal and Sunil is 3 : 2, while the ratio of the shares of Sunil and Mita is 4 : 5. If the difference between the largest and the smallest of these three shares is Rs 400, then Sunil's share, in rupees, is __
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Ans:
Let the amount received by them be a, s and m respectively.
Given, a: s= 3:2 ands: m = 4:5
a:s:m=6:4:5
Amal's share = 6x and
Sunil's share = 4x.
Given, difference between the largest and the smallest of these shares is Rs 400.
=> 6x - 4x = 400
=> x = 200.
.: Sunil's share = 4x = Rs. 800.
Ques: In an examination, Rama's score was one-twelfth of the sum of the scores of Mohan and Anjali. After a review, the score of each of them increased by 6. The revised scores of Anjali, Mohan, and Rama were in the ratio 11 : 10 : 3. Then Anjali's score exceeded Rama's score by
- 26
- 32
- 24
- 35
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Ans: (B)
It is given that the scores of Anjali, Mohan and Rama after review were in the ratio 11: 10 : 3
let their values be 11x, 10x and 3x of Anjali, Mohan and Rama respectively.
Score increased by 6 after review.
So, scores before review = 11x - 6, 10x - 6 and 3x - 6 respectively
From the data given :
=> (11x - 6 + 10x - 6) x 1/12 = 3x - 6
=> 21x - 12 = 36x - 72
=> 60 = 15x
x=4
Marks after revision are 44, 40 and 12 respectively.
So, Anjali's score exceeded Rama's by 44 - 12 = 32 marks
Ques: Raju and Lalitha originally had marbles in the ratio 4 : 9. Then Lalitha gave some of her marbles to Raju. As a result, the ratio of the number of marbles with Raju to that with Lalitha became 5 : 6. What fraction of her original number of marbles was given by Lalitha to Raju?
- 6/19
- 15
- 7/33
- 1/4
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Ans: (C)
Let Raju and Lalitha start with (4x) and (9x) marbles. After that, Lalitha gave (y) marbles to Raju.
4x+y/9x-y=5/6
Solving the above equation:-
We get
y=21x/11
Required fraction = \(\frac{\frac{21x}{11}}{9x}= \frac{7}{33}\)
Ques: The scores of Amal and Bimal in an examination are in the ratio 11 : 14. After an appeal, their scores increase by the same amount and their new scores are in the ratio 47 : 56. The ratio of Bimal's new score to that of his original score is
- 4:3
- 3:2
- 8:5
- 5:4
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Ans: (A)
Suppose the original scores of Amal and Bimal are 11x and 14x respectively.
Suppose 'y' is the increase in marks.
Therefore the new marks of Amal and Bimal are '11x + y' and '14x + y' respectively.
Therefore, we have
616x + 56y = 658x + 47y
we get 14x = 3y
Therefore Bimal's old marks = 14x = 3y
So, new marks = 3y + y = 4y.
Therefore the required ratio = 4:3
Ques: Suppose, C1, C2, C3, CA, and Cs are five companies. The profits made by C1, C2, and C3 are in the ratio 9 : 10 : 8 while the profits made by C2, C4, and C5 are in the ratio 18: 19 : 20. If C5 has made a profit of Rs 19 crore more than C1, then the total profit (in Rs) made by all five companies is: (CAT 2017 Slot 1)
- 438 crore
- 435 crore
- 348 crore
- 345 crore
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Ans: (A)
As C2 is common in both the ratios. Taking LCM of 10 and 18 which is 90.
C1, C2, C3, C4 and C5 are in the ration 81:90:72:95:100
So,
C5 - C1=19
Total profit = 438 crore.
Ques: A stall sells popcorn and chips in packets of three sizes: large, super, and jumbo. The numbers of large, super, and jumbo packets in its stock are in the ratio 7: 17: 16 for popcorn and 6: 15 : 14 for chips. If the total number of popcorn packets in its stock is the same as that of chips packets, then the numbers of jumbo popcorn packets and jumbo chips packets are in the ratio:
- 1:1
- 8:7
- 4:3
- 6:5
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Ans: (A)
Adding all types of packets for popcorn = 7+17+16 =40
Adding all types of packets for chips = 6+15+14 = 35
Ratio of Jumbo packet for popcorn = 16/40
Ratio of Jumbo packet for chips = 14/35
Required ratio = (16/40)/(14/35) = 1:1
Ques: If a, b, c are three positive integers such that a and b are in the ratio 3: 4 while b and c are in the ratio 2: 1, then which one of the following is a possible value of (a + b+ c)?
- 201
- 205
- 207
- 210
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Ans: (C)
To find a common ratio, taking L.C.M(2, 4) = 4
multiplying each of b and c by 2
Hence, ratio of b: c = 4: 2
:. Ratio of b: c= 4:2
Now
let a = 3x,
ratio of a:b: c = 3:4:2
So
a + b + c=3x + 4x + 2x = 9x
Since a, b, and c are all positive numbers, this means that a, b, and c added together will be a positive integer that is a multiple of 9.
Out of the given options, only 207 is a multiple of 9
Ques: The total cost of 2 pencils, 5 erasers and 7 sharpeners is Rs. 30, while 3 pencils and 5 sharpeners cost Rs. 15 more than 6 erasers. By are what amount (in Rs.) does the cost of 39 erasers and 1 sharpener exceed the cost of 6 pencils? (CAT 2016)
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Ans:
Let the cost of pencil be p, eraser be e and sharpener be s .
According to the question,
2p+5e+ 7s=30 ...(i)
3p − 6 e + 5s = 15 ...(ii)
To find the answer to the question, the following equation will help us to answer.
E = − 6 p + 39 e + s
multiplying Eq. (i) by x and Eq. (ii) by y and adding we get the equation E.
2x + 3y = − 6 (i)
5x−6y= 39 (ii)
Multiplying 2 with Eq.(i) and adding with Eq.(ii), We get
x= (- 6) (- 6) - 3(39)/ (2) (- 6) – (5) (3)
x= 3 and y= – 4
E= 3(30) – 4(15) = 30
Ques: Balram, the local shoe shop owner, sells four types of footwear - Slippers (S), Canvas Shoes (C), Leather Shoes (L) and Joggers (J ). The following information is known regarding the cost prices and selling prices of these four types of footwear
(i) L sells for Rs. 500 less than J, which costs Rs. 300 more than S, which in turn, sells for Rs. 200 more than L.
(ii) L cost Rs. 300 less than C, which sells for Rs. 100 more than S, which in turn, costs Rs. 100 less than C.
If it is known that Balram never sells any item at a loss, then which of the following is true regarding the profit percentages earned by Balram on the items L, S, C and J represented by l, s, c and j, respectively? (CAT 2016)
- l ≥ c ≥ s ≥ j
- c ≥ s ≥ l ≥ j
- l ≥ s ≥ c ≥ j
- s ≥ l ≥ j ≥c
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Ans: (C)
At cost price ‘y’ selling price of item is (x-300)
At cost price ‘y+100’ selling price of item is (x-200)
At cost price ‘y-200’ selling price of item is (x-500)
At cost price ‘y+300’ selling price of item is (x)
Comparing profit percentages, we can compare SP/CP
x-300/y , x-200/ y+100, x-500/ y-200, x/y+300
The above fractions can be written as
a/b, a+100/ b+100, a-200/b-200, a+300/b+300
Now, no item sells at a loss and given the identity
That m/n =m+k/n+k whenever m/n ≥ 1 and k is a +ve quantity.
a/b ≥ a+100/b+100 ≥ a – 200/ b- 200 ≥ a+300/b+300
l ≥ s ≥ c ≥ j
Ques: A milkman mixes 20 L of water with 80 L of milk. After selling one-fourth of this mixture, he adds water to replenish the quantity that he has sold. What is the current proportion of water to milk? (CAT 2010)
- 2 : 3
- 1:2
- 1 : 3
- 3:4
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Ans: (A)
Out of the mixture of 100L, 20L is water and 80L is milk. When the milkman sells \(\frac{1}{4}\) th part of mixture i.e., 25 L , water will be 15 L and 60 L of milk in total 75 L of mixture.
When he adds 25 L water in it, total water will be 25 + 15 =40 L and milk is 60 L.
Ratio becomes= 40 : 60 = 2 : 3.
How to Prepare Ratio & Proportion Questions for CAT?
- Try to be thorough with the rules of ratio and proportion.
- Practice as much questions as possible as many time it gets confusing leading to wring anwers.
- Questions based on proportionality and continued proportion are often asked in the CAT exam.
- With practice try to reduce the time taken to solve question.
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