Proportion
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We have indirect variation if one going up causes the other to go down. An example of this might be speed and time to do a particular journey; so the higher the speed, the shorter the time.
Direct Variation:
When x and y are directly proportional, then doubling x will double the value of y; and if we divide these variables we get a constant result.
Example:
Given that y and x are directly proportional, and y = 2 when x = 5, find the value of y when x = 15.
We first find value of k, using
Now use this constant value in the equation y = kx for situation when x = 15.
Indirect or Inverse Variation:
We know that 'the higher the speed, the shorter the time.'
In this case, if we double the speed, we halve the time. So the product, speed × time = constant.
In general, if x and y are inversely proportional, then the product x × y will be constant.
Example:
If it takes 4 hours at an average speed of 90 km/
hour to do a certain journey, how long would it take at 120km.hour?
k = speed × time = 90 × 4 = 360 (k in this case is the distance.)
Then time 3 hours.
Some important results of proportions:
If a ∝ b and b ∝ c, then a ∝ c
If a ∝ c and b ∝ c, then a + b ∝ c and
∝ c
If a ∝ bc, then b ∝ and c ∝
If a ∝ b and c ∝ d, then ac ∝ bd
If a ∝ b, then an ∝ bn
If a ∝ b then ap ∝ bp, where p is any quantity variable or constant.
Application of ratios:
Partnership:
Partnership is an association of two or more persons who invest their money in order to carry on a certain business.
A partnership is called a simple partnership, if the capitals of the partners are invested for the same time period.
In case of simple partnership, the profit or loss is divided in the ratio of their investments.
If A and B are partners in a business, then:
or
If there are three partners A, B and C, then:
Investment of A: Investment of B: Investment of C
= Profit of A: Profit of B: Profit of C OR
Loss of A: Loss of B: Loss of C Example:
Three partners A, B and C invest Rs. 3200, Rs. 3600 and Rs. 4600 respectively in a business. The total profit is Rs.798.
Then profit is to be divided in the ratio 32:36:46 or 16:18:23
Now, A's share of profit = = Rs. 224 B's share of profit = = Rs. 252 C's share of profit = = Rs. 322
A partnership is called a compound partnership, if the capitals of the partners are invested for different time periods.
In this case we calculate the product of the capital invested and the period for which it is invested for all the partners, this product is called monthly equivalent investment, Now the profit or loss is divided
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in the ratio of their monthly equivalent investment.
If A and B are partners in a business, then:
Or
Example:
A, B, C enter into a partnership. A invests Rs. 2400 for 4 months, B invests Rs. 2800 for 8 months and C invests Rs. 2000 for 10 months. They gain Rs.
1170 altogether.
In this case Monthly equivalent investment of A = 2400 × 4 = 9600
Monthly equivalent investment of B = 2800 × 8
= 22400
Monthly equivalent investment of C = 2000 × 10 = 20000 The profit is divided in the ratio 96: 224: 200 i.e.
12:28:25
Now, A's share of profit = = Rs. 216 B's share of profit = = Rs. 504 C's share of profit = = Rs. 450 Races:
The concepts of ratio are also used in solving problems on races. For example: The statement "In a race of 1000 meters, A can give B a start of 100 meters"
means when A runs 1000 meters, B runs 900 meters. Thus the ratio of the speeds of A and B is 10:9.
Time and Work:
Example:
A can do a piece of work in 21 days, B is 40 % more efficient than A. In how many days B will complete the same work?
Solution:
Ratios of the efficiencies of A and B = 100: 140 or 5:7 Since efficiency is inversely proportional to the number of days, the ratio of days taken to complete the job is 7: 5
So number of days taken by B = (5/7)×21 = 15 days.
Unitary Method:
Consider the following example:
9 dozens of bananas cost Rs. 540, what will be the cost of 4 dozens of bananas? Here cost of 9 dozens of bananas is given. This part of question is called a statement. From this the cost of 1 dozen of bananas is to be obtained by compound division and from this value of one dozen of bananas the cost of 4 dozens of bananas is obtained by compound multiplication.
This method in which first the value of one unit is to be found is called the unitary method.
Example:
12 men or 18 women can reap a field in 7 days, in what time can 4 men and 8 women can reap the field.
Solution:
12 men = 18 women
1 man = (18/12) = (3/2) women 4 men = 4 × (3/2) women = 6 women 4 men + 8 women = (6 + 8) = 14 women
The more the women less the number of days, i.e.
indirect proportion Hence 18: 14 : : x: 7
i.e. x = (18/14)×7 = 9 days (Answer) Example:
If 25 elephants consume 18 quintals in 36 days, how long will 28 quintals last for 30 elephants?
Solution:
Let required number of days be m
More elephants less the number of days (indirect proportion)
More food more the number of days (direct proportion)
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Hence Elephant 30: 25 :: 36: m Food 18: 28
Hence 30 × 18 × m = 25 × 28 × 36 Or m = days (Answer)
Example:
If the wages of 5 men for 12 days be Rs. 6000 then what will be the wages of 6 men for 20 days?
Solution:
Let the required wages be m
More men more wages (direct proportion) More days more wages (direct proportion) Hence Men 5: 6
:: 6000: m Days 12:20
Hence 5 × 12 × m = 6 × 20 × 6000 Or m = 12,000 (Answer)
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1. A and B are in a ratio 5:7 and sum of these two numbers is 108. Find the difference of these two numbers?
a. 18 b. 8
c. 9 d. 12
2. Raja, Raju and Ram have invested Rs. 5,00,000 Rs. 7,00,000 and Rs. 9,00,000 respectively in a project for 4 months, 5 months and 6 months respectively. If the total income at the end of complete project was s Rs. 2,18,000 then find the share of Raju?
a. Rs. 40,000 b. Rs. 70,000
c. Rs. 1,08,000 d. None of the above 3. If a: b = 4:5 and b: c = 3:4, then find the value
of a: b: c?
a. 8:10:15 b. 6:9:12 c. 12:15:20 d. 12:15:22
4. If a:b = 4:7 then b is how much per cent more than a?
a. 42 % b. 75 % c. 27 % d. 100 %
5. If n: 4=9: n, then what is the value of n?
a. 6 b. 1/6
c. 8 d. 1/8
6. Two dozen of glasses were being transferred from one place to another when a few of these glasses fell on the floor and broke down. Which of the following cannot be the ratio of broken and unbroken glasses?
a. 1:2 b. 1:3
c. 1:4 d. 1:5
7. If price of sugar increases by 25 %, then by what percentage sugar consumption should be decreased to maintain the same expenditure?
a. 20 % b. 25 %
c. 33.33 % d. None of the above
8. While organizing a party, average contribution per head comes out to be Rs 100 when there are exactly 100 boys attending the party. But the average contribution increases to Rs 120 when there were 80 boys attending the party. What shall be the approximate average contribution if exactly 120 boys are attending the party?
a. Rs 80 b. Rs 85 c. Rs 70 d. Rs 75
9. In an exam, Ramu and Kallu have scored in the ratio of 1:2. But had both of them scored 3 more marks each, then they would have scored in the ratio of 3:5. How much marks did Ramu score?
a. 6 b. 9
c. 12 d. 15
10. If a: b = 3:5, then find the value of : a. 3:5 b. 11/29
c. 29/11 d. 9/20
11. If , then find
?
a. 1 b. 1/2
c. 1/3 d. 1/4
12. There is a fruit juice which contains 20 % fruit pulp. How much pure pulp should be added to 1 kg of juice to increase the concentration of fruit pulp to 25 %?
a. 33.33 grams b. 50 grams c. 66.66 grams d. 100 grams
13. Price of a piece of diamond is directly proportional to the square of the weight of the
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diamond. If a diamond of Rs. 5,000 breaks into two equal pieces then find the loss incurred?
a. Rs 1,250 b. Rs 2,500
c. Rs 3,750 d. None of the above 14. Two gears are working with each other in such a
way that first gear has 80 cogs and second gear has 16 cogs. If first gear takes 100 revolutions in 1 minute then second gear will take how many revolutions in 15 seconds?
a. 5 b. 125
c. 2000 d. 80
15. There are two shopkeepers one of whom is giving a discount of 20 % and the second one is giving 20 % extra on each and every purchase.
Which of these is a better deal to choose from if both claim the same price?
a. First shopkeeper b. Second shopkeeper c. Both are equally good d. None of the above
16. What is the mean proportion of 8 and 162?
a. 24 b. 36
c. 54 d. 72
17. In a map, if 20 cm represents 400 km. What will be actual area of a country which is represented by 40 square cm on this map?
a. 16000 sq Km b. 1600 sq Km c. 8000 sq Km d. 80000 sq Km
18. A group of students decided to go for a sea trip whose cost is constant. If the number of students decrease by 25 then averages contribution per student increases by Rs 20. What will be the further increase in average contribution if 20 more students back out from the trip?
a. Rs 25 b. Rs 29
c. Rs 15 d. Cannot be determined 19. Chaman, Raman and Karan started a business
by investing Rs 4,500, Rs 6,750 and Rs 10,125 respectively. If their return on complete investment was in the ratio of 1: 2: 3 then find the ratio of time for which they invested their money.
a. 3:4:4 b. 4:3:3 c. 4:4:3 d. 3:4:3
20. Ramu, Kallu and Billu started a business in partnership which required an investment of Rs 1,00,000 for one full year. So on 1st of January, Ramu invested Rs 50,000, Kallu invested Rs 30,000 and Billu invested Rs 20,000 in the year but on 1st of April, when Ramu required Rs 20,000, both Kallu and Billu gave him Rs 10,000 each and again on 1st of July Ramu required another 20,000 and again Kallu and Billu gave him Rs 10,000 each to rescue him.
Finally on 1st of October Ramu gave back Rs 10,000 to both Kallu and Billu. Find the ratio in which they should divide their return on the investment?
a. 3:3:4 b. 3:4:3 c. 4:3:3 d. 2:3:4
21. In what ratio should a 20% sugar solution be mixed with a 50% sugar solution so that the resultant solution has 40% sugar in it?
a. 1 : 2 b. 2 : 1 c. 1 : 3 d. 3 : 1
22. 50 men were hired to complete a job in 50 days. But after 10 days, it was realized that only 10 % of the job is finished. How many men should be hired with effect from 11th day to finish the job at least 5 days in advance?
a. 78 b. 79
c. 128 d. 129
23. Ramu, Kallu and Billu started a business by investing some money in the ratio of
. They had invested for some time in the ratio of . If net income from this investment was Rs 1950, then what is the share of Kallu in the profit?
a. Rs 650 b. Rs 600
c. Rs. 702 d. None of the above 24. If price of sugar increases by 20 %, then Ramu
can purchase 4 Kg less sugar for Rs 120. What was the original price of sugar per Kg.?
a. Rs 4 per Kg b. Rs 5 per Kg
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c. Rs 6 per kg d. Rs 7 per Kg
25. Ramu can complete a piece of work in 120 days. Kallu is 20 % more efficient than Ramu.
In how many days, Kallu can complete the same work?
a. 100 days b. 96 days c. 90 days d. 80 days
26. Increase in height of a person is in proportion with the square root of his age till he attains the age of 16 years. And after 16 years of age, they will become 1:2. Which of the following is one of those two numbers?
a. 3 b. 4 return the bottles after emptying. If you want to keep the empty bottles 5 bottles of "Fumes
Up" will cost Rs. 35. Pooja wants to buy 6 bottles. Had she promised to return 4 of the empty bottles, how much should she pay?
a. 55 b. 36
c. 42 d. 44
32. The ratio of the density of oil : petrol : diesel is 9 : 7 : 5. The density of diesel is 5 gm/cc.
They are mixed in the ratio of 12 : 10 : 8 by weight. If a litre of petrol costs Rs. 25.2, then what is the value of petrol in 300 kg of the mixture?
a. Rs. 326 b. Rs. 375 c. Rs. 360 d. Rs. 355
33. Pipe A is connected to Vessel A and Pipe B is connected to Vessel B. If the volumes of vessel A : volume of vessel B : : 26 : 9 and they get filled in 2 minutes and 13 minutes respectively, find the ratio of the diameter of pipe A : pipe B, assuming that volume of water flowing through a pipe is proportional to the day's bill which amounts to Rs. 45.75, B the second day's bill which amounts to Rs. 59.60 and C the third day's bill which amounts to Rs.
74.65. When they settle their accounts, which of the following is going to happen?
a. C will pay A Rs. 2.25 b. B will pay C Rs. 0.40 c. A will pay C Rs. 1.35 d. B will pay A Rs. 1.60
Directions (Questions 35–36) based on the following information.
A monkey distributed ladoos to two cats, cheating them of some and eating these himself. At the end of the distribution, if the black cat were to give some ladoos to the white cat, the white cat would have 5 times as many whole ladoos as the black cat. If the white cat were to give the same number of ladoos to
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the black cat, the white cat would have 3 times the number of ladoos as the black cat.
35. What is the ratio of ladoos initially distributed to the white cat and black cat respectively?
a. 10 : 3 b. 19 : 5
c. 5 : 21 d. Cannot be determined 36. If the total number of ladoos was thirty, what
was the number of ladoos eaten by the monkey?
a. 19 b. 5 following pairs contains a number that is not an integer?
a. b.
c. d.
39. A person spent half of the money he had. Now he finds that he has just as many paisa as he had rupees and half as many rupees as he had paisa in the beginning. If 1% error is allowed what should be your nearest guess about his money in the beginning?
a. Rs. 50 b. Rs. 80 c. Rs. 90 d. Rs. 100
40. 1/3rd of the contents of a container evaporated on the 1st day. 3/4th of the remaining contents of the container evaporated on the second day.
What part of he contents of the container is left at the end of the second day?
a. 1/4 b. 1/2
c. 1/18 d. 1/6
41. Mohan ate half a pizza on Monday. He ate half of what was left on Tuesday and so on.
He followed this pattern for one week. How much of the pizza would he have eaten during the seven days of the week?
a. 99.22% b. 95% to its side, the diagonals of the rhombus are in the ratio
a. b.
c. 3 : 1 d. 2 : 1
44. Rs. 1360 have been divided among A, B, C such that A gets (2/3) of what B gets and B gets (1/4) of what C gets, then B's share is
a. Rs. 120 b. Rs. 160 c. Rs. 240 d. Rs. 300
45. The Binary Ice Cream Shoppe sells two flavours, vanilla and chocolate. On Friday, the ratio of vanilla cones sold to chocolate cones sold was 2 : 3. If the store had sold 4 more vanilla cones, the ratio of vanilla cones sold to chocolate cones sold would have been 3 : 4. How many vanilla cones did the store sell on Friday?
a. 32 b. 35
c. 42 d. 48
46. After reading 3/5 of the biology homework on Monday night, Sanjay read 1/3 of the remaining homework on Tuesday night. What fraction of the original homework would Sanjay have to read on Wednesday night to complete the biology assignment?
a. 1/15 b. 2/15 c. 4/15 d. 2/5
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47. If the A : B: C is a. 4 : 3 : 5 b. 5 : 4 : 3 c. 3 : 4 : 5 d. 20 :15 : 12
48. Visitors to a show were charged Rs. 15 each on the first day, Rs. 7.50 on the second day and Rs. 2.50 on the third day and the attendance on the three days was in the ratio 2:5:13. The average charge per person for the whole show was
a. Rs. 6.33 b. Rs. 9 c. Rs. 5 d. Rs. 7.50
49. In what proportion must water be added to spirit to gain 20% by selling it at the cost price?
a. 2 : 5 b. 1 : 5 c. 3 : 5 d. 4 : 5
50. Salaries of A, B and C were in the ratio 3 : 5 : 7, respectively. If their salaries were increased by 50%, 60% and 50% respectively, what will be the new ratio of their respective salaries?
a. 4 : 5 : 7 b. 3 : 6 : 7 c. 4 : 15 : 18 d. 9 : 16 : 21
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Allegation is a rule that enables us to find the ratio in which the two or more ingredients at different prices must be mixed to find a mixture of the desired price.
The cost price of a unit quantity of the mixture is called the Mean Price.
Let's consider the following problem:
Example:
In what ratio should two qualities of Custard Powder having the rates of Rs. 47 per kg and Rs. 32 per kg be mixed in order to get a mixture that would have a rate of Rs. 37 per kg?
Solution:
The conventional method is:
Let x kilograms of custard powder of Rs. 47 per kg be mixed with y kilograms of custard powder of Rs.
32 per kg.
Hence x (47) + y (32) = (x + y) (37) (47 – 37) x = (37 – 32) y
Allegations Rule
Let Qc = Cheaper quantity Qd = Dearer quantity
M = Mean price or Price of one unit of mixture Pc = Cost price of one unit of cheaper variety Pd = Cost price of one unit of dearer variety hen
Mean price of one unit of mixture cannot be more than the cost price of dearer item and cannot be less than the cost price of one unit of cheaper item
Mixing a pure component to a solution:
Example:
A jar contains a mixture of two liquids A and B in the ratio 5:1. When 10 litres of liquid B is poured into the jar the ratio becomes 3:2. How many litres of liquid A were contained in the jar?
Solution:
Conventional Method:
Let the quantities of A and B in the original mixture be 5x and x litres, then
10x = 3x + 30 x = (30 / 7) liters
Hence quantity of A in the original mixture = (150 / 7) liters
By Allegation Rule:
The average composition of B in the first mixture
=(1/6)
The average composition of B in the second mixture = 1 The average composition of B in the final mixture
= (2/5)
B in B in First Second
Mixture of Mixtures:
Example:
Two jars A and B contain mixture of water and milk in the ratio 1:2 and 3:2 respectively. In what ratio one must mix these two mixtures in order to get a
7 Allegations
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final mixture which contains water and milk in the ratio 1:1.
Solution:
Average composition of milk in jar A = 1/3 Average composition of milk in jar B = 3/5 Average composition of milk in final mixture = ½
So one must mix these two mixtures in the ratio 3:5 Case of more than two varieties:
When a mixture of three ingredients P, Q
& R is given, take any two ingredients such that cost of the mixture is between the costs of the two chosen ones and find the ratio.
Once again, take two more ingredients and find their ratio. Then find the combined ratio. This will give an infinite number of solutions.
Example:
Let there be varieties of rice A, B & C priced at Rs.
10 per kg, Rs. 15 per kg & Rs. 20 per kg respectively.
In what ratio these three varieties should be mixed to get a mixture which would have a rate of Rs. 16 per kg?
Solution:
We can get a mixture of Rs. 16 per kg
(i) By mixing the varieties of Rs. 10 per kg and Rs. 20 per kg.
(ii) By mixing the varieties of Rs. 15 per kg and Rs. 20 per kg.
In the first case:
Hence we should mix A and C in the ratio 4:6 or 2:3 In the second case:
Hence we should mix B and C in the ratio of 4:1 Now if we mix these two mixtures in any ratio the cost price of one unit of the final mixture will be Rs 16 in either case
Suppose we take 5 kg of first mixture and 5 kg of second mixture, then in the final mixture:
Quantity of A = 2 kg Quantity of B = 4 kg
& Quantity of C = (1 + 3) = 4 kg
Hence we should mix A, B and C in the ratio 2:4:4 or 1:2:2
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1. If x litres of 20 % milk solution is added to 50 liters of 50 % milk solution then we get a 40 % milk solution. What is the value of x?
a. 25 litres b. 75 litres c. 100 litres d. 150 litres
2. We have 20 litres of 50% sprit solution. How much pure sprit should be added to this to make it a 75% sprit solution?
a. 15 litres b. 20 litres c. 25 litres d. 30 litres
3. An unpuffed cigarette burns completely in 15 minutes. But once puffed, rate of burning increases to thrice the original speed. If Ramu smokes a complete cigarette in 12 minutes then what part of the cigarette was puffed?
a. 30% b. 70%
c. 60% d. 50%
4. In the previous question, cigarette was puffed for how much time?
a. 1.5 minutes b. 4.5 minutes c. 3 minutes d. 3.6 minutes
5. Petrol costs Rs 50 per litres and kerosene cost Rs 20 per litres. In what ratio petrol and kerosene
5. Petrol costs Rs 50 per litres and kerosene cost Rs 20 per litres. In what ratio petrol and kerosene