qa 10307

© VistaMind Education Pvt Ltd, 2012 | QA 10307 | of 4

SUMMARY

Proportion:

A statement expressing the equality of two ratios is called a proportion, i.e a:b :: c:d or a c

b d Then a, b, c & d are

said to be in proportion, where a is the first proportion, b is the second proportion, c is the third proportion, and d is

the fourth proportion.

If the four quantities are in proportion then a & d are called the extremes and b & c are called means and the following

relation always holds true:

Product of means = product of extremes i.e. ad = bc

Properties of proportion:

If x / y = a / b, then

y/x = b/a (Invertendo)

x/a = y/b (Alternendo)

x y a b

y b

(Componendo)

x y a b

y b

(Dividendo)

x y a b

x y a b

(Componendo & Dividendo)

Continued proportion:

Three quantities x, y and z are said to be in continued proportion if x : y = y : z, then y2 = xz. Here y is also known as

the mean proportional of x and z. x, y and z are known as the first, second and the third proportional’s respectively

Variation:

Direct Variation - if with the increase in one quantity other quantity increases, then it is known as direct variation. i.e,

x∝ y (x is directly proportional to y), we can convert the relation to an equation as x/y = k, where k is a constant;

known as constant of proportionality.

Indirect Variation - if with the increase in one quantity other quantity decreases then it is known as indirect variation.

i.e, x 1

y (x is inversely proportional to y), we can convert the relation to an equation as xy = k, where k is a

constant and is known as constant of proportionality.

CLASS DISCUSSION PROBLEMS

1. What should be added to each of 3, 15, 38

and 134 so that they should become

proportionate to each other?

(1) 1 (2) 2

(3) 3 (4) 4

2. The mean proportional between 64 and 16 is

(1) 36 (2) 32

(3) 48 (4) 40

3. The third proportional to the numbers 0.9

and 0.45 is

(1) 0.2 (2) 0.25

(3) 0.225 (4) 0.3

4. What should be subtracted from each of 6,

18, 41 and 137 so that they become

proportionate to each other?

(1) 1 (2) ½

(3) 2 (4) 1/3

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QA 10307 Proportion & Variation


5. A milkman mixes 20 liters of water with 80

liters 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?

(1) 2: 3 (2) 1:2

(3) 1:3 (4) 3:4

6. The cost of a diamond varies directly with

the cube of its weight. A diamond merchant

accidently dropped a diamond and it broke

into four pieces with weights in the ratio

1:2:3:4. When the pieces were sold

individually the merchant got `9,00,000 less

as compared to what he would have got had

he sold the original diamond. Find the

original price of the diamond? (CAT1996)

(1) `10 lakh (2) `20 lakh

(3) `15 lakh (4) `25 lakh

7. Pressure varies inversely with volume while

temperature varies directly with volume.

When Volume= 50 m3, Temperature = 25 K

and Pressure = 1 atmosphere. If the volume

is increased to 300 m3 and the pressure

remains unchanged, then the temperature

will be

(1) 100 K (2) 50 K

(3) 125 K (4) 150 K

8. In what proportion must a number be

divided so that 1

4of the first part and 2

3of

the second part are together equal to 1

2

of

the original number?

(1) 3:2 (2) 2:5

(3) 5:3 (4) 2:3

9. The reduction in the speed of a railway

engine is directly proportional to the square

of the number of bogies attached to it. The

speed of the train is 200 km/hr when there

are 4 bogies and 155 kmph when there are 5

bogies. What is the maximum number of

bogies that can be attached to the train so

that it can move?

(1) 6 (2) 5

(3) 4 (4) 7

10. Given that a, b, c are in continued

proportion; b, c, d are in continued

proportion; and c, d, e are also in continued

proportion. If b:c = 3:4 and all the five

numbers are positive integers, what is the

minimum possible value of (a + b+ e)?

(1) 444 (2) 445

(3) 592 (4) cannot be determined

11. The duration of a railway journey varies as

the distance and inversely as the velocity;

the velocity varies directly as the square root

of the quantity of fuel used, and inversely as

the number bogies in the train. In a journey

of 50 km in half an hour with 18 bogies, 100

kg of fuel is required. How much fuel will

be consumed in a journey of 42 km in 28

minutes with 16 bogies?

(1) 48 kgs (2) 56 kgs

(3) 64 kgs (4) none of these

12. There are two containers: the first contains

1000 m1 of milk, while the second contains

1000 m1 of water. Three cups of milk from

the first container is taken out and is mixed

well in the second container. Then, three

cups of this mixture is taken out from the

second container and is mixed in the first

container. Let X denotes the proportion of

water in the first container and Y denote the

proportion of milk in the second container.

Then

(1) X >Y (2) X < Y

(3) X = Y (4) cannot be determined

13. The price of Assam tea (in rupees per

kilogram) is 100 + 0.10n on the nth day of

2011 (n = 1, 2…..... 100), and then remains

constant. On the other hand, the price of

Shillong tea (in rupees per kilogram) is 89 +

0.15n on the nth day of 2011 (n = 1, 2, ...,

365). On which date in 2011 will the prices

of these two varieties of tea be equal?

(CAT-2007)

(1) May 21 (2) April 11

(3) May 20 (4) April 10

14. The fee (per student in `) charged by Rahul

for the “Training” Program is inversely

proportional to the number of interested

students. The normal price is `40 per student

if there are 600 students. But for every

increase of ` 1 in the fee, he loses 5 students.

What is the maximum amount that he can

earn?

(1) `28,000 (2) `30,000

(3) `32,000 (4) `36,000

15. ‘The Pastry Shop’ sells pastries in boxes of

different sizes. The pastries are priced at `20

per pastry upto 200 pastries. For every

additional 20 pastries, the price of the whole

lot goes down by `1 per pastry. What would

be the number of pastries in the box such

that the revenue of ‘The Pastry Shop’ from

that customer is the maximum? (CAT 2002)

(1) 240 (2) 280

(3) 300 (4) None of these

16. A vendor of selling Candies at Koramangala

in Bangalore prices his candies at 85 paise

per candy, but found that there were no

buyers at this price. Thereafter, he reduced

the price of candies and managed to sell all


of them, realizing ` 77.28. What is the total

number of candies he sold?

(1) 92 (2) 47

(3) 84 (4) 81

Directions for questions 17 & 18: Refer to

the data below and answer the questions that

follow.

According to Bright’s equation, the purity P

of milk is directly proportional to the square

root of the ratio of pure milk to water in

volume terms in the sample. According to

Young’s equation, the same purity P of milk

is inversely proportional to the cube root of

the ratio of water to the total sample volume.

17. If the pure milk to water ratio is 2:1, then

what is the ratio of the proportionality

constant of the Bright equation to the

proportionality constant of the Young

equation?

(1) 8/9 (2) 3

2

3

2

(3) 9/8 (4) None of these

18. For which sample will the proportionality

constants of both the equations be equal?

(1) Pure Milk

(2) Pure Water

(3) For both pure milk and pure water

(4) None of these

19. A Bus is to be driven 300 kms on a highway

at a constant speed of x kmph. Speed rules

of the highway require that 30 x 60.

The fuel costs ` 10 per litre and is consumed

at the rate of 2 +2

600

x litres/hour. The wages

of the driver are ` 200 per hour. The most

economical speed to drive the Bus, in kmph,

is

(1) 30 (2) 60

(3) 30 33 (4) 20 33

20. The demand for a commodity linearly

decreases with price and it vanishes when

the price is set at `60 and equals to the half

of the difference of its equilibrium price and

this threshold price. The supply of the

commodity vanishes when the price is set at

`25 and equals the square root of the price in

excess of this threshold price. The

equilibrium price at which the supply

coincides with the demand is (CAT-1999)

(1) `58 (2) `50

(3) `79 (4) `62

21. A weight weighing 40 kg is accidentally

broken into four pieces such that by using

these four weights on a balance a man can

weigh all integral weights from 1 kg to 40

kg. The weights of the broken pieces were?

(1) 1, 4, 11, 24 (2) 1, 3, 9, 27

(3) 1, 5, 10, 24 (4) Not possible

22. The speed of light is directly proportional to

the refractive index of the medium it travels

through. There is a 60 cm thick slab of glass,

followed by a 50 cm slab of ice and a 40 cm

slab of plastic A ray of light is sent across

from one side of three slabs to another. If the

refractive index of glass, ice and plastic are

in G.P. refractive index of glass being the

maximum, then what is the ratio of time

taken for the ray of light to travel through

the first two slabs and the last two slabs?

(Assume the common ratio to be 2 between

the refractive indices).

(1) 28/11 (2) 8/13

(3) 13/20 (4) 31/21

23. Grass grows in a field at constant rate ‘r’

where ‘r’ is the units of grass grown per day.

It is known that, if 10 sheep are turned out in

the field, they will eat the grass in 20 days

and on the other hand, if 15 sheep are turned

out in the field, the same grass will be eaten

in 10 days. For how many days will the

grass last if 25 sheep are turned out in the

field? (Assume that some grass was present

initially)

(1) 5 days (2) 9 days

(3) 8 days (4) 12 days

24. In a certain exam, the number of students

passing is four times the number of students

failing. If the number of students appearing

for the exam is decreased by 35, and the

number of students failing is increased by 9,

then the number of students passing would

have been twice the number of students

failing. What is the number of students

appearing for the exam after the decrease?

(1) 140 (2) 160

(3) 120 (4) 155

25. The square of the time of revolution of a

planet varies as the cube of its distance from

the Sun. The time of Venus’s revolution,

assuming the distance of Earth and Venus

from the Sun to be 191

4 and 66 million

miles respectively is

(1) 250 days (2) 225 days

(3) 210 days (4) 150 days


Directions for Questions 26 to 30:

Choose (1), if the question can be answered by one of

the statements alone, but cannot be answered by using

the other statement alone.

Choose (2), if the question can be answered by using

either statement alone

Choose (3), if the question can be answered by using

both the statements together but cannot be answered

by using either statement alone

Choose (4), if the question cannot be answered even

by using both the statement together.

26. What is the total number of type-2 diabetic

patient in Bangalore, if the total number of

Diabetic patient (Type 1 and Type 2) in

Bangalore is 2, 00,000?

I. If the type-1 diabetic patients increase by

10% and type-2 diabetic patients decrease by

6%, than the total number of diabetic

patient’s remains the same.

II. The ratio of type-1 and type-2 diabetic

patient in Bangalore is 2:1

27. Total expenditure of VistaMind, Mumbai

Center is partly fixed and partly varying

linearly with the number of faculty members

in the Academic Team. What is the total

expenditure of VistaMind, Mumbai center,

when the Academic team comprises of 20

faculty members?

I. Total expenditure is `15 lakhs when the

Academic team comprises of 5 faculty

members.

II. Total expenditure is `25 lakhs when the

Academic team comprises of 15 faculty

member

28. What is the ratio of X: Y: Z?

I. Z = 1 and XY = 32

II. X/Y = 2 and Z/Y = ¼

29. A sequence of numbers A1, A2 ….. varies in

such a way i.e 1

2

n nA A Does 3 appear in

the sequence?

I. A1 = 2

II. A3 = 16

30. Ajeet and Sujeet share a room on rent in the

proportion to their period of stay. How much

would each pay for the month of September

2012?

I. Ajeet was away for the first 10 days and

Sujeet stayed alone till he returned and then

both stayed for the remaining part of the

month.

II. Sujeet paid `15 for his stay.

qa 10307

Technology

b y b dividendo x y

b y b componendo x y

continued proportion

proportion variation

proportion of milk

summary proportion

yb alternendo x y

quantities x