Table Of ContentQUANTITATIVE
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APTITUDE
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VOLUME-II
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Dinesh Khattar
Cover image: Shutterstock.com www.pearson.co.in
The Pearson Guide To
Quantitative Aptitude
for
Competitive Examinations
Volume 2
A01_VOL 2_9997_SE_FM.indd 1 2/4/2016 1:01:00 PM
Copyright © 2016 Pearson India Education Services Pvt. Ltd
Published by Pearson India Education Services Pvt. Ltd, CIN: U72200TN2005PTC057128, formerly
known as TutorVista Global Pvt. Ltd, licensee of Pearson Education in South Asia.
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ISBN: 9789332570009
e-ISBN: 9789332578661
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Table of ConTenTs
Chapter 1 Stocks, Shares and Debentures 1
Chapter 2 Discount: True and Banker’s 16
Chapter 3 Binary Number System 27
Chapter 4 Series 33
Chapter 5 Clocks and Calendar 61
Chapter 6 Polynomials 73
Chapter 7 H.C.F. and L.C.M. of Polynomials 82
Chapter 8 Linear Equations 89
Chapter 9 Quadratic Equations 110
Chapter 10 Progressions 126
Chapter 11 Set Theory 154
Chapter 12 Permutations and Combinations 175
Chapter 13 Probability 200
Chapter 14 Mensuration I: Area and Perimeter 230
Chapter 15 Mensuration II: Volume and Surface Area 296
Chapter 16 Trigonometric Ratios 334
Chapter 17 Heights and Distances 350
Chapter 18 Plane Geometry 369
Chapter 19 Co-ordinate Geometry 419
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1
Stocks, Shares and Debentures
IntroductIon
Sl. Par value Number of Rate of dividend
To start a big business or an industry, a large sum of money No. of a share Common Shares declared on a
Common Share
is required. It may not be possible for one or two persons to
arrange for the requisite finance and expertise required for (i) `10 500 10% per annum
the project. So a number of individuals join hands to form a (ii) `10 800 5% semi-annually
company called a Joint Stock Company. It is a registered
(iii) `100 1500 5% quarterly
body under the Companies Act. The persons who join
(iv) `10 2500 2% per month
together to form the company are called its Promoters. The
total amount of money required by the company is called
Solution: (i) Annual dividend on one share
the Capital.
10
The promoters of the company issue a circular giving = 10% of `10 = ` ×10 = Re.1
100
the details of the project, its benefits and drawbacks; and
invite the public to come forward and subscribe towards the Annual dividend on 500 shares
capital of the company. The company divides the required
= `(500 × 1) = `500.
capital into small units of equal amount. Each unit is called
(ii) Annual dividend on one share
a share. Each person who purchases one or more shares
of the company is called a shareholder of the company. = `(2 × 5)% of `10
The company issues a share certificate to each of its
10
shareholders stating the number of shares alloted to the = ` ×10 = Re.1
100
person and the value of each share. The value of a share as
stated on the share certificate is called the nominal value \ Annual dividend on 800 shares
(face value or par value) of the share. = `(800 × 1) = `800.
When a company earns a profit during a financial year,
(iii) Annual dividend on one share
a part of it is used in paying for working expenses, taxes,
= (4 × 5)% of `100
interest on loans and keeping some part of it as reserve fund
for future expansion of the project, the remaining profit is 20
= ` ×100 = `20
distributed among the shareholders. The distributed profit is 100
called the dividend.
\ Annual dividend on 1500 shares
Dividends are declared annually, semi-annually,
= `(1500 × 20) = `30000.
quarterly as per regulations of the company. The dividend
on a share is expressed as certain percentage of its face (iv) Annual dividend on one share
value which is printed on the share certificate. Sometimes
= (12 × 2)% of `10
it is also expressed as a specified amount per share. For
example, we may say that dividend on a share is 12% of its = ` 24 ×10 = `2.40
face value or the dividend is `2 per share. 100
Illustration 1 Find the annual dividend paid in each of the \ Annual dividend on 2500 shares
following cases: = `(2500 × 2.40) = `6000.
Chapter_1.indd 1 01-02-2016 11:12:19
2 Chapter 1
tYPES oF SHArES Illustration 2 Find the cost of purchasing 150 shares of a
company, each of par value `10, quoted at `16 each in the
The shares are generally of two types:
market from the original shareholder. Also, find the gain to
(i) Preferred shares
the new shareholder if he sells each share at a premium of
(ii) Common (ordinary) shares `10.
(i) Preferred shares These shares get preference in Solution: Market value of share = `16
terms of payment of dividend and return of capital \ Market value of 150 shares
over ordinary shares. The rate of dividend for these = `(150 × 16) = `2400
shares is decided when they are issued and dividend Thus, the new shareholder spent `2400 for buying 150
to preferred shareholders is paid before any dividend
shares. The new shareholder sold the shares at a premium
is paid to common shareholders. of `10.
(ii) Ordinary shares Ordinary shareholders are paid \ Now, market value of a share
dividend only when profits are left after preferred = `(10 + 10) = `20
shareholders have been paid dividend at specified
The selling price of 150 shares at the new market value
rate. The rate of dividend on these shares is also not
= `(150 × 20) = `3000
fixed and depends upon the amount of available profit.
\ Gain of the new shareholder in the transaction
FAcE VALuE And MArKEt VALuE oF A SHArE = `(3000 – 2400) = `600.
The price at which the shares are initially issued by the Illustration 3 Raja buys 200 shares, each of par value `10
company to its shareholders is called the face value of a of a company which pays annual dividend of 15% at such
share (This is also called nominal or par value of a share). a price that he gets 12% on his investment. Find the market
In fact, this is that value of a share which is mentioned in the value of a share.
share certificate issued by the company to its shareholders.
Solution: Par value of 200 shares
As other things, shares are also sold in (or purchased = `(200 × 10) = `2000.
from) the market. The value of a share quoted in the market
Dividend received by Raja
is called the market value of the share. The market value
of a share keeps on changing according to its demand and 2000×15
= ` = `300.
supply changes. 100
If the market value of a share is equal to the par value Let the market value of 200 shares be `x. We have to
of the share, the share is said to be at par. If the market find x such that 12% of x = 300,
value of a share is more than its face (or par) value, the
12 100×300
share is said to be at premium. On other hand, if the market i.e., × x = 300 \ x = = 2500
100 12
value of a share is less than its face value, the share is said
to be at discount (or below par). i.e., Market value of 200 shares = `2500
For example If the market value of a `100 share is `130, it Hence, the market value of one share = `12.50.
is said to be at 30% premium.
If the market value of a `100 share is `90, it is said to StocKS And BroKErAgE
be at 10% discount. If `100 share is quoted at 45 premium
Stock
then its market value is
In the previous section, we have learnt about shares which
`(100 + 45) = `145.
can be sold and purchased by the public. The nominal value
Every company declares dividend on the face value of or face value of shares is fixed (usually `10 or `100) but
its shares irrespective of the market value of the share.
their market value varies.
Notes: The statement “32%, `100 shares at `125” means: Sometimes, joint stock companies or the government
(i) Face value of each share is `100 also raises loans from the market by issuing bonds or
(ii) Dividend on each share is 32% of the face value promisory notes. They promise to pay a fixed amount (called
(iii) Market value of each share is `125 redemption value) on a future date and interest payments at
(iv) An investment of `125 earns an annual income of fixed periods until that time. The money paid to company or
`32. government for buying such bonds is called stock.
Chapter_1.indd 2 01-02-2016 11:12:19
Stocks, Shares and Debentures 3
The stocks are usually known by their rates of dividend. Illustration 5 Find the income on 10% stock of `25000
Thus by 9% stock we mean that the dividend on a `100 purchased at `120.
stock is `9.
Solution: Face value of stock = `25000
If the market value of `100 stock, which yields a
Income on `100 stock = `10
dividend of `5, is `115, the stock is called 5% stock at 115.
10
Similarly, 10% stock at 120 means that a stock of face value Income on `1 stock = `
`100 gives a dividend of `10 and is available in the market 100
of 120. Income on `25000 stock
Note: There can be stocks in units different from `100, say 25000×10
`500, `1000, etc., but the phrase “8% stock at 90” can be = ` 100
used only in case of that stock whose face value is `100.
= `2500.
Dividend on a stock is fixed (declared at the time of issue)
whereas for a share it varies with time. Usually, the date
coMPutAtIon oF InVEStMEnt or MArKEt
of maturity of the stock is fixed. In case the holder of the
VALuE oF A StocK
stock requires money before the due date, he can sell his
stock to some other person, whereby his claim of interest is If the face value of a stock is given, the market value of the
transferred to that person. stock can be found on the basis of market value of each unit
of stock.
Brokerage
Illustration 6 Find the investment required to purchase
The sale and purchase of stock is generally executed through `75000 of 10% stock at 95.
a stockbroker, who charges some money, called Brokerage
Solution: Market value of `100 stock = `95
from both the seller and purchaser. The brokerage is charged
\ Market value of `75000 stock
either as some fixed amount on each unit of stock or as some
percentage of the market value of unit of stock. 95
= ` ×75000 = `71250
Thus, the brokerage of `x means that x rupees are to 100
be added or subtracted from the market value of the stock. \ An investment of `71250 is required to purchase
Similarly, brokerage 2% means that the brokerage equal to `75000 of 10% stock at 95.
2% of the market value of a unit of stock and be added to (or
Illustration 7 Find the investment required to get an
subtracted from) the market value of a unit of stock.
1
Notes: income of `4200 from 10 % stock at 80 (Brokerage: 2%).
2
(i) The brokerage is added to the market value when the
2
stock is purchased. Solution: Brokerage = 2% of `80 = ` ×80 = `1.60
100
(ii) The brokerage is subtracted from the market value
\ Investment needed to buy `100 stock
when the stock is sold.
1
= `81.60 on which the income is `10
2
cALcuLAtIon oF IncoME on A StocK
1
When the face value of the total stock is given, the income For income of `10 , the investment = `81.60
2
can be calculated on the assumption that the face value of
For income of `4200, the investment
each unit of stock is `100. On the contrary, if the market
value of the total investment is given, the income can be = `81.60× 2 ×4200 = `32640.
calculated on the basis of the market value of a unit of stock. 21
Illustration 4 Find the income from `2875 of 4% stock.
coMPutAtIon oF gAIn or LoSS In tHE SALE
Solution: By 4% stock, we mean a stock of `100 will fetch
And PurcHASE oF A StocK
a dividend of `4 p.a.
Hence, the income from `2875 of 4% stock When the market is favourable to stock holders, i.e., they
are likely to get better proceeds for their stock, they sell the
2875×4
= = `115. stock and may reinvest the money so obtained in another
100
stock which may give them more income.
Chapter_1.indd 3 01-02-2016 11:12:19
4 Chapter 1
Illustration 8 Ram bought `12000 of 8% stock at 92 and period of time and at a fixed rate of interest by dividing
sold it when the price rose to 98. Find his total gain and gain the amount required into small parts. These small parts are
per cent. called debentures.
Solution: Investment made by Ram in buying `12000 of The debenture holders are creditors of the company
8% stock at 92 and do not have any right on the profits declared by the
92 company. However, interest at fixed rate and fixed time
= ` 12000× = `11040.
is payable to debenture holders, irrespective of the fact
100
whether the company is running in profits or losses.
When the price rose to `98, Ram sold the stock, thus
Like shares, debentures can also be sold in or purchased
money realised from selling the stock
from the market. The terms used in case of shares, are also
= `12000× 98 = `11760. used with the same meaning in case of debentures. Thus
100 we use the terms ‘debentures at premium’, ‘debentures at
\ Gain realised in the transaction discount’ etc. Furthermore, the rules for calculating the
brokerage on debenture are also the same as those in case
= `(11760 – 11040) = `720
of shares.
(720×100) 12
\ Gain per cent = = 6 %.
11040 23 dIFFErEncE BEtWEEn SHArES And
dEBEnturES
cHAngE In IncoME on SALE or rEInVEStMEnt
Shares Debentures
A person having one type of stock may sell it to buy another (a) Share money forms a (a) Debentures are a mere
which gives higher income. In such problems, the income in part of the capital of the debt.
two cases is calculated and change is found out. company. (b) Debenture holders are
(b) Shareholders have right creditors of the company
Illustration 9 Ram invests `46500 in 6% stock at 93 and
on the profit declared by and do not have any right
sells the stock when its price rose to `95. He invests the the company. on the profit declared by
sale proceeds in 9% stock at 95. Find the change in Ram’s the company.
income. (c) Shareholders may re- (c) Debenture holders re-
6 ceive different dividend ceive interest at a fixed
Solution: Income from first stock = ` ×46500
93 according as profit is rate.
more or less.
= `3000
We have to find the amount realized on selling this Illustration 10 Find the income percent of a buyer on 8%
stock. debentures of face value `120 and available in the market
Amount realized on selling `93 stock = `95 at `180.
\ Amount realized on selling `46500 stock Solution: The market value of a debenture is `180
95 \ Income on `180 is `8
= ` × 4650 = `47500
93 \ Income on `120 is ` 8 ×120 = `51
180 3
This amount is invested in 9% stock at 95
\ Income from 2nd stock \ Per cent income on the debenture is 51%.
3
9
= `95×47500 = `4500 Illustration 11 Ram has 500 shares of par value `10 each
of a company and 500 debentures of par value `100 each.
Hence, increase in income
The company pays a dividend of 8% on the shares and pays
= `(4500 – 3000) = `1500. an interest of 10% on its debentures. Find the total annual
income of Ram and the rate of return on her investments.
dEBEnturES
Solution: Annual dividend on 500 shares
Sometimes a running joint stock company may require more (500×10×8)
= `
capital for its further expansion. The company borrows the 100
required sum of money from the general public for a fixed = `400.
Chapter_1.indd 4 01-02-2016 11:12:19
Stocks, Shares and Debentures 5
Annual interest on 500 debentures Total investment of Ram
(500×100×10) = `(500 × 10 + 500 × 100) = `55000.
= `
100 Rate of return on Ram’s investment
= `5000. 5400
= ×100%
\ Total annual income of Ram 55000
= `(5000 + 400)
108 9
= % or 9 %.
= `5400.
11 11
Multiple ChoiCe QueStionS
1. A man who owned 25% of the equity capital of a certain Arun drives twice as many miles as Balu. What is the ratio
company sold one-third of his holding last year and five- of the number of hours that Arun spends in driving to the
twelfths of the remaining this year. What part of the meeting to the number of the hours that Balu spends in
business does he now own ? driving to the meeting?
(a) 1/5 (b) 5/144 (a) 3:2 (b) 8:3
(c) 7/72 (d) 65/72 (c) 2:3 (d) 4:3
[Based on MAT, 2004] 7. A man wants to secure an annual income of `1,500 by
2. Arun has 800 shares of par value of `50 each, and 600 investing in 15% debentures of face value `100 each and
debentures of par value `100 each of a company. The available for `104 each. If the brokerage is 1%, then the
company pays an annual dividend of 6% the shares and sum of money he should invest is
interest of 12% on the debentures. The rate of return of his (a) `19,642 (b) `10,784
investment is (c) `10,504 (d) `15,000
(a) 8% (b) 9.6% 8. A person invests `5508 in ‘4% stock at 102’. He afterwards
(c) 10.6% (d) 8.6% sells out at 105 and reinvest in ‘5% stock at 126’. What is
[Based on MAT, 2008] the change in his income?
3. The capital of a company is made up of 50,000 preferred (a) `20 (b) `7
shares with dividend of 20% and 20,000 common shares, (c) `10 (d) `9
the par value of each type of share being `10. The company [Based on IIFT, 2005]
had a total profit of `1,80,000 out of which `30,000 was
9. Shyam, Gopal and Madhur are three partners in a business.
kept in reserve fund and the remaining distributed to
Their capitals are respectively `4000, `8000 and `6000.
shareholders. Find the dividend per cent to the common
Shyam gets 20% of total profit for managing the business.
share-holders.
The remaining profit is divided among the three in the
(a) 20% (b) 24% ratio of their capitals. At the end of the year, the profit of
(c) 25% (d) 30% Shyam is `2200 less than the sum of the profit of Gopal
and Madhur. How much profit, Madhur will get?
4. At what price should I buy a share, the value of which
is `100, paying a dividend of 8%, so that my yield is (a) `1600 (b) `2400
11%? (c) `3000 (d) `5000
(a) `70 (b) `72.72 [Based on IIFT, 2010]
(c) `75 (d) `84 10. A sum of `2236 is divided among A, B and C such that A
receives 25% more than C and C receives 25% less than
5. A person had deposited `13,200 in a bank which pays
B. What is A’s share in the amount?
14% interest. He withdraws the money and invests in
(a) `460 (b) `890
`100 stock at `110 which pays a dividend of 15%. How
(c) `780 (d) `1280
much does he gain or lose?
[Based on Indian Overseas Bank PO, 2009]
(a) loses `48 (b) gains `48
(c) loses `132 (d) gains `132 11. A sum of money is divided among A, B, C and D in the
ratio of 2:3:7:11, respectively. If the share of C is `2755
6. Arun and Balu drive separately for a meeting. Arun’s more than the share of A, then what is the total amount of
average driving speed is one-third greater than Balu’s and money of B and D together?
Chapter_1.indd 5 01-02-2016 11:12:20
