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© VistaMind Education Pvt Ltd, 2012 | QA 10307 | Page 1 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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Email Address: [email protected]
QA 10307 Proportion & Variation
© VistaMind Education Pvt Ltd, 2012 | QA 10307 | Page 2 of 4
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
© VistaMind Education Pvt Ltd, 2012 | QA 10307 | Page 3 of 4
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
© VistaMind Education Pvt Ltd, 2012 | QA 10307 | Page 4 of 4
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.