IFBI Solved Maths 7

Solved Examples

Question 1

Successive discounts of 10% and 20% are equivalent to a single discount of  (1) 30% (2) 15% (3) 72% (4) 28% Solution: (4) Let P be the original price. Then a discount of 10% gives a new price P - .1 P = .9 P. Following this by a discount of 20%, we have .9P - .2 (.9P) = .72 P Thus, the net discount is P – .72 P = .28P or 28%

Question 2 After selling a watch, Shyam found that he had made a loss of 10%. He also found that had he sold it for Rs 27 more, he would have made a profit of 5%. The initial loss was … (1) Rs 2.70 (2) Rs 16.65 (3) Rs 18.00 (4) Data insufficient Solution: (3) 15% of the cost price of the watch = Rs 27 ∴10% of the cost price of the watch =

Rs

27

×

10

15

=

Rs 18

Question 3 The value of a share of stock P and the value of a share of stock Q each increased by 16%. If the value of a share of stock P increased by 16% cents and the value of a share of stock Q increased by $1.68, what is the difference between the value of stock Q and the value of stock P before the increases? (1) $8.00 (2) $9.50 (3) $10.00 (4) $10.50

Solution: (2) Note: Only one basic insight is needed to handle the problem. Knowing the amount and percent of an increase is sufficient to allow you to calculate the starting and ending amounts. Here we need the starting amounts. For stock P we know that an increase of 16 cents is equal to 16% of the original value of the stock: $0.16 = 16% of Original Value $0.16 = 0.16 × OV

$0.16 0.16

=

OV

Original Value = $1.00 For stock Q we know that an increase of $1.68 is equal to 16% of the original value of the stock: $1.68 = 16% of Original Value $1.68 = 0.16 × OV $1.68 0.16

=

OV

Original Value = $10.50 Now we find the difference between the original values: $10.50 - $1.0 = $9.50 Directions for Questions. 4 to 6 : Refer to the data below and answer the questions that follow.

`A, B and C form a partnership investing in the ratio 1 : 2: 3. A agrees to work as the Director for a payment of Rs 500 along with 10% of the total profit as incentive. The partnership is for a period of 3 years. Half the profits after  payments to A are out in a corpus fund for investment, e.g., at the end of 1 year, a sum is invested. Similarly at the end of the second year an amount is invested. This fund earns 10% simple interest per annum. The total profit in each year is 50% greater than that of the previous year. At the end of 3 years, the interest accumulates to Rs 240. The total profit less wages and incentive to A is net profit. While half the net profit is saved every year, the other half is shared as per  partnership investment. The invested amount is not shared nor the interest shared. Question 4 What is the profit made in the third year? (1) 4000 (2) 4500 (3) 5000 Solution: (2) Now we get profit in one third year = 2.25 x  = 4500

(4) 6500

$0.16 0.16

=

OV

Original Value = $1.00 For stock Q we know that an increase of $1.68 is equal to 16% of the original value of the stock: $1.68 = 16% of Original Value $1.68 = 0.16 × OV $1.68 0.16

=

OV

Original Value = $10.50 Now we find the difference between the original values: $10.50 - $1.0 = $9.50 Directions for Questions. 4 to 6 : Refer to the data below and answer the questions that follow.

`A, B and C form a partnership investing in the ratio 1 : 2: 3. A agrees to work as the Director for a payment of Rs 500 along with 10% of the total profit as incentive. The partnership is for a period of 3 years. Half the profits after  payments to A are out in a corpus fund for investment, e.g., at the end of 1 year, a sum is invested. Similarly at the end of the second year an amount is invested. This fund earns 10% simple interest per annum. The total profit in each year is 50% greater than that of the previous year. At the end of 3 years, the interest accumulates to Rs 240. The total profit less wages and incentive to A is net profit. While half the net profit is saved every year, the other half is shared as per  partnership investment. The invested amount is not shared nor the interest shared. Question 4 What is the profit made in the third year? (1) 4000 (2) 4500 (3) 5000 Solution: (2) Now we get profit in one third year = 2.25 x  = 4500

(4) 6500

Question 5 Find A’s earnings in the 3 year both as Director as well as partner? (1) 340 (2) 2400 (3) 1100 (4) 2600 Solution: (1) A’s total earning =

1500

=

1500

+

0.475 x 

+

950

+

(

1 1300 +

+

2200

+

3550

)

6

587.5

=

3037 .5

=

340

Question 6 What is the amount earned by C over the 3 years? (1) 6600 (2) 3300 (3) 2200

(4) 1770

Solution: (4) Amount earned by C 1 =

2

(1300

+

2200

+

3550

)

3 ×

6

1 =

4

×

7050

DIRECTIONS for questions 7 and 8:

=

1762.5

≈

1770

Refer to the data below and answer the

questions that follow. Per capita rice consumption of a country is 40 kg. (per capita = Total consumed/Total population.) Question 7 If only 60% of the population eat rice, find the consumption per rice-eater. (1) 200/3 kg (2) 400/3 kg (3) 60 kg (4) population statistic needed to solve the question Solution: (1) Let population = N. Total consumption = 40N kg. Number of rice eaters = 0.6N. ∴Consumption per rice eater  = 40N/(0.6)N = 400/6 = 200/3 kg.

Question 8 If the number of rice-eaters increases by 20% and total population increases by 15%. while the consumption per rice-eater remains unchanged, find the new per  capita rice consumption. (1) 41.7 (2) 48 (3) 54.2 (4) Insufficient data Solution: (1) New number of rice eaters = (0.6)N(1.2) (20% increase) Total population = N1 = (1.15)N New per capita rice consumption (0.6)(1.2N × 400/6 / (1.15)N = 41.7 kg. Hence, (1).

Question 9 The income of a person becomes 3 times itself. What is the percentage change in income? (1) 200 % (2) 100 % (3) 50 % (4) 50%

Solution: (1) Let the income be Rs. 100. If it becomes 3 times (i.e. Rs. 300), then the percentage change of the income = ((300 - 100)/100)100 = 200 % increase.

Question 10 The price of an item increases first by 25 percent and then decreases by x %. If  the final price is the same as the initial price, then the value of x in % is (1) 40 (2) 25 (3) 20 % (4) Insufficient data Solution: (3) If the old price = Rs. 100, then the price after the first increase = 125. Then the decrease % required so that it becomes equal to the original = 20 %.

6ROYHG([DPSOHV

4XHVWLRQ

7ZRKLJKVFKRROFODVVHVWRRNWKHVDPHWHVW2QHFODVVRIVWXGHQWVPDGHDQ DYHUDJHJUDGHRIWKHRWKHUFODVVRIVWXGHQWVPDGHDQDYHUDJHJUDGHRI 7KHDYHUDJHJUDGHIRUDOOVWXGHQWVLQERWKFODVVHVLV  

 

 

 QRQH RI WKHVH

6ROXWLRQ

6XSSRVHWKHPD[LPXPPDUNVIRUWKHWHVWZHUH6RWKHFODVVRIVWXGHQWV VFRUHVDWRWDORIPDUNVZKLOHWKHFODVVRIVWXGHQWVVFRUHVD WRWDORIPDUNV7DNHQWRJHWKHUWKHILIW\VWXGHQWVVFRUHDWRWDORI PDUNV6RRQDQDYHUDJHHDFKRIWKHVHVWXGHQWVVFRUHV  

PDUNVRUPDUNVVLQFHZHKDYHDVVXPHGWKHPD[LPXPPDUNV

RIWKHWHVWWREH 4XHVWLRQ

 $VDOHVPDQDYHUDJHGIRUHDFKGD\KHZDVRQWKHURDGEXWIRUWZRRIWKHVH GD\VKHPDGHRQO\DOWRJHWKHU$SDUWIRUPWKHVHWZREDGGD\VKHKDG DYHUDJHGIRUWKHRWKHUGD\VKHZRUNHG+RZPXFKGLGWKHVDOHVPDQHDUQ WKDWPRQWK"  

 

 

 

6ROXWLRQ

/HWWKHQXPEHURIGD\VLQWKHPRQWKEHµQ¶7RWDOHDUQLQJLQWKHPRQWKQ 7RWDOHDUQLQJIRUQ±GD\VQ± Q±       Q Q Q Q      Q Q 7RWDOHDUQLQJ     

+HQF +H QFH H  

Q Q  

4XHVWLRQ

,QDFODVVRIVWXGHQWVDYHUDJHPDUNVLQ(QJOLVKZHUHDQGDYHUDJHPDUNV LQ0DWKVZHUH$YHUDJHWRWDOPDUNVSHUVWXGHQWLQ(QJOLVK0DWKVDQG+LQGL DUH)LQGWKHDYHUDJHPDUNVRIWKHFODVVLQ+LQGL  

 

 

 &DQQRW EH

GHWHUPLQHG 6ROXWLRQ

7RWDOLQ(QJOLVK7RWDOLQ0DWKV 2YHUDOOWRWDO 7RWDOPDNHVLQ+LQGL  ±  $YHUDJHLQ+LQGL



'LUHFWLRQVIRU  4DQG4 5HIHUWRWKHGDWDEHORZDQGDQVZHUWKHTXHVWLRQV

WKDWIROORZ 7KUHHGLVWLQFWRQHGLJLWQXPEHUVDUHWDNHQDQGDOOWKHQXPEHUVWKDWFDQEH IRUPHGE\GLIIHUHQWFRPELQDWLRQVRIWKHVHSULPHVWDNHQDOODWDWLPHDUHOLVWHG 7KHGLIIHUHQFHEHWZHHQWKHODUJHVWDQGVPDOOHVWQXPEHUVIRUPHGLV7KH VXPRIGLJLWLVPRUHWKDQ 4XHVWLRQ

:KDWLVWKHDYHUDJHRIWKHYDULRXVQXPEHUVWKDWFDQEHIRXQG"  

 

 

 

6ROXWLRQV

QXPEHUVFDQEHIRUPHG $YHUDJH 

               

 











 





























     































4XHVWLRQ

7KHDJHVRISHUVRQVIURPDQ$3LQDVFHQGLQJRUGHURIPDJQLWXGH7KH \RXQJHVWWZRDQGWKHHOGHVWWZROHIWWKHJURXS:KDWZDVWKHFKDQJHLQWKH FKDQJHLQWKHDYHUDJHDJH" 7KHDYHUDJHLQFUHDVHV 7KHDYHUDJHGHFUHDVHV 7KHWZRDYHUDJHVDUHHTXDO FDQQRWEHGHWHUPLQHG 6ROXWLRQ

,I$3LVD±G «D±G DDG «DG DG WKHROGDYHUDJHDVZHOO DVWKHQHZDYHUDJHLVD 4XHVWLRQ

 $JURXSRIVWXGHQWVGHFLGHGWREX\DWDSHUHFRUGHULQWKHUDQJHRIWR UXSHHV%XWDWWKHODVWPRPHQWVWXGHQWVEDFNHGRXWRIWKHGHFLVLRQVRWKDWWKH UHPDLQLQJVWXGHQWVKDGWRSD\5HHDFKPRUHWKDQWKH\KDGSODQQHG:KDW ZDVWKHSULFHRIWKHWDSHUHFRUGHULIWKHVWXGHQWVSDLGHTXDOVKDUHV"  5V

 5V

 5V

 5V

6ROXWLRQ

3ULFH5V [ 1XPEHURIVWXGHQWVQ 7KHQWKHFRQWULEXWLRQPDGHE\HDFKVWXGHQWLQWKHWZRFDVHVDUH5V [ QDQG5V  [ QUHVSHFWLYHO\

*LYHQ [ Q±± [ Q! [ Q±Q 1RZ [  !Q±Q !QQ±!!Q!DQGQQ±!Q ! [  %XWµLL¶LVDQDWXUDOQXPEHUZHKDYHQ &RVWRIWKHWDSHUHFRUGHUQ



±Q5V+HQFH

4XHVWLRQ 

 $PLONPDQEX\VPLONWKDWLVSXUH+HWKHQGLOXWHVLWIXUWKHUE\DGGLQJZDWHU ,IKHDGGHGDOLWUHRIZDWHUWROLWUHVRIPLONERXJKWDWWKHUDWHRI5VSHUOLWUH WKHQZKDWLVKLVSURILWDVVXPLQJWKDWKHVHOOVDOOWKDWKHVRREWDLQVDWWKHUDWHRI 5VSHUOLWUH"  

 

 

 

6ROXWLRQ

%RXJKWDW5VVROGDW5V3URILW5V 3HUFHQWDJHRISURILW+HQFH

4XHVWLRQ

OLWUHVRIDFHUWDLQPL[WXUHRIDOFRKRODQGZDWHULVDW VWUHQJWK,WLVDGGHG ZLWKRQHOLWUHRIZDWHU7KHDOFRKROVWUHQJWKRIWKHQHZPL[WXUHLV   6ROXWLRQ

OLWUHVOLWUHVDOFRKROOLWUHRIZDWHULVDGGHG 1RZOLWUHVDOFRKROLQDWRWDORIOLWUHV$OFRKROVWUHQJWK

4XHVWLRQ

 $FXSRIPLONFRQWDLQVSDUWVSXUHPLONDQGSDUWZDWHU+RZPXFKRIWKH PL[WXUHPXVWEHZLWKGUDZQDQGVXEVWLWXWHGZLWKZDWHULQRUGHUWKDWWKHUHVXOWLQJ PL[WXUHPD\EHKDOIPLONDQGKDOIZDWHU" òó  6ROXWLRQ

/HWWKHUHEH [ SDUWVSXUHPLONDQG [ SDUWVZDWHURULJLQDOO\ /HWWKHSDUWRIWKHPL[WXUHWKDWLVZLWKGUDZQEH\ 1RZWKHPLONOHIW [ ±\ôôRI\LVPLON

:DWHUOHIW [ ±\ó 1RZ [\ô [ \ó\  [\ [ \\  [ \DQG\ [   [  \ [  \LVUGRIWKHPL[WXUH+HQFH 4XHVWLRQ

 $YHUDJHZHLJKWRIDVWXGHQWLQDFODVVLVNJ)RXUQHZVWXGHQWVDUHDGPLWWHG WRWKHFODVVZKRVHZHLJKWVDUHNJNJNJDQGNJUHVSHFWLYHO\ 1RZWKHLUDYHUDJHZHLJKWLVNJ)LQGWKHQXPEHURIVWXGHQWVLQFODVVLQWKH EHJLQQLQJ   

 



 

6ROXWLRQ

/HWWKHQXPEHURIVWXGHQWVRULJLQDOO\EHµQ¶,IWKHDYHUDJHZHLJKWLVWRWDO ZHLJKWRIDOOVWXGHQWVLVQ 1RZQQ QQQ

6ROYHG([DPSOHV 4XHVWLRQ 5VZDVGLYLGHGDPRQJ$UXQ%KDVNDUDQG&KHHQXLQDFHUWDLQUDWLR+DG HDFKRIWKHPUHFHLYHG5VOHVVWKDQWKHLUDFWXDOVKDUHWKHQIRUHYHU\SDUW$UXQ JRW&KHHQXZRXOGKDYHJRWSDUWVDQGIRUHYHU\SDUWV$UXQJRW%KDVNDU ZRXOGKDYHJRWSDUWV+RZPXFKZDV%KDVNDU¶VDFWXDOVKDUH"  5V 

 5V 

 5V 

 QRQH RI WKHVH

6ROXWLRQ 7RWDO5V :KHQHDFKVKDUHLVGHFUHDVHGE\5VWRWDOUHGXFWLRQî5V QHZWRWDOWRZKLFKWKHUDWLRVFRUUHVSRQGV±



*LYHQWKDW$&DQG$% $%& %î %¶VWRWDOVKDUH5V



4XHVWLRQ 7KHUDWLRRILQFRPHRI$QXMDQG%KDUDWDQGDOVRWKRVHRI%KDUDWDQG&KDPSDN DUHLQWKHUDWLRRI$WKLUGRI&KDPSDN¶VLQFRPHH[FHHGVKDOIRI$QXM¶V LQFRPHE\5V,IHDFKRIWKHPVSHQGWKHVDPHDPRXQWRIPRQH\WKHQWKHLU VDYLQJVDUHLQWKHUDWLRRI:KDWLVWKHLUFRPELQHGH[SHQGLWXUH"  5V 

 5V 

 5V 

 QRQH RI WKHVH

6ROXWLRQ  $%DQG%& $%& /HWLQFRPHVNNNJLYHQWKDWN±òN N



LQFRPHVDUH$N%N&N



/HWSEHWKHH[SHQGLWXUH

1RZ±S±S±SEHLQWKHUDWLRRI   S



  S



±S±S







  S



S  

WRWDOVSHQGLQJî



4XHVWLRQ ,QDFODVVWKHUHDUHVRPHER\VDQGJLUOV7KHUDWLRVRIJLUOVWRER\VZKHQ ER\VDUHWDNHQRXWDQGZKHQER\VDUHDGGHGWKHUHFLSURFDOVRIRQHDQRWKHULI WKHQXPEHURIJLUOVGRXEOHVWKHQWKHUDWLRRIJLUOVWRER\VEHFRPHV:KDWLV WKHVWUHQJWKRIWKHFODVV"

 

 

 

 

6ROXWLRQ *%±[\DVVXPH*%\[ *%±%* *%%± *% ± % ± 



*LYHQWKDW*%ô *%%

«««

«««

3XWLQ 

%  



%





%  



%  



%     

%\VROYLQJZHJHW%DQG*% VWUHQJWK

4XHVWLRQ ,QDIDUPZKLFKKDVRQO\FRFNVDQGEXOOVWRWDOFRXQWRIOHJVZDVOHVVWKDQ WLPHVWKHWRWDOFRXQWRIKHDGV+RZPDQ\OHJVDUHFRXQWHGLQWRWDO"  

 

 

G FDQQRW EH GHWHUPLQHG

6ROXWLRQ /HWWKHQXPEHURIFRFNV&ZKLFKKDVKHDGOHJV%XOOKDV%KHDG OHJV &%&%&%&% &&OHJV&%6LQFHZHFDQQRWNQRZWKHQXPEHURI



EXOOVZHFDQQRWILQGWKHYDOXH

4XHVWLRQ ,ISSRWWHUVZRUNLQJSKRXUVSHUGD\IRUSGD\VSURGXFHSSRWVWKHQKRZPDQ\ SRWVFDQEHPDGHE\TSRWWHUVZRUNLQJTKRXUVDGD\IRUTGD\V" ST

 TS

 ST

 ST

6ROXWLRQ 0'+ 



0 ' + 

:

:

 S  S  S



TTT

 S

:





:



 S  T  T  T  S  S  S



:



T S

4XHVWLRQ 7ZRQXPEHUVDUHVXFKWKDWWKHLUGLIIHUHQFHVWKHVXPDQGWKHLUSURGXFHDUHLQ WKHUDWLRRI)LQGWKHGLIIHUHQFHRIVTXDUHVRIWKHQXPEHUV    

 

6ROXWLRQ /HWWKHQXPEHUVEH[\ [±\[\[\ [\ [\



 

 

 



[\    

[\ [\



«« «««



[\[±\ \\\±\ \\ \ [\ [\±

4XHVWLRQ ,I5VLVWREHGLVWULEXWHGWRWKUHHPHQ$P\%XQW\DQG&DVWRULQVXFKD ZD\ZKHQ5V5VDQG5VDUHDGGHGWRWKHLUUHVSHFWLYHVKDUHVWKHUDWLR EHFRPHV:KDWLV%XQW\¶VVKDUH"  5V 

 5V 

 5V 

 5V 

6ROXWLRQ 7KHWRWDODPRXQW 6KDUHRI%XQW\LQ5Vî5V %XQW\¶VVKDUHEHIRUHDGGLQJ5V±5V

4XHVWLRQ ,QWZRDOOR\VFRSSHUDQGWLQDUHPL[HGLQWKHUDWLRDQG,INJRIWKH ILUVWDOOR\NJRIWKHVHFRQGDOOR\DQGVRPHSXUHFRSSHUDUHPHOWHGWRJHWKHU WKHQDQHZDOOR\LVIRUPHGLQZKLFKWKHUDWLRRIFRSSHUWRWLQLV)LQGWKH ZHLJKWRIWKHQHZDOOR\  NJ

 NJ

 NJ

  NJ

6ROXWLRQ NJRIILUVWDOOR\KDVNJRIFRSSHUDQGNJRIWLQNJ RIVHFRQGDOOR\KDVNJRIFRSSHUDQGNJRIWLQ

,IWKHZHLJKWRISXUHFRSSHUPHOWHGDORQJZLWKWKHVHWZRDOOR\VLV[WKHQ [ [:HLJKWRIQHZDOOR\ NJ

4XHVWLRQ ,Q'HOKLDQ$PEDVVDGRUFDUZLWKRQO\WKHGULYHULQVLGHPRYHVDWDVSHHGRI NPKU,WVVSHHGJHWVUHGXFHGE\DTXDQWLW\ZKLFKLVGLUHFWO\SURSRUWLRQDOWRWKH QXPEHURISDVVHQJHUVVHDWHGLQVLGH7KHVSHHGRIWKHFDUUHGXFHVE\NPKULI WKHUHDUHSDVVHQJHUV$PLQLPXPRIKRZPDQ\SDVVHQJHUVVKRXOGEHVHDWHG VXFKWKDWWKHFDUGRHVQRWPRYHDWDOO"  

 

 

 QRQH RI WKHVH

6ROXWLRQ ,I5LVWKHUHGXFWLRQLQVSHHGDQG1LVWKHQXPEHURISDVVHQJHUVWKHQ5 1 51.ZKHUH.LVWKHFRQVWDQWRISURSRUWLRQDOLW\*LYHQGDWD5ZKHQ1 KHQFH.DQG517KHHTXDWLRQIRUVSHHGLV6±5 6±1 &DUVKRXOGQRWPRYH611

4XHVWLRQ :KDWQXPEHUVKRXOGEHVXEWUDFWHGIURPHDFKRIVXFKWKDWWKH UHPDLQGHUDUHSURSRUWLRQDO"    

   

   

 QRQH RI WKHVH

6ROXWLRQ 6D\[LVWKHQXPEHU 

  [   [



  [   [



±[±[±[±[7U\LQJRXWRSWLRQVZHILQGQRQHRIWKHP



VDWLVI\EXW[VDWLVILHV

6ROYHG([DPSOHV

4XHVWLRQ  ,IWKHVXPRIWKH FXEHRIDQXPEHU [ DQGWKHFXEHRILWVUHFLSURFDOLVWLPHVWKH VXPRIWKHQXPEHUDQGLWVUHFLSURFDOILQG [  [          6ROXWLRQ [[[[ [[[[[[ [[[[ [[![[

+HQFH

4XHVWLRQ :KHQDVNHGWRZULWHDE [ \ DVDVXPRIWZRSHUIHFWVTXDUHVDVWXGHQW JDYHWKHIROORZLQJDQVZHUV D[ E\ E[ D\   D[ ±E\ E[ ±D\  D[ E\ E[ ±D\   D[ ±E\ E[ D\  6ROXWLRQ DE [ \ 

 D [ E\ E [ D\   D [ E\ DE[\ E [ D\ ±DE[\   D[ E\ E[ ±D\ 

 $OVR DE [ \ 

 D [ E\ ±DE[\ E [ D\ DE[\   D[ ±E\ E[ D\  7KXVDE [ \ D[ E\ E[ ±D\ RUD[ ±E\ E[ D\ 

4XHVWLRQ ,I

D



E



F

D  E E  F F D

  

  ILQGWKHYDOXHRI   

6ROXWLRQ

 D   E   F                D  E   E  F   F  D  

E D E



F EF



D F D



E D E



 ±

F



D

E F F D

 

4XHVWLRQ  7KHVXPRIWZRSRVLWLYHQXPEHUVLV DQGWKHVXPRIWKHLUUHFLSURFDOVLV)LQG  WKHQXPEHUV        DQG   DQG  DQG  DQG       

6ROXWLRQ  /HWWKHQXPEHUVEH [ DQG\   [ \       

  [ 

 



 

\   [  \

 

[ \   [\   [\  [\ 

[\



 



  [ \ ± [\  [ ±\       [  \       [  \       [  \     [ \     [  \     [  





 [



 



  

\ 



7KH QXPEHUV DUH  DQG

 [  \



 [ 

 [   

DQG\ 

 [ \ 

 

%RWKJLYHSRVLWLYHDQVZHUV

4XHVWLRQ )LQGWKHYDOXHRI.LI [ LVDIDFWRURI [ . [  ± [       

 

6ROXWLRQ  S [  [ . [  [   SO .  .+HQFH>@ .

4XHVWLRQ ,IDEDQGDEILQGWKHYDOXHRIDE         6ROXWLRQ :HZLOOXVHWKHLGHQWLW\

 

 [   \    [   \    [\ \   [  ,QWKLVFDVH [ D\ E

 D  E   D  E  DE D  E  D  E   D  E  DED  E              4XHVWLRQ 5HGXFH

           WRLWVVLPSOHVWIRUPJLYHQWKDWDQGDUHDOOHTXDO  

 

 

6ROXWLRQ   

  



     

         

 



    

                   

 

4XHVWLRQ I  [   N[    J  [    [    ,I  I  R J  J R I ILQG N         

 ,QGHWHUPLQDWH

6ROXWLRQ I  R J  I J  [   I  [      [   N    L JR I   J I  [   J N[     N[      LL L  LL  N[   N     N[       N    DQG N     4XHVWLRQ ,IWKHYDOXHRIDDWKHYDOXHRIDDLV  

6ROXWLRQ DD DD    DD DD D±D   D ± D    DD    1RZ D±DD±D±DDD [ D D±D±D±DD±D±D±D+HQFH

4XHVWLRQ ,II [  [ WKHQI^III [ `"  [  [     [  [   

 [  [     1RQH RI WKHVH

6ROXWLRQ I [  [ I [  [  [ [  I [  [  [  [  [  [  I [  [  [  [  [  [ +HQFH

6ROYHG([DPSOHV

4XHVWLRQ :KDWLVWKHQXPHULFDOGLIIHUHQFHEHWZHHQWKHSHULPHWHUDQGWKHDUHDRID UHFWDQJOHZKRVHVLGHVDUHWKHURRWVRIWKHHTXDWLRQ [   

 

 

   [     "

 

6ROXWLRQ 5RRWVRIWKHHTXDWLRQ  [ 

    [   

6TXDULQJ[±[±[

[±[[[ DUHDîSHULPHWHU

GLIIHUHQFH± 4XHVWLRQ

DQGDUHGLVWLQFWURRWVRIDTXDGUDWLFHTXDWLRQZKHUH DQG  ±)LQGWKHHTXDWLRQZKRVHURRWVDUH> @DQG>@ [ ± [ ±   

 [ [

[ ± [    

 [ [±

6ROXWLRQ ,WLVJLYHQWKDWDQG

ERWKDQGVDWLVI\WKHHTXDWLRQ[±[  DQG WKHQHZURRWVDUH DQG  VXP

        

SURGXFWRIWKHURRWV









        

WKHUHTXLUHGHTXDWLRQLV[±[

4XHVWLRQ  $VWDQGDUGTXDGUDWLFHTXDWLRQKDVDEDQGF)LQGWKHGLIIHUHQFH EHWZHHQWKHSURGXFWRIWKHURRWVDQGWKHVXPRIWKHURRWV  

 

 

 

6ROXWLRQ 7KHSURGXFWRIWKHURRWVFD 7KHVXPRIWKHURRWVED 7KXVWKHGLIIHUHQFH

4XHVWLRQ 7KHURRWVRIWKHTXDGUDWLFHTXDWLRQD[ E[FDUHLQWHJHUVDQGGLIIHUE\ 7KHUHVXOWREWDLQHGE\PXOWLSO\LQJWKHVXPRIWKHURRWVDQGWKHSURGXFWRIWKH URRWVLVDOZD\VGLYLVLEOHE\  

 

 

 

6ROXWLRQ OHWWKHURRWVEHDQG6XPRIURRWV    3URGXFWRIURRWV   

««

«««

î DQGWKLVLVGLYLVLEOHE\ VLQFHDQGDUHFRQVHFXWLYHQXPEHUVDQGWKHLUSURGXFWLV GLYLVLEOHE\

4XHVWLRQ [±[NDQG[±[±NKDYHDFRPPRQURRWIRUVRPHYDOXHRIN )LQGWKHFRPPRQURRW   ò

 

 

 

6ROXWLRQ /HWEHWKHFRPPRQURRW VKDOOVDWLVI\HDFKRIWKHHTXDWLRQV

N«««DQG N«««

±îJLYHVWKHHTXDWLRQ N

 N VXEVWLWXWLQJLQ  

RU 4XHVWLRQ )LQGWKHUDWLRRIWKH/&0DQGWKH+&)RIWKHURRWVRIWKHHTXDWLRQ[ ±[     

   

   

   

6ROXWLRQ (TXDWLRQLV[±[ 5RRWV

 

   



   



 



 



5RRWVDUH /&0

+&)

/&0  +&)   +&)  /&0   















/&0 +&)



    

 

4XHVWLRQ 7KHURRWVRIWKHHTXDWLRQ[ [DUH UDWLRQDODQGXQHTXDO

LUUDWLRQDODQGXQHTXDO

 UDWLRQDO DQG HTXDO

 UHDODQGHTXDO

6ROXWLRQ EDF6LQFH EDFWKHURRWVDUHUHDO DQGHTXDO+HQFH

4XHVWLRQ

 E[ P   KDVURRWVZKLFKDUHHTXDOLQPDJQLWXGHEXWRSSRVLWH D[  F P  

7KHHTXDWLRQ

[

LQVLJQ7KHUHIRUHP"  



DE



D E

D E DE



6ROXWLRQ  [ E[ PD[ ±F P  [ PE[ PD[ PE[ F P  [ P [ EPDPDEF P ,IURRWVDUHHTXDODQGRSSRVLWHWKDQEPDPDE EPDPDEP

D E D E

+HQFH >@

4XHVWLRQ 6ROYH [  [ ± [ ± [ ±ò  

6ROXWLRQ  [  [ ±[ ± [ ±ò[  [ ò [ ± [   [  [  [  [ ± [ ±  [  [  [ [   [  ; ± ; ± [  [  [ O   [ ² ;   ; +HQFH  $OWHUQDWLYHO\ 7KHYDOXHVFRXOGDOVREHFKHFNHGE\VXEVWLWXWLQJWKHYDOXHVIURPWKHFKRLFHV

4XHVWLRQ 7KHURRWVRIWKHHTXDWLRQ [ P [ ZLOOEHLQWKHUDWLRLIPHTXDOV   

6ROXWLRQ 6XPRIWKHURRWV P 3URGXFWRIWKHURRWV 

RU  1RZ

««L

 $OVRPPLL P P[P+HQFH

6ROYHGH[DPSOHV

4XHVWLRQ  $FDNHZHLJKLQJJLVFXWLQWRWKUHHXQHTXDOSLHFHV7KHODUJHVWSLHFHZDVWZLFHWKH ZHLJKWRIWKHPLGGOHVL]HGSLHFHDQGWKHVPDOOHVWSLHFHLVJPVOLJKWHUWKDQWKHODUJHVW SLHFH)LQGWKHZHLJKWRIWKHVPDOOHVWSLHFH  JPV

 JPV

 JPV

 JPV

6ROXWLRQ ,IWKHZHLJKWRIWKHVPDOOHVWSLHFHLV[JPWKHQWKHZHLJKWRIWKHODUJHVWSLHFHLV[

 [     JP7KHQ     

JPDQGWKHPLGGOHVL]HGSLHFHKDVDZHLJKWRI  [

 [

  

[JP

4XHVWLRQ /HW5J6±:KHQ65:KHQ65LVHTXDOWR  

 

 

 

6ROXWLRQ )URPWKHJLYHQFRQGLWLRQV J±

Jò 5ò±

4XHVWLRQ )RUWKHVLPXOWDQHRXVHTXDWLRQ[±\\±[  [   \  

 [   \  

 [   \  

 WKHUH LV QR VROXWLRQ

6ROXWLRQ :KHQWKHWZRJLYHQHTXDWLRQVDUHSORWWHGWKH\UHSUHVHQWWZRSDUDOOHOOLQHV+HQFHWKH\ QHYHUPHHW7KLVLQWXUQPHDQVWKDWWKHUHLVDLQILQLWHRIVHWRIDQVZHUVZKLFKZRXOGVDWLVI\ WKHHTXDWLRQV  $OJHEUDLFDOO\LID[E\FDQGD[E\FWKHQPXOWLSO\WKHILUVWHTXDWLRQ E\EWKHVHFRQGE\EDQGVXEWUDFWLQJZHKDYH

DE±DE[FE±FE

[ 

F E   F  E D  E   D  E

WKXVLIDE±DEDQGF F [LVXQGHILQHGDQGQRVROXWLRQH[LVWV

4XHVWLRQ )URPDJURXSRIER\VDQGJLUOVJLUOVOHDYH7KHUHDUHWKHQOHIWWZRER\VIRUHDFKJLUO  $IWHUWKLVER\VOHDYH7KHUHDUHWKHQJLUOVIRUHDFKER\7KHQXPEHURIJLUOVLQWKH EHJLQQLQJZDV  

 

 

 

6ROXWLRQ ,I%LVWKHRULJLQDOQXPEHURIER\VDQG*WKHRULJLQDOQXPEHURIJLUOVWKHQ % *  

 

%   *  



 

*

4XHVWLRQ ,I[\[\WKHQYDOXHRI[DQG\LV  [   \  

 [   \  

 [   \  

 [   \  

6ROXWLRQ [   \  

««««

[  \  

««««

DGGLQJDQG [\  [  \  

««««

VXEWUDFWLQJIURP\±[

\±[

««««

DGGLQJDQGZHJHW\ \  

«««

VXEWUDFWLQJ\LQ(TZHJHW[±  $OWHUQDWLYHO\RQHFDQSXWWKHYDOXHVLQWKHSDUHQWHTXDWLRQVDQGFKHFNZKLFKRIWKHSDLURI YDOXHVVDWLVI\WKHHTXDWLRQV

4XHVWLRQ )LYH\HDUVDJRWKHDJHRIDPRWKHUZDVIRXUWLPHVWKHDJHRIKHUGDXJKWHU)LIWHHQ\HDUV KHQFHWKHPRWKHU¶VDJHZLOOEHWZLFHWKHDJHRIWKHGDXJKWHU)LQGWKHSUHVHQWDJHRIWKH GDXJKWHU

  \HDUV    \HDUV  

  \HDUV  

  \HDUV

6ROXWLRQ /HWWKHSUHVHQWDJHRIWKHPRWKHUDQGWKHGDXJKWHUEH0DQG'UHVSHFWLYHO\WHQ 0 ±    ' ± 

«««

0     '  

«««

(T(TZHJHW  ±'±'\HDUV

4XHVWLRQ 7KHH[FHVVRIDWZR±GLJLWQXPEHURYHUWKHQXPEHUIRUPHGE\UHYHUVLQJLWVGLJLWVLV,I WKHVXPRILWVGLJLWVLVWLPHVWKHGLIIHUHQFHRIWKHGLJLWVILQGWKHQXPEHU  

 

 

 

6ROXWLRQ /HWWKHWZRGLJLWQXPEHUEHDE)URPWKHFRQGLWLRQRIWKHSUREOHPZHJHW DED±ED±ERUE±D$VD!EDED±EDE+HQFH DPXVWEHDPXOWLSOHRI$VDLVDVLQJOHGLJLWLQWHJHUD+HQFHE

4XHVWLRQ 7KHWRWDODJHRI$PDUDQG%KDZDQ\HDUVDJRDQGWKHWRWDORIWKHLUSUHVHQWDJHVDUHLQ WKHUDWLRRI,IWKHUDWLRRIWKHLUSUHVHQWDJHVLVILQGWKHSUHVHQWDJHLQ\HDUVRI  $PDU  

 

 

 

6ROXWLRQ /HWWKHSUHVHQWDJHRI$PDUDQG%KDZDQEH$DQG%WKHQ  $±%±$%$% +HQFH$ô

4XHVWLRQ ,QVWHDGRIPXOWLSO\LQJDQXPEHUZLWK$MD\PXOWLSOLHGLWZLWKLWVUHFLSURFDO$VDUHVXOWKH JRWDQDQVZHUZKLFKLVPRUHWKDQWKHH[SHFWHGUHVXOW)LQGWKHQXPEHU  

6ROXWLRQ

 

 

 

/HWWKHQXPEHUEH[$MD\ZDVH[SHFWHGWRILQG[+HIRXQG[7KHLUGLIIHUHQFHLV JLYHQE\  

[

 

[   

 

[  

[

4XHVWLRQ 7KHDPRXQWRIPRQH\ZLWK5DPHVKDQG5DMKDYHLVLQWKHUDWLRRI,I5DPHVKKDV5V PRUHWKDQ5DMILQGWKHWRWDODPRXQWZLWKERWK  5V 

 5V 

 5V 

 5V 

6ROXWLRQ /HWWKHDPRXQWRIPRQH\ZLWK5DPHVKDQG5DMEH[DQG[UHVSHFWLYHO\7KHQ [±[[[ WRWDODPRXQWZLWKERWK[5V

6ROYHGH[DPSOHV

4XHVWLRQ  $FDNHZHLJKLQJJLVFXWLQWRWKUHHXQHTXDOSLHFHV7KHODUJHVWSLHFHZDVWZLFHWKH ZHLJKWRIWKHPLGGOHVL]HGSLHFHDQGWKHVPDOOHVWSLHFHLVJPVOLJKWHUWKDQWKHODUJHVW SLHFH)LQGWKHZHLJKWRIWKHVPDOOHVWSLHFH  JPV

 JPV

 JPV

 JPV

6ROXWLRQ ,IWKHZHLJKWRIWKHVPDOOHVWSLHFHLV[JPWKHQWKHZHLJKWRIWKHODUJHVWSLHFHLV[

 [     JP7KHQ     

JPDQGWKHPLGGOHVL]HGSLHFHKDVDZHLJKWRI  [

 [

  

[JP

4XHVWLRQ /HW5J6±:KHQ65:KHQ65LVHTXDOWR  

 

 

 

6ROXWLRQ )URPWKHJLYHQFRQGLWLRQV J±

Jò 5ò±

4XHVWLRQ )RUWKHVLPXOWDQHRXVHTXDWLRQ[±\\±[  [   \  

 [   \  

 [   \  

 WKHUH LV QR VROXWLRQ

6ROXWLRQ :KHQWKHWZRJLYHQHTXDWLRQVDUHSORWWHGWKH\UHSUHVHQWWZRSDUDOOHOOLQHV+HQFHWKH\ QHYHUPHHW7KLVLQWXUQPHDQVWKDWWKHUHLVDLQILQLWHRIVHWRIDQVZHUVZKLFKZRXOGVDWLVI\ WKHHTXDWLRQV  $OJHEUDLFDOO\LID[E\FDQGD[E\FWKHQPXOWLSO\WKHILUVWHTXDWLRQ E\EWKHVHFRQGE\EDQGVXEWUDFWLQJZHKDYH

DE±DE[FE±FE

[ 

F E   F  E D  E   D  E

WKXVLIDE±DEDQGF F [LVXQGHILQHGDQGQRVROXWLRQH[LVWV

4XHVWLRQ )URPDJURXSRIER\VDQGJLUOVJLUOVOHDYH7KHUHDUHWKHQOHIWWZRER\VIRUHDFKJLUO  $IWHUWKLVER\VOHDYH7KHUHDUHWKHQJLUOVIRUHDFKER\7KHQXPEHURIJLUOVLQWKH EHJLQQLQJZDV  

 

 

 

6ROXWLRQ ,I%LVWKHRULJLQDOQXPEHURIER\VDQG*WKHRULJLQDOQXPEHURIJLUOVWKHQ % *  

 

%   *  



 

*

4XHVWLRQ ,I[\[\WKHQYDOXHRI[DQG\LV  [   \  

 [   \  

 [   \  

 [   \  

6ROXWLRQ [   \  

««««

[  \  

««««

DGGLQJDQG [\  [  \  

««««

VXEWUDFWLQJIURP\±[

\±[

««««

DGGLQJDQGZHJHW\ \  

«««

VXEWUDFWLQJ\LQ(TZHJHW[±  $OWHUQDWLYHO\RQHFDQSXWWKHYDOXHVLQWKHSDUHQWHTXDWLRQVDQGFKHFNZKLFKRIWKHSDLURI YDOXHVVDWLVI\WKHHTXDWLRQV

4XHVWLRQ )LYH\HDUVDJRWKHDJHRIDPRWKHUZDVIRXUWLPHVWKHDJHRIKHUGDXJKWHU)LIWHHQ\HDUV KHQFHWKHPRWKHU¶VDJHZLOOEHWZLFHWKHDJHRIWKHGDXJKWHU)LQGWKHSUHVHQWDJHRIWKH GDXJKWHU

  \HDUV    \HDUV  

  \HDUV  

  \HDUV

6ROXWLRQ /HWWKHSUHVHQWDJHRIWKHPRWKHUDQGWKHGDXJKWHUEH0DQG'UHVSHFWLYHO\WHQ 0 ±    ' ± 

«««

0     '  

«««

(T(TZHJHW  ±'±'\HDUV

4XHVWLRQ 7KHH[FHVVRIDWZR±GLJLWQXPEHURYHUWKHQXPEHUIRUPHGE\UHYHUVLQJLWVGLJLWVLV,I WKHVXPRILWVGLJLWVLVWLPHVWKHGLIIHUHQFHRIWKHGLJLWVILQGWKHQXPEHU  

 

 

 

6ROXWLRQ /HWWKHWZRGLJLWQXPEHUEHDE)URPWKHFRQGLWLRQRIWKHSUREOHPZHJHW DED±ED±ERUE±D$VD!EDED±EDE+HQFH DPXVWEHDPXOWLSOHRI$VDLVDVLQJOHGLJLWLQWHJHUD+HQFHE

4XHVWLRQ 7KHWRWDODJHRI$PDUDQG%KDZDQ\HDUVDJRDQGWKHWRWDORIWKHLUSUHVHQWDJHVDUHLQ WKHUDWLRRI,IWKHUDWLRRIWKHLUSUHVHQWDJHVLVILQGWKHSUHVHQWDJHLQ\HDUVRI  $PDU  

 

 

 

6ROXWLRQ /HWWKHSUHVHQWDJHRI$PDUDQG%KDZDQEH$DQG%WKHQ  $±%±$%$% +HQFH$ô

4XHVWLRQ ,QVWHDGRIPXOWLSO\LQJDQXPEHUZLWK$MD\PXOWLSOLHGLWZLWKLWVUHFLSURFDO$VDUHVXOWKH JRWDQDQVZHUZKLFKLVPRUHWKDQWKHH[SHFWHGUHVXOW)LQGWKHQXPEHU  

6ROXWLRQ

 

 

 

/HWWKHQXPEHUEH[$MD\ZDVH[SHFWHGWRILQG[+HIRXQG[7KHLUGLIIHUHQFHLV JLYHQE\  

[

 

[   

 

[  

[

4XHVWLRQ 7KHDPRXQWRIPRQH\ZLWK5DPHVKDQG5DMKDYHLVLQWKHUDWLRRI,I5DPHVKKDV5V PRUHWKDQ5DMILQGWKHWRWDODPRXQWZLWKERWK  5V 

 5V 

 5V 

 5V 

6ROXWLRQ /HWWKHDPRXQWRIPRQH\ZLWK5DPHVKDQG5DMEH[DQG[UHVSHFWLYHO\7KHQ [±[[[ WRWDODPRXQWZLWKERWK[5V

6ROYHG([DPSOHV

4XHVWLRQ ,ID [ F TEWKHQF \ D] G WKHQ  [\  T]



[ \



T

 [  \  T  ]

]

 [ ± \  T ± ]

6ROXWLRQ 6LQFHF D [ F D]\ 

 F T  G ]  \  T  D ]  [ 

]T \

 [\  T]

4XHVWLRQ  $UUDQJHWKHIROORZLQJQXPEHUVLQLQFUHDVLQJRUGHU

    ,     



    ,,     

 , ,, ,,, ,9



,,, 

 ,, ,,, , ,9

6ROXWLRQ 

,

            

,,

             

,,,



,9

 



+HQFH ,  ,,,  ,,  ,9

,9 

 , ,,, ,, ,9

 ,9 ,, ,,, ,

4XHVWLRQ ,I  [     [     WKHQ ILQG WKH YDOXH RI [

   

 ± 

 ± 

 ± 

6ROXWLRQ  [     [   

[±[  ±  [±± [ [ [± 4XHVWLRQ ,I

[ 

 P

[

 



 WKHQ ILQG WKH YDOXH RI P JLYHQ [  DQG[

 

 

 

6ROXWLRQ 

[ 

 P

[





îPîîî P 4XHVWLRQ  $UUDQJHLQDVFHQGLQJRUGHURIPDJQLWXGHV 

, 

,,

 ,,, ,, ,

 ,, ,,, ,



,,,





 , ,,, ,,

 , ,, ,,,

6ROXWLRQ 

,    

,,  

,,,   5DLVHHDFKWRWKHWKSRZHUEHLQJWKH/&0RI 

             

             

             WKHRUGHULV,,,,,, 4XHVWLRQ 7KHYDOXHRI  



  

 

  



 









  LV 



  

6ROXWLRQ

              *LYHQ                                

     

 

  

    

                        



 

4XHVWLRQ ,ID [ E\ F ] DQGEDF WKHQµ\ ¶HTXDOV



 [] 

 [] 



 [  ] 



 [  ] 

 []    [  ] 

6ROXWLRQ  [ 

\ 

  \ 

  [ 

] 

/HWD E F  N WKHQD N  E N 



F  N    ] 

EDF  



 





  N  N  N  N [   ]   \









\

 

[





[



 ] 

[  ] 

\

 \ 

 ] 

[]    []   [  ] 

4XHVWLRQ ,I  D    DQG  E   WKHQWKHYDOXHRIDELV  





 



6ROXWLRQ *LYHQ  

D





 



  RU     D





            RU          



E

E



 E   E &RPELQLQJ          

D

(TXDWLQJWKHLQGLFHV D 

 E

RU DE 

 





 



 []  ]  ]  [

4XHVWLRQ ,I«;WKHYDOXHRI[LV

 

 

 

 

6ROXWLRQ [ LH«[

[[  [[ [[ [RU[ [  



4XHVWLRQ ,I



[

   ILQG WKH YDOXH RI [

 

 

6ROXWLRQ

 



[



   [

 

[

 

 

6ROYHG([DPSOHV

4XHVWLRQ ,ILVUHSUHVHQWHGE\E\ E\DQGE\WKHQWKHYDOXHRIWKHIROORZLQJ H[SUHVVLRQLV«

              

 

 

 

6ROXWLRQ 7KHH[SUHVVLRQFDQEHZULWWHQDV^>^±`  @î`  ±

4XHVWLRQ ,IWKHVXPRIWKHILUVWµ Q¶QDWXUDOQXPEHUVLVILQGWKHVXPRIWKHFXEHVRIWKH ILUVWQQDWXUDOQXPEHUV

  

  

  

  

6ROXWLRQ 7KHVXPRIWKHILUVWQQDWXUDOQXPEHUVLV

Q

  Q



 



 Q  Q     7KHVXPRIWKHLUFXEHVLV          

4XHVWLRQ 0XOWLSO\E\DVXLWDEOHOHDVWQXPEHUWRPDNHLWDSHUIHFWVTXDUHDQGILQG LWVVTXDUHURRW



 

 

 

6ROXWLRQ

 îîî ,IWKLVLVPXOWLSOLHGE\LWEHFRPHVDSHUIHFWVTXDUH îîî 7KHVTXDUHURRWRIWKHDERYHH[SUHVVLRQLVîîî

4XHVWLRQ :KDWPXVWEHDGGHGWRWKHSURGXFW D±DDDWRPDNHLWD SHUIHFWVTXDUH"

 

 

 

 

6ROXWLRQ D±DDDDD±DD  S SZKHUH SDD  S S ,IEHDGGHGWKHQ S SEHFRPHV S

4XHVWLRQ :KDWLVWKHORZHVWQXPEHUZKLFKZKHQGLYLGHGVHSDUDWHO\E\DQG ZLOOLQHDFKFDVHOHDYHDVWKHUHPDLQGHU"

 

 

 

 

6ROXWLRQ 

   /&0îîîî  6LQFHLWOHDYHVDVUHPDLQGHUWKHUHTXLUHGORZHVWQXPEHULV

4XHVWLRQ )LQGWKHSURGXFWî

 

 

 

 

6ROXWLRQ î±  ± ±  

4XHVWLRQ :KLFKLVFORVHVWWR" 





  

     







    



6ROXWLRQ    

 

  QHDUO\

 

  EHFDXVH



   

   

   

 

  EHFDXVH!

LVFORVHVWWR 4XHVWLRQ ,IDEF G DUHIRXUFRQVHFXWLYHQDWXUDOQXPEHUVZKDWPXVWEHDGGHGWRWKH SURGXFWDEFG WRPDNHLWDSHUIHFWVTXDUH"

 

 

 

 

6ROXWLRQ /HWWKHQXPEHUVEHDDDDQGD 3URGXFWDDDDDDDD  DDDD  S SSXWWLQJ SDD  S S 7RPDNHWKLVDSHUIHFWVTXDUHZHPXVWDGG

4XHVWLRQ 7KHSURGXFWRIDWZRGLJLWQXPEHUE\WKHQXPEHURIWKHVDPHGLJLWVZULWWHQLQWKH UHYHUVHRUGHULV)LQGWKHVXPRIWKHVTXDUHVRIWKHWZRGLJLWV

 

 

 

 

6ROXWLRQ 7KHWZRGLJLWQXPEHUPXVWEHHLWKHURURURUIRUWKHQRQO\WKHXQLW GLJLWLQWKHSURGXFWZLOOEH]HUR ZLOODJUHHDVî

UHTXLUHGVXP 4XHVWLRQ  $QXPEHUZKHQGLYLGHGE\DFHUWDLQGLYLVRUOHDYHVDUHPDLQGHUZKHQWZLFH WKHVDPHQXPEHULVGLYLGHGE\WKHVDPHGLYLVRUWKHUHPDLQGHULV7KHQWKH GLYLVRULV«

 

 

 



6ROXWLRQ /HWWKHGLYLGHQGEH1DQGWKHGLYLVRU [ 7KHUHPDLQGHULV

1T[  [  1T[  :KHQT[ LVGLYLGHGE\ [ ZHJHWWKHTXRWLHQW [ DQXPEHUZKLFKPXVW FRQWDLQ [ RQFHDQG  [ 

[ 

6ROYHG([DPSOHV 4XHVWLRQ 7KHQXPEHU [ DQG\ KDYHSULPHIDFWRUVDVDQGRQO\DQGWKHLUUDWLR

 \

LV

 [

:KLFKRIWKHVWDWHPHQWVEHORZLVWUXH"  7KH +&) LV 

 7KH /&0 LV  [

 7KH +&) LV [

 7KH +&) LV \

6ROXWLRQ 6LQFH [ LVDIDFWRURI\ +&)RIWKHWZRQXPEHUVLV [ DQG/&0LV\ 

4XHVWLRQ )RXUEHOOVDUHKHDUGDWLQWHUYDOVRIVHFRQGVUHVSHFWLYHO\+RZPDQ\ WLPHVFDQWKH\EHKHDUGZLWKLQPLQXWHVRIWLPHH[FOXGLQJWKHLQLWLDORQH"  

 

 

 

6ROXWLRQ /&0RIDQGLV 7KHUHIRUHDWWKHHQGRIHYHU\VHFRQGVWKHEHOOVZLOOEHKHDUG VLPXOWDQHRXVO\ PLQXWHVVHFRQGVWKHUHIRUHWKHEHOOVZLOOEHKHDUGVLPXOWDQHRXVO\ WLPHV

4XHVWLRQ 7KH/&0RIWZRQXPEHUVLVDQGWKHLU+&)LV,IWKHVXPRIWKHWZR QXPEHUVLVWKHQXPEHUVDUH

  

  

  

  

6ROXWLRQ  /HWRQHRIWKHQXPEHUVEH [ 7KHRWKHUQXPEHULV± [ 3URGXFWRIQXPEHUVWKHLU/&0î+&) LH [ ± [ î  [  [ 

 

 [ ± [ 

 [ ± [ ±JLYLQJ [ RU



7KHUHIRUHWKHQXPEHUVDUHDQG

4XHVWLRQ :KDWLVWKHJUHDWHVWQXPEHURIWKUHHGLJLWVZKLFKZKHQGLYLGHGE\DQG OHDYHWKHVDPHUHPDLQGHU"

 

 

 

 

6ROXWLRQ 7KH/&0RILV7KHJUHDWHVWPXOWLSOHRIDQGRIWKUHHGLJLWVLV LH0XOWLSOHRI 0



7KHUHTXLUHGQXPEHULV

4XHVWLRQ )LQGWKHJUHDWHVWQXPEHUWKDWZLOOGLYLGHVRDVWROHDYH UHPDLQGHUVUHVSHFWLYHO\

 

 

 

6ROXWLRQ )LQGWKH+&)RI ±±DQG±

 

LHDQG 7KH+&)LV

4XHVWLRQ )LQGWKHJUHDWHVWQXPEHUZKLFKZLOOGLYLGHDQGOHDYLQJWKHVDPH UHPDLQGHULQHDFKFDVH

 

 

 

 

6ROXWLRQ 6LQFHWKHUHPDLQGHUVDUHWKHVDPHWKHUHTXLUHGGLYLVRUZLOOH[DFWO\GLYLGHWKH GLIIHUHQFHRIDQ\WZRRIWKHJLYHQWKUHHQXPEHUV 7KHQXPEHULVWKH+&)RI±DQG±ZKLFKFDQEH



FRPSXWHGDV

4XHVWLRQ :ULWHWKHIUDFWLRQ 

 



 

 

LQLWVVLPSOHVWIRUP 

 



 

6ROXWLRQ :KHUH\RXFDQQRWJLYHWKHIDFWRUFRPPRQWRWKHQXPHUDWRUDQGWKH GHQRPLQDWRUILQGWKHLU+&) +&)RIDQGLV7KHIUDFWLRQLV

  

 

 



4XHVWLRQ 7KH/&0DQG+&)RIWZRQXPEHUVDUHUHVSHFWLYHO\DQG7KH GLIIHUHQFHRIWKHQXPEHUVLV

 

 

 

 

6ROXWLRQ  îîîDQG îî ,I+DDQG+EDUHWKHQXPEHUVZLWK+&)DV+WKHQ/&0+DE +DE 

 

+



JLYLQJDERUî

7DNLQJDDVDQGEDVWKHQXPEHUVDUHDQG7KHLUGLIIHUHQFHLV 

4XHVWLRQ 7ZRQXPEHUVDUHVXFKWKDWWKH+&)RIWKHQXPEHUVIRUPHGE\VTXDULQJWKHVXP RIWKHWZRDQGWKHGLIIHUHQFHEHWZHHQWKHLUVTXDUHVLV$OVRWKH/&0RIWKHVH QHZQXPEHUVLVWLPHVWKH+&))LQGWKHVHQXPEHUV

  

  

  

  

6ROXWLRQ /HWWKHQXPEHUVEH [ DQG\ 7KHQHZQXPEHUVDUH [ \ DQG [  ±\  7KH+&)LV [ \ DQG/&0LV [ \  [ \  )URPWKHTXHVWLRQ [ \ DQG [ \  [ \ ]   [ \ 



4XHVWLRQ ,ILQDVHWRIWKUHHQXPEHUVRQHQXPEHULVDOVRWKH+&)RIDOOWKHWKUHH QXPEHUVWKHQ«  2QHRIWKHRWKHUWZRQXPEHUVZLOODOVREHWKH/&0RIDOOWKHWKUHH )RUWKHRWKHUWZRQXPEHUVRQHZLOOEHWKH+&) 7KH/&0RIWKHWKUHHQXPEHUVZLOOEHWKH/&0RIWKHRWKHUWZR 1RQHRIWKHDERYH 

6ROXWLRQ /HWWKHWKUHHQXPEHUVEHDEDQGF  D+&)ENDF ODZKHUHN O DUHLQWHJHUV /&0RIDEDQGF/&0RIN DQGO  D/&0RIEDQGF 



Solved Examples

Question 1  x and y are positive integers and 6 is a factor of  xy . Which of the following must

always be true? (1) 6 is a factor of  x but only of y  (2) 6 is a factor of both  x and y  (3) 2 is a factor of  x and y 

(4) 3 is a factor of at least one of  x and y 

Solution: (4)  An actual example will be easier to grasp the situation. If  xy is 48, 48 × 1 or 16 × 3 or 12 × 4 or 24 × 2, clearly one of the two numbers in each must have 3 as a factor. The other three choices are not applicable.

Question 2 If it were possible to mark on the number line the points exactly midway between (1) (3)

1 60

and

1 12

1

1 30

(4)

1 24

1 40

Solution: (1) The required number =

1§ 1

60

and

will represent the number 

(2)

20

1

1 · 1 1 1  ¸ u  ¨ 2 © 60 12 ¹ 2 10 20

1 12

, the point

Question 3 The number 88179 is not divisible by: (1) 7

(2) 17

(3) 13

(4) 11

Solution: (4) The divisibility rule for 11 is the easiest to try. Trying it out first, we get our  answer.

Question 4  A 2-digit number is divisible by 4 but not by 8. When the digits are interchanged, the number is also divisible by 8. The absolute value of the difference between such a number and its reverse is: (1) 24 (2) 36 (3) 40 (4) 32

Solution: (2) Both the digits have to be even. All numbers between 10 and 99 divisible by 4 but not by 8 should be odd multiples of 4. The possible numbers are 28, 44, 68 and 84. (No need to consider 20 and 60 because when the digits of these numbers are interchanged, they form a single digit number.) When reversed, the numbers are 82, 44, 86 and 48. Only 48 is divisible by 8. Hence the difference is 84 – 48 = 36

Question 5  ‘xyz ’ denotes a 3-digit number. If ‘ x ’ and ‘y ’ are interchanged, the value of the

number decreases by 90. How many possible values exist for  ‘ x ’ and ‘ y ’? (1) 10 (2) 9 (3) Depends on value of ‘y’ (4) None of these

Solution: (2) Take 452 for example. If numbers are swapped it becomes 542. Take 361 if swapped it becomes 631. It is obvious that  x , y must be consecutive numbers for difference to be 90. Since the value of numbers decreases, y < x . Therefore possible pairs 10, 21, 32…98 which are 9 pairs.  Alternatively  100 x + 10y + z –(100y + 10 x + z )= 90



90 x – 90y = 90

? x – y = 1  Also, x > 0 and x and y both are less then 10. There are 9 such pairs of ( x , y ) satisfying equation (i).

Question 6  A shopkeeper is very particular that the amount for which he buys and sells goods should always include the digit ‘9’ in it. Moreover, the digits should not add up to 13 or a multiple of 13. If the lowest price that he can buy an item at is Rs 400 and the highest price he can sell it for is Rs 899, the maximum profit possible is

(1) 499 (2) 498 (3) 489 (4) 479

Solution: (4) Take the lowest number and add 9, i.e., 400 + 9 = 409 But the digits of this number add up to 13. So the number should be 419. Similarly, 899 adds up to 26 which is divisible by 13.

? the number is 898. Maximum profit = 898 – 419 = 479.

Question 7 In the following series each term after the first two is a sum of previous two term 1, 1, 2, 3 … What is the ratio of the number of even terms to the number of odd ones in the first 90 terms?

(1) 2 :7

(2) 1 :3

(3) 1 : 2

(4) 1 : 1

Solution: (3) We expand the series 1, 1, 2, 3, 5, 13, 21, 34, 55… We find that we have two odds followed by an even, and then again two odds and so on. First two being odds, their sum is even, which when added to its previous odd number gives an odd number. This odd number added to its previous even number gives another odd. This last odd number, when added to the number before it that is odd, gives an even number and thus the cycle goes on as OOEOOEOOE…O being odd and E being even

?EVEN : ODD = 1 : 2 Question 8 a, b, c , d and e are five consecutive given numbers. What is the maximum power 

of 2 by which the product of the above numbers be necessarily divisible?

(1) 6

(2) 7

(3) 8

(4) 9

Solution: (3) a

,

b

,

c 

,

d 

2 2 2 2

and

e

2

are five consecutive integers. At least, one of the five is divisible by 4, one by 2 and the rest are odd. Therefore product should be divisible by 2 to the power of (5 + 2 + 1).

Question 9 For a 3-digit number of which one digit is 6, the sum of the number and its mirror  image is a 3-digit number divisible by 111. What is this sum?

(1) 666

(2) 777

(3) 888

(4) 999

Solution: (3) Note that in the sum, the middle digit is added to itself and the extreme digits are added to each other. Also note that the sum has all digits same. Now, 6 cannot be the middle digit as 6 + 6 = 12 and 222 is not a choice. Notice, however, that the middle digit in the sum has to be an even number. So the viable choices are 666 and 888. But as 6 is one of the extreme digits, the sum cannot be 666. A little inspection will reveal that the number is 642 or 246.

Question 10 There is one number which is formed by writing one digit 6 times (e.g. 111111, 444444 etc.) Such a number is always divisible by

(1) 7, 33 and 13

(2) 7and 33 only

(3)13 and 33 only

(4) 7 and 13 only

Solution: (1)  A six-digit number with all identical digits, e.g., 888888 = 888 u 1001. 1001 is divisible by 7, 11 and 13. Also, the three-digit number with identical digits will be divisible by 3. Hence, the number will be divisible by 7, 13 and 33.