Education

Telugu Grammer( తెలుగు వ్యాకరణం )

మన మనస్సు లోని భావాలను , అనుభూతులను పైకి చెప్పడానికి  మాతృభాష ఎంతో ఉపయోగం , అందుకే మాతృ భాష తల్లి లాంటిది అంటారు. మనం మన మాతృ భాషని గౌరవించాలి .

గ్రహణ సామర్థ్యం పెరగడానికి మాతృ భాష  లో విద్యా బోధన ఎంతగానో ఉపయోగ పడుతుంది.  మాతృ భాష లో బోధించడం వలన విద్యార్థుల్లో సృజనాత్మకత పెరుగుతుంది.


భాషా భాగాలు :

భాషకు ప్రాణం భావ ప్రసరణ . ఈ భావ ప్రసరణ ఒకరి నుండి మరొకరికి చేరాలి . ఇలా చేరడానికి కొన్ని పదాలు వాక్యాలు అవసరం. ఇలా వాక్యం లోని ఉపయోగాన్ని బట్టి భాషకు ఐదు ప్రధాన భాగాలుగా విభజించారు.

అవి :− 1. నామవాచకం   2. సర్వనామం  3. విశేషణం  4. క్రియ  5. అవ్యయము

  1. నామవాచకం : నామము అనగా పేరు.ఒక వ్యక్తిని గాని, వస్తువుని గాని ,గుణమును గాని, జాతిని గాని తెలుపును.

ఉదా :- ధర్మరాజు , హైదరాబాద్ , బంతిపువ్వు , ఆవు మొ|| నవి .

  1. సర్వనామము : నామ వాచకాలకు బదులుగా వాడే వాటిని “ సర్వనామాలు “ అంటారు. “సర్వ” అనగా సమస్తము.

ఉదా :- అది, ఇది, అతడు , ఆమె ,అన్ని ,కొన్ని మొ|| నవి.

  1.  విశేషణం  : నామవాచకము మరియు సర్వనామముల యొక్క గుణమును తెలియజేయునది .

ఉదా :- మంచి , చెడు , లావు , పొట్టి , పొడుగు  , ఎత్తు  మొ|| నవి.

  1. క్రియ : పనులను, స్తితిగతులను తెలియజేయునది .

ఉదా :- రాస్తున్నాడు , వెళ్తున్నాడు , పాడుతున్నాడు  మొ|| నవి.

  1. అవ్యయము : వ్యయము అనగా నశించేది, అవయము అనగా నశించనిది .
  • లింగ, వచన, విభక్తుల ప్రసక్తిగాని, వచన ఆకాంక్ష లేని వాటిని అవ్యయములు అంటారు .

ఉదా :-  అక్కడ, ఇక్కడ, ఆహా , భళా  మొ|| నవి.

సంధులు 

  • వ్యాకరణ భాషలో రెండు  స్వరాల కలయికను  సంధి  అంటారు .
  • రెండు అచ్చుల మధ్య  జరిగే మార్పును  సంధి కార్యం  అంటారు.
  • సంధి జరిగే మొదటి పదo చివరి అక్షరం లోని అచ్చును ‘పూర్వ పదం’ అంటారు .
  • సంధి జరిగే రెండవ పదం మొదటి అక్షరం లోని అచ్చును ‘పర పదం ‘ అంటారు .

ఉదా :- రామ + అయ్యా:  ‘ రామ’  లోని     ‘మ’  లో  ‘అ’ పూర్వ పదం  ‘అయ్యా’  లోని  ‘అ ‘ పర పదం .

  1. అత్వ సంధి(అకార సంధి );  అత్తునకు సంధి  బహుళంగా వస్తుంది .

ఉదా :-1) మేఅల్లుడు = మేన + అల్లుడు                     3) లేకేమి  = లేక + ఏమి

2) రాకుంటే  = రాక + ఉంటే                          4) పోవుటెట్లు =  పోవుట  + ఎట్లు

  1. ఇత్వ సంధి ( ఇకార సంధి): ఏమ్యాదులకు ఇత్తునకు సంధి .

ఉదా :- 1) ఏమంటివి = ఏమి + అంటివి                       3) పైకెత్తినారు  = పైకి + ఎత్తినారు

2) వచ్చిరిపుడు  = వచ్చిరి + ఇపుడు                4) మనిషన్నవాడు  = మనిషి + అన్నవాడు

  1. ఉత్వ సంధి ( ఉకార సంధి ): ఉత్తునకు అచ్చు పరమైనపుడు సంధి నిత్యంగా వస్తుంది .

ఉదా :- 1) రాముడతడు = రాముడు + అతడు            3) మనమున్నాము = మనము + ఉన్నాము

 2) అతడెక్కడ = అతడు + ఎక్కడ                    

  4) మనసైన = మనసు + ఐన 

  1. యదగామ సంధి : అంది లేని చోట అచ్చుల మద్య ‘య్’ వచ్చి  చేరడాన్ని “యడాగమం” అంటారు .

ఉదా :- 1) మాయమ్మ = మా + అమ్మ                             3) హరియతడు = హరి + అతడు

2) మాయిల్లు = మా + ఇల్లు

  1. ఆమ్రేడిత సంధి : అచ్చునకు ఆమ్రేడితం పరమైతే సంధి తరచుగా వస్తుంది .

ఉదా :- 1)  ఆహాహా = ఆహా + ఆహా                                  3)  ఔరౌర = ఔర  + ఔర

2) అరెరే  = అరె + అరె                                           4)  ఏమిటేమిటి = ఏమిటి + ఏమిటి

  1. గసడదవాదేశ సంధి : ప్రథమ మీది పరుషాలకు గ, స ,డ , ద ,వ   లు  బహుళంగా వస్తాయి .

ఉదా :-  1)  కొలువుసేసి = కొలువు + చేసి                   3) కూరగాయలు = కూర + కాయ

2) పాలువోయక = ఆలు + పోయక                4) తల్లిదండ్రులు  = తల్లి + తండ్రి

  1. త్రిక సంధి : త్రికము మీది అసంయుక్త హల్లునకు దిత్వం బహుళంగా వస్తుంది .
ఆ , ఈ , ఏ  లు త్రికంఅనబడుతాయి

ద్విరుక్తమైన హల్లు పరమైనపుడు, అచ్చికమైన దీర్ఘానికి  హ్రస్వం వస్తుంది.

ఉదా :-  1)  ఇక్కాలము  = ఈ + కాలము                       3)  అక్కోమరుండు =  ఆ + కొమరుండు

2) ఎవ్వాడు = ఏ + వాడు                                    4) అచ్చోట = ఆ + చోట

  1. రుగాగమ సంధి : పేదాది శబ్దాలకు  ‘ఆల‘ శబ్దo పరమైతే కర్మదారాయం లో రుగాగం వస్తుంది .

ఉదా :- 1) మనుమరాలు = మనుమా + ఆలు                3) ధీరురాలు  = దీరు + ఆలు

2) పేదరాలు = పేద + ఆలు                                     4)  బాలెంతరాలు = బాలెంత + ఆలు

5 ) ముద్దరాలు = ముద్ద +ఆలు                               6) జవరాలు = జావా + ఆలు

9 . సవర్నదీర్గ సంధి: అ, ఇ , ఉ,  ఋ లకు అవే అచ్చులు పరమైతే వాని దీర్గాలు ఎకాదేశంగా వస్తాయి .

ఉదా :-    1)  రామానుజుడు = రామ + అనుజుడు    2 )  రామాలయం = రామ + ఆలయం                                                                                3)  భానూఉదయం  = భాను + ఉదయం                  4)  కవీంద్రుడు = కవి + ఇంద్రుడు                  5 ) పితౄణం = పితృ +ఋణం        6) వదూపేతుడు  = వధు + ఉపేతుడు

10 . గుణసంధి :  ఇ , ఉ , ఋ  పరమైతే ఏ, ఓ , ఆర్  లు క్రమంగా ఎకాదేసంగా వస్తాయి .

ఉదా :-    1)  రాజేంద్రుడు  = రాజ  + ఇంద్రుడు          2 )  పరోపకారం  = పర  + ఉపకారం

3)  రాజర్షి   = రాజ  + ఋషి                                     4)   మహోన్నతి = మహా  + ఉన్నతి

11.యణాదేశ సంధి: ఇ , ఉ, ఋ లకు అసవర్ణ అచ్చులు పరమైతే య , వ ,ర  లు వస్తాయి .

ఉదా :-    1)  అత్యవసరం  = అతి  + అవసరం          2 )  ప్రత్యేకం  = ప్రతి  + ఏకం                       3)  అణ్వస్త్రం   = అణు  + అస్త్రం                                                4)  పిత్రార్జితం  = పితృ  + ఆర్జితం       5 ) పితౄణం = పితృ +ఋణం

12 . వృద్ధి సంధి :  అకారినికి ఏ , ఐ  లు పరమైతే ‘ఐ ‘ కారము , ఓ , ఔ లు పరమైతే  ‘ఔ’ కారము వస్తాయి.

ఉదా :-    1)  వసుధైక  = వసుధ   + ఏక           2 )  సమైక్యం   = సమ   + ఐక్యం

3)  వనౌసది    = వన   + ఔసది                                                  4)  పిత్రార్జితం  = పితృ  + ఆర్జితం

13 . అనునాసిక సంధి :  వర్గ ప్రతమాక్షరాలకు ‘న’ గాని , ‘మ’ గాని ప్రమైతే అనునాసికాలు.

 

ఉదా :-    1)  వాజ్మయం   = వాక్   + మయం           2 )  జగన్నాథుడు   = జగత్  + నాథుడు

3)  అణ్వస్త్రం   = అణు  + అస్త్రం               4)  పిత్రార్జితం  = పితృ  + ఆర్జితం       5 ) తన్మయం  = తత్  + మయం

 సమాసాలు 

  • సమాసం : వేరు వేరు అర్థాలు కల రెండు పదాలు కలిసి , ఏకంగా ఏర్పడితే దాన్ని ‘సమాసం’ అంటారు .
  1. దంద్వ సమాసం: రెండు కాని,  అంతకంటే ఎక్కువ కాని నామవాచకాల మద్య ఏర్పడే సమాసాన్ని ‘దంద్వ సమాసం ‘ అంటారు.

ఉదా : – 1) అన్నదమ్ములు  – అన్న, తమ్ముడు    2) తల్లిదడ్రులు – తల్లి, తండ్రి    3) మంచిచెడులు – మంచి, చెడు                                      4 ) కష్టసుఖాలు – కష్ట , సుఖము

  1. ద్విగు సామాసం : సమాసంలో మొదటి పదంలో సంఖ్య గల సమాసాన్ని ‘ద్విగు’ సమాసం అంటారు.

ఉదా : – 1) నవరసాలు   – నవ సంఖ్య  గల రసాలు    2) రెండుజడలు  – రెండు సంఖ్య  గల జడలు     3) నాలుగు వేదాలు – సంఖ్య గల వేదాలు

   3. తత్పురుష సమాసం : విభక్తి ప్రత్యాలు విగ్రహ వాక్యంలో ఉపయోగించే సమాసాలు ‘ తత్పురుష  సమాసాలు .

విభక్తులు

  • ఒక వాక్యం లోని వేరు వేరు పదాలకు అన్వయం కలిగించు పదాలను “ విభక్తులు” అంటారు
విభక్తులు ప్రత్యయాలు
ప్రథమా విభక్తి డు – ము – వు – లు
ద్వితీయ విభక్తి నిన్ – నన్ – లన్ – కూర్చి – గురించి
తృతీయ విభక్తి చేతన్ – చెన్ –  తోడన్ – తోన్
చతుర్థి విభక్తి కొరకున్  – కై
పంచమ విభక్తి వలనన్ – కంటెన్ – పట్టి
షష్ఠివిభక్తి యొక్క – లోన్ – లోపలన్
సప్తమి విభక్తి అందున్ – నన్
సంబోధన ప్రథమా  విభక్తి ఓరి – ఓయి – ఓసి
   
సమాస పదం విగ్రహ వాక్యం సమాసం పేరు
మద్యాహ్నము అన్నము యొక్క మద్య ప్రథమా తత్పురుష
జటాధారి జడలను ధరించినవాడు ద్వితీయ  తత్పురుష
రాజ పూజితుడు రాజు చే పూజితుడు తృతీయ  తత్పురుష
       వంట కట్టెలు వంట కొరకు కట్టెలు చతుర్థి  తత్పురుష
              అగ్నిభయం                      అగ్ని వల్ల భయం పంచమ  తత్పురుష
భుజభలం భుజాల యొక్క భలం షష్ఠి తత్పురుష
పుర జనులు పురమునందు  జనులు సప్తమి తత్పురుష

విశేషణ పూర్వపద కర్మధారయ సమాసం : విశేషణం పూర్వపదంగా (మొదటి) ఉండే సమాసం .

ఉదా : – తెల్ల గుర్రం – తెల్లదైన  గుర్రం,  ఇస్టార్థములు – ఇష్టమైన అర్థములు

సంభావన  పూర్వపద కర్మధారయ సమాసం :  సమాసం లోని పూర్వపడం సంజ్ఞావాచాకంగా , ఉత్తరపదం జాతి వాచకంగా ఉంటుంది .

 

ఉదా :- కాశిక పట్టణం – కాశిక అను పేరు గల పట్టణం, తెలంగాణా రాష్ట్రము – తెలంగాణ అను పేరు గల రాష్ట్రం

నైతత్పురుష సమాసం :  వ్యతిరేఖ పదాన్ని ఇచ్చే పదం .

ఉదా :- అసత్యం – సత్యం కానిది , నిరాదారం – ఆదారం కానిది , అనుచితం – ఉచితం కానుది .


అలంకారాలు
  

అలంకారం : చెప్పదలచిన విషయాన్ని అందంగా మలిచేది.

  1. అంత్యాను ప్రాస అలంకారం:  ఒకే అక్షరం లేదా రెండు , మూడు అక్షరాలు వాక్యం చివర మాటి మాటికి వస్తే దాన్ని  అంత్యాను ప్రాస అలంకారం అంటారు .

ఉదా : – భాగవతమున భక్తి – భారతమున యుక్తి – రామ కథయే రక్తి  ఓ కూనలమ్మ .

  1. వృత్యానుప్రాస అలంకారం: ఒకటి గాని అంతకంటే ఎక్కువ గాని హల్లులు పలుమార్లు వచ్చునట్లు చెబితే  వృత్యానుప్రాస అవితుంది.

ఉదా : – వీరు పొమ్మను వారు  వారు పోగబెట్టు వారు

కాకి కోకికాదు దా

  1. చేకానుప్రాస అలంకారం: అర్థ భేదం తో కూడిన హల్లుల జంట వెంట వెంటనే వస్తే చేక్కనుప్రాస అనబడుతుంది .

ఉదా : – అ నాథ నాథ  నంద నంద న నీకు వందనం

నీకు వంద వందనాలు.

  1. లాటాను ప్రాస అలంకారం : అర్థంలో భేధం లేకపోయినా , తాత్పర్యంలో భేదం ఉండేటట్లు , ఒక పదం రెండు సార్లు ప్రయోగించబడితే లాతానుప్రాస అనబడుతుంది.

ఉదా : – కమలాక్షునకు అర్పించు కరములు కరములు

  1. యమకం : అచ్చులలో హల్లులలో మార్పు లేనట్టి అక్షరాల సమూహం అర్థ భేదంతో మళ్ళీ  ప్రయోగిన్చినట్లయితే యమకం అనబడుతుంది.

ఉదా : –  పురము నందు నంతిపురము 

  1. ముక్త పద గ్రస్తo : విడిచి పెట్టబడ్డ పద భాగాలను వ్యవదానం లేకుండా వెంటనే ప్రయోగించి చెబితే ముక్త ప్రదగ్రస్తం .

ఉదా :- సుదతీ సూదన మదనా

మదనా గ తురంగ పూర్ణ మణి మాయ సదనా

సదనా మయ గజ రాదనా .

  1. ఉపమాలంకారం : ఉపమేయానికి ఉపమానం తో చక్కని పోలిక వర్ణించబడిన యెడల ఉపమాన అలంకారం అనబడుతుంది .

ఉదా : – 1) చేనేత కార్మికులు ఎలుకల్ల మాడి పోతున్నారు    2) నీ కీర్తి హంష  లాగ ఆకాశ గంగలో మునుగుతుంది

  1. రూపకాలoకారం : ఉపమానానినికి , ఉపమేయానినికి భేదం లేనట్లు వర్ణించి చెబితే రూపకాలoకారం అంటారు.

ఉదా : – 1) సంసార సాగరాన్ని తరించడం మిక్కిలి కష్టం   2 ) మౌనికి తేనె పలుకులు అందరికి ఇష్టమే

  1. ఉత్ప్రేక్ష అలంకారం : ఉపమానానినికి ఉన్న ధర్మాలు  ఉపమేయంలో ఉండడం చేత , ఉపమేయాన్ని  ఉపమానo గా ఊహించి చెబితే  ఉత్ప్రేక్ష  అలంకారం అంటారు.

ఉదా : –  1) ఆ మేడలు ఆకాశాన్ని  ముద్దడుతున్నాయా అన్నట్లు ఉన్నాయి    2) ఈ వెన్నెల పాలవెళ్లి యో  అన్నట్లుంది .

  1. అతి శయోక్తి అలంకారం : ఒక వస్తువు గురించి కాని సందర్భాన్ని గురించి కాని ఉన్నదాని కంటే ఎక్కువ చేసి చెబితే అతి శయోక్తి అంటారు.

ఉదా : – మా నగరం లోని మేడలు ఆకాశాన్ని అంటుతున్నాయి

  1. శ్లేషాలo కారం : అనేకమైన అర్థాలు కల శబ్దాలను ఉపయోగించి చెబితే శ్లేష అనబడుతుంది

ఉదా : – రాజు కువలయానంద  కరుడు

రాజు = ప్రభువు , చంద్రుడు       కువలయం = భూమి , కలువ పూలు

  1. స్వభావోక్తి అలంకారం :  జాతి గుణం క్రియాదు లలో ఉన్నది ఉన్నట్లు చెప్పడం

ఉదా : – చెట్ల ఆకులు గాలికి కదులుతున్నాయి

ఛందస్సు

  • పద్య లక్షణాన్ని తెలిపే శాస్త్రాన్ని ఛందస్ శాస్త్రం అంటారు .
  • ఒక మాత్ర కాలం లో ఉచ్చరించబడేది లఘువు (I )
  • రెండు మాత్రల  కాలం లో ఉచ్చరించబడేది  గురువు ( U  )

తెలుగు ఛందస్సు

             య  గణం   IUU                                            జ  గణం    IUI

                                 మ  గణం  UUU                                           భ  గణం    UII

 త   గణం   UUI                                             న  గణం III

   ర   గణం    UIU                                              స  గణం  IIU

 

పద్యం పేరు గణాలు యతి స్తానం అక్షరాల సంఖ్య
ఉత్పల మాల భ, ర, న, భ, భ, ర, వ 10 20
చంపక మాల న, జ, భ, జ, జ, జ, ర 11 21
శార్దూలం మ, స, జ, స, త, త, గ 13 19
మత్తేభం స, భ, ర, న, మ, య, వ 14 20

వాఖ్య నిర్మాణము – రకాలు 

వాక్యాలు మూడు రకాలు : 1 ) సామాన్య వాక్యము 2) సంశ్లిష్ట వాక్యము       3) సంయుక్త వాక్యము .

  • సామాన్య వాక్యము :- క్రియ ఉన్నా  లేకున్నా ఒకే ఒక్క భావాన్ని ప్రకటించే వాక్యాలను సామాన్య వాక్యాలు అంటారు .

ఉదా : (i ) సీత బజారుకు వెళ్ళింది .    (ii) పాము కాటేసింది   (iii ) మురళి మంచి బాలుడు

  • సంశ్లిష్ట వాక్యము:- ఒక సమాపక క్రియ , ఒకటి గాని అంతకన్నా ఎక్కువ గాని అసామాపక క్రియలు ఉంటే  ఆ వాక్యాన్ని    సంశ్లిష్ట వాక్యము అంటారు .

ఉదా :- (i )  రాము అన్నము తిని , పడుకున్నాడు         (ii )  సీత బజారుకు వెళ్లి , సరుకులు కొన్నది

  • సంయుక్త వాక్యము:- సమ ప్రాధాన్యం కల వాక్యాలను కలపడం వల్ల ఏర్పడే వాక్యాo ను సంయుక్త వాక్యము అంటారు.

ఉదా :-  (i ) అతడు నటుడు, రచయిత    (ii ) రాము మరియు సిత హైదరాబాద్ వెళ్లారు    (iii ) సీత చదువుతుంది ,

కర్తరి – కర్మణి వాఖ్యాలు 

  • కర్తరి వాక్యము :- ఒక వాక్యంలో కర్తకు ప్రాధాన్యం ఇచ్చి , కర్మకు ద్వితీయ విభక్తి (నిన్ , నున్ , లన్ , కూర్చి , గురించి ) చేరితే ఆ వాక్యాని కర్తరి వాక్యం అంటారు.
  • కర్మణి వాక్యము :-  ఒక వాక్యంలో క్రియకు ధాతువు చేరి  , కర్మకు తృతీయ  విభక్తి  ( చేతస్ , చేన్ , తోన్ , తోడన్ ) చేరితే ఆ వాక్యాని కర్తరి వాక్యం అంటారు.

ఉదా :-

  కర్తరి వాక్యము కర్మణి  వాక్యము
1  ప్రజలు శాంతిని కోరుతున్నారు ప్రజలచే శాంతి కోరబడుతుంది
2 మేం పెద్దలను గౌరవిస్తాము మాచే పెద్దలు గౌరవించ బడతారు
3 రాజు రైలును నడిపాడు రైలు రాజు చే నడపబడింది
4 భీముడు కొండలను పిండి చేసాడు కొండలు భీముని చే పిండి చేయబడెను
5 నా మీద రాళ్ళు విసురుతారు నా మీద రాళ్ళు విసరబడతాయి

ప్రత్యక్ష – పరోక్ష కథనాలు 
ప్రతక్ష కథనం : ఒకరు చెప్పిన విషయాన్ని ఉన్నది ఉన్నట్లుగా చెప్పడం . ఒకరు చెప్పిన విషయం  “     “  చిహ్నాల మద్య ఉందును.

ఉదా:-   i )  “ నేను రస జీవిని “  అని చాసో అన్నాడు    (ii ) అంబేత్కర్  “  నేను ఎవరిని యాచిన్చను  “ అని అన్నాడు

  • పరోక్ష కథనం :  ఒకరు చెప్పిన విషయాన్ని మన మాటల్లో చెప్పడం .    ఇందులో   “    “  చిహ్నాలు ఉండవు .

ఉదా:-  (i )  తాను రస  జీవినని చాసో అన్నాడు    (ii ) అబ్మేత్కర్ తాను ఎవరినీ  యాచిన్చనని  అన్నాడు

 

గమనిక : ప్రత్యక్ష కథనం  నుండి పరోక్షం లోకి మార్చునప్పుడు  జరుగు మార్పులు :

ప్రత్యక్షం పరోక్షం
నేను తాను
ఆయన అతను, వాడు
అది ఇది
నాకు తనకు
నా తన
నన్ను తనను
మేము తాము
మాకు తమకు
ఇది అది

 

 

1594987672651

Inter Second Year Accountancy Material

This note is designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate second-year Accountancy students.
These notes cover all the topics covered in the TS I.P.E second year Accountancy syllabus and concept to help you solve all the types of Inter Accountancy problems asked in the I.P.E and entrance examinations.


1. DEPRECIATION

DEPRECIATION: It is permanent, continuous and gradual shrinking in the book value of a fixed asset.

DEPLETION: It refers to the physical deterioration by the exhaustion of natural resources. (e.g. ore deposits in mines, oil wells, quarries, etc.,)

FIXED INSTAL MENT METHOD: A method under which the depreciation provided annually on the fixed asset remains the same throughout the life span of the asset.

NONCASH EXPENSE: Expenses which may be operational in nature but that do not affect the payment of cash (e.g. depreciation).

OBSOLESCENCE: diminution in the value of fixed assets due to new inventions, new improvement, change in fashions, change in customer’s tastes and preferences.

RESIDUAL VALUE (scrap value): The realisable value of fixed assets after the expiry of its estimated economic life.

SINKING FUND: A fund created for the repayment of along-term liability or the replacement of an asset at a set date in the future.

DEPRECIATIONFUND: A fund created for the replacement of an asset at a set date in the future.

WRITTEN DOWN VALUE: The value of a fixed asset after depreciation.

WRITTEN DOWN VALUE METHOD: A method under which depreciation is calculated at a fixed percentage on the original cost of the asset in the first year and on written down value in the subsequent year.

DOUBLE ENTRY SYSTEM: The accounting system of recording both the receiving (debit) and giving (credit) aspects of a business transaction is called a double-entry system.

SINGLE ENTRY SYSTEM: It is a mixture of a double-entry, single entry and no entry. It is an incomplete double-entry system.

STATEMENT AFFAIRS: To find out capital, a statement showing various assets and liabilities of a business concern is prepared on a particular date, which is called a statement of affairs. It is similar to a balance sheet.

CAPITAL COMPARISON METHOD: Under this method, the profit/ loss for a particular period is ascertained by comparing the closing capital of the opening capital.

Meaning of depreciation

 Depreciation means a fall in the value or quality of an asset. The word depreciation is derived from the Latin word “depretium”. ‘De’ means decline and ‘pretium’ means price.  it is the decline in the price or value of the fixed assets. Depreciation is described as a permanent, continuing and gradual shrinkage in the value of fixed assets. It is based on the cost of the asset consumed in a business and not on its market value.

         The depreciation is that part of the original cost of a fixed asset that is consumed during its period of use by the business. Thus, the depreciation is the loss of value of a fixed asset arising from use, effluxion of time or obsolescence. Depreciation sometimes restricted to fixed tangible assets but in the UK, it also usually includes the amortization of intangible assets. However, the tangible fixed assets lose their value over a period of time as they are used in the business operation and they do not last forever. If any amount is received on the sale of the fixed asset is deducted from the cost of it. Then the remaining value of the fixed asset is said to have” depreciated value” by that amount over its period of usefulness to the business.    

Quantitative Aptitude For Competitive Exams

This note is designed by the ‘Basics in Maths’ team. These notes to do help the  students for any Competitive Exam.

These notes cover all the topics covered in the any Competitive Exam syllabus and include plenty of formulae and concept to help you solve all the  Competitive examinations.


Number System

Number: A number is a mathematical object used to count and measure.1,2,3……. etc.

the ten thousand places in 5432 are greater than that in 4978.

Order of numbers

Ascending Order: arrange the numbers from smallest to the greatest; this order is called Ascending order.

Ex: – 23, 44, 65, 79, 100

Descending Order: arrange the numbers from greatest to the smallest, this order is called Ascending order.   Ex: – 100,79, 65, 33, 23

Formations of numbers

Form the largest and smallest possible numbers using the digits 3, 2, 4, 1 without repetition:

Largest number formed by arranging the given digits in descending order _ 4321.

Smallest number formed by arranging the given digits in ascending order _ 1234.

Greatest two-digit number is 99.

Greatest three-digit number is 999.

Greatest four-digit number is 9999.

Place value

Place value is the positional notation, which defines a digit’s position.

Ex: – 1234 ⟶   4 is one’s place, 3 is tens place, 2 is hundreds place and 1 is thousands place.

Face value

The face value of a digit in a numeral is its own value.

Ex: – 1234

Face value of r is 200

Face value of 3 is 30

Place value table for Indian system:

Q.A place value chart


Indian system of numeration: –
in Indian system of numeration we use ones, tens, hundreds, thousands, lakhs and crores. The first comma comes after three digits from the right, the second comma comes two digits latter and third comma comes after another two digits. 

Ex: – “three crores thirty-five lakh seventeen thousand four hundred thirty” can be written   as 3,35,17,430

International system of numeration: – In International system of numeration we use ones, tens, hundreds, thousands, millions and billions.

Ex: – “six hundred thirty-five million two hundred eighteen thousand nine hundred twenty-four” can be written as 635,218,924.

Types of Numbers:

Natural numbers: All the counting numbers starting from 1 are called Natural numbers.

1, 2, 3…

Whole numbers: Whole numbers are the collection of natural numbers.

0, 1, 2, 3 …

Note:  All natural numbers are whole number but all whole numbers need not to be natural numbers.

Integers: integers are the collection of whole numbers and negative numbers.

…., -3, -2, -1, 0, 1, 2, 3….

Rational numbers: The numbers which are written in the form of TS IX Maths Rational numbers 1  , where p, q are integers and q ≠ 0 are called rational numbers. Rational numbers are denoted by Q.

Ex: – 2, 3, 0.3,TS IX Maths Surface area and Volumes 6 TS IX Maths Surface area and Volumes 5 TS IX Maths Rational numbers 2 and so on

Even numbers: The numbers which are divisible by 2 successfully are called even numbers.

EX: 2, 4, 6, 8, 10, …

Odd numbers:  The numbers which are not divisible by 2 are called Odd numbers.

Ex: 1, 3, 5,7, 9, 11, …

Note:

  • Sum of two even numbers is an even number.
  • Sum of two odd numbers is an odd number.
  • Sum of even and odd number is an odd number.
  • Difference of two even numbers is an even number.
  • Difference of two odd numbers is an even number.
  • Difference of even and odd numbers is an odd number.
  • Product of two even numbers is even number.
  • Product of two odd numbers is odd number.
  • Product of even and odd number is even number.

multiplication of signs (+ or -)

Prime numbers: The numbers, which have only two factors 1, and itself are called prime numbers.

2, 3, 5, 7, …. Are prime numbers

Composite numbers: The number, which have more than two factors are called composite numbers.

4, 6,8,9…. are composite numbers.

Note: – 1) 1 is neither prime nor composite

2) 2 is the smallest prime number

3) 4 is the smallest composite number.

Co – prime number: The number which has no common factor except 1 is called co – prime number.

Ex: – (2, 3), (4,5) ……

Twin – primes: If the difference of two prime numbers is 2, then those numbers are called twin prime numbers.

Ex: – (2,3), (3,5), (17,19) ….

Special products:

(a + b)2 = a2 + 2ab + b2

(a − b)2 = a2 − 2ab + b2

(a + b) (a – b) = a2 − b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3 = a3 + b3 + 3ab (a + b)

(a – b)3 = a3 – 3a2b + 3ab2 – b3 = a3 – b3 – 3ab (a – b)

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

a3 + b3 = (a + b) (a2 – ab + b2)

a3 – b3 = (a – b) (a2 + ab + b2)

if a + b + c = 0, then a3 + b3 + c3 = 3abc.


Divisibility Rules

The process of checking whether a number is divisible by a given number or not without actual division is called divisibility rule for that number.

Divisibility by 2:  A number is divisible by 2 if its once place is either 0, 2, 4, 6 or 8.

Ex:  26 is divisible by 2. 35 not divisible by 2.

Divisibility by 3: If the sum of the digits of a number is divisible by 3, then that number is divisible by 3.

Ex:  231 → 2 + 3 +1 =6, 6 is divisible by 3

∴ 231 is divisible by 3

436 → 4 + 3 + 6 = 13, 13 is not divisible by 3

∴ 436 is not divisible by 3.

Divisibility by 4:  If the last two digits of a number is divisible by 4, then that number is divisible by 4.

Ex:  436, 36 is divisible by 4            ∴ 436 is divisible by 4

623, 23 is not divisible by 4      ∴ 623 is not divisible by 4.

Divisibility by 5:  A number is divisible by 5, if its once place is either 0 or 5.

Ex: 20, 25 are divisible by 5. 22, 46 are not divisible by 5.

Divisibility by 6: A number is divisible by 6, if it is divisible by both 3 and 2.

Ex: 242 is divisible by both 2 and 3     ∴ 242 is divisible by 6

232 is divisible by 3 but not 2        ∴ 232 is not divisible by 6

Divisibility rule by 7: –

Fist multiply the last digit of given number by 2,

subtract this result from the number formed by remaining digits of given number.

If that result is divisible by 7, then the given number is divisible by 7.

Ex: 112

Last digit is 2 ⇒ 2 × 2 = 4

Now 11 – 4 = 7

7 is divisible by 7

∴ 112 is divisible by 7.

Divisibility by 8:  if the last three digits of a number is divisible by 8, then that number is divisible by 8.

Ex: 4232, last three digits 232 are divisible by 8

∴ 4232 is divisible by 8.

Divisibility by 9:  if the sum of the digits of a number is divisible by 9, then that number is divisible by 9.

Ex:  459, 4 + 5 + 9 = 18 → 18 is divisible by 9       ∴ 459 is divisible by 9

532, 5 + 3 + 2 = 10 → 10 is not divisible by 9       ∴ 532 is not divisible by 9.

Divisibility by 10: – a number is divisible by 10, if its once place is 0.

Ex: 20 is divisible by 10. 22, 45 are not divisible by 10.

Factorial of a number: For positive integer ‘n’, the continued product of first natural numbers is called factorial n and is denoted by n!

Ex: 4! = 4 × 3 × 2 × 1 = 24

7! = 7 × 6 × 5× 4 × 3 × 2 × 1 = 5040

Absolute value or Modulus of a number:   Absolute value of a number always gives a positive number.

Modulus of x = x if x ≥ 0

Modulus of x = −x if x < 0

Ex: modulus of a numbers 1

Greatest integral value: The greatest integral value of a number ‘n’ is denoted by [n]

n <[n]< n + 1

Ex: [3.2] = 3; [5.6] = 5; [– 5. 4] = – 6


HCF and LCM

Factors: a number which divides the other number exactly is called factor of that number.

6 = 1×6

= 2×3      ⟹ factors of 6 are: 1, 2, 3 and 6

Common factors:  Common factors are those numbers, which are factors of all the given numbers.

Ex: 12, 9

Factors of 12 are:  1, 2, 3, 4, 6 and 12

Factors of 9 are:  1, 3 and 9

∴ Common factors of 12, 9 are 1,3

Perfect number: If the sum of the factors of a number (except the number itself) is equal to that number, then it is called Perfect number.

Ex: 6

The factors of 6 are: 1, 2, 3, 6

1 + 2 + 3 = 6

Therefore 6 is a perfect number.

Highest Common Factor (H.C.F):- The highest common factor of two or more numbers is the highest of their common factors. It is also called ad Greatest Common Divisor (G.C.D).

Ex: H.C.F of 12, 9

Factors of 12 = 1, 2, 3,4, 6, 12

  Factors of 9 = 1, 3, 9

Common factors of 12, 9 = 1,3

Highest common factor is 3

∴ H.C.F of 12, 9 is 3

Method of finding HCF:

Prime factorisation method: The HCF of 9, 12 can be found by prime factorisation method as follows.

Ex: – HCF of 9, 12

LCM of numbers 1

The common factor of 12, 9 is 3

∴ H.C.F of 12, 9 is 3

Continue division method: Euclid invented this method. Divide larger number by smaller and then divide the previous divisor by the remainder until the remainder zero. The last divisor is the HCF of given numbers.

HCF 1

∴ HCF of 9, 12 is 3

Common multiple: multiples of 3 are 3, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39,42…

Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52….

Common multiples of 3 and 4 are: 12, 24, 36….

Least common multiple (LCM): The least common multiple of two or more given numbers is the lowest of their common multiple.

Ex: LCM of 3 and 4

Multiples of 3 =    3, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39,42…

Multiples of 4 =   4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52….

Common multiples of 3 and 4 = 12, 24, 36….

∴ LCM of 3, 4 is 12.

Methods of finding LCM:

Prime factorisation method: –

The LCM of 6, 15 by using prime factorisation method is as follows:

Prime factors of 6 =   2 × 3

Prime factors of 15 = 5× 3

1.Express each number as product of prime factors

2.Take the common factors both: 3

3.Take the extra factors of both 6 and 15 i.e., 2 and 5

4.The product of all common factors of two numbers and extra common factors of both finds LCM.

           ∴ LCM of 6 and 15 = (3) × 2 × 5 = 30.

Division method: – To find LCM of 6 and 15:

Arrange the given numbers in a row.

Then divide the least prime number, which divides at least two of the given numbers, and carry forward the numbers, which are not divisible by that number if any.

Repeat the process till no numbers have a common factor other than 1.

LCM is the product of the divisors and the remaining numbers.

Ex: –    Find the LCM of 6, 15

LCM2

                            ∴ LCM of 6, 15 = 3 × 2 × 5 = 30.

Note:  the product of LCM and HCF of given numbers = the product of given numbers.

            Ex: – 6 × 15 = 30 × 3 = 90.

∎ HCF of co – prime numbers is 1 and LCM is their product

∎ HCF of fractions = HCF of Fractions

∎ LCM of fractions =LCM of fractions


Fractions

Fraction: Fraction is a part of whole. The whole may be a single object or group of objects.

In Fractions 1 , 3 is the numerator and 4 is the denominator.

Proper fraction: if the numerator of a fraction is less than the denominator, then the fraction is called proper fraction.

Ex:Fractions 2   etc.

Improper fraction: if the numerator of a fraction is greater than the denominator, then the fraction is called improper fraction.

Ex: Fractions 3 etc.

Mixed fraction: An improper fraction can be written as a combination of a whole and part. Such fraction is called mixed fraction.

Ex:Fractions 4

Decimal fractions: If the denominators of fractions are the powers of 10, then those fractions are called decimal fractions. We can write decimal fraction with a decimal point (.). it makes easier to do addition, subtraction and multiplication on fractions.

Ex:decimal fraction example  etc.

Parts in a decimal fraction:

In a decimal fraction, a dot(.) or a decimal point separate the whole part of the number from the fractional part.

The part right side of decimal point is called the decimal part of the number as it represents a part of 1. The part left to the decimal point is called the integral part of the number.

In 2.35, 2 is integral part and 35 is decimal part

It is read as two point three five

Addition and Subtracting of Decimal Fractions:

while adding or subtracting decimal numbers, the digits in the same places must be added or subtracted.

While writing the numbers one below the other, the decimal points must become one below other. Decimal places made equal by placing zeroes on the right side of the decimal numbers.

Ex: Add 2.63, 4.1, 7.625

addition of decimal fractions

Subtract 3.12 from 5.327

subtraction of fractions

Comparison of decimal numbers:

while comparing decimal numbers, first we compare the integral parts. If the integral parts are same, then compare the decimal part.

Ex: – which is bigger: 13.5 or 14.5

Ans: 14.5

Which is bigger: 13.53 or 13. 25

Ans: 13.53

Multiplication of decimal fractions:

For example, we multiply 0.1 × 0.1

multiplication of decimal fractions

Multiplication of decimal fractions by 10, 100, and 1000: –

multiplication of decimal fractions 1

Here, we notice that the decimal point in the product shift to the right side by as many zeroes as in 10, 100 and 1000.

Division of decimal fractions:

division of fractions 1

Division of decimal number by 10,100 and 1000:

Here, we notice that the decimal point in the product shift to the left side by as many zeroes as in 10, 100 and 1000.

BODMAS Rule

B ⟶ Bracket

O ⟶ Off (Product)

D ⟶ Division

A ⟶ Addition

S ⟶ Subtraction

Ex:  Solve:  7 – 12 ÷ 3 of 12

Sol: 7 – 12 ÷ 3 of 12 = 72 – 12 ÷ 3 of 2    ( 3 of 2 = 3× 2 = 6)

= 72 – 12 ÷ 6

= 72 – 2

= 70


Squares and Square roots, Cubes and Cube roots

Square:  Square number is the number raised to the power 2. Number obtained by the number multiplied by itself.

Ex: – 1) square of 5 = 52 = 5 × 5 = 25, 2) square of 3 = 32 = 3× 3 = 9

Perfect Square:  A natural number is called a perfect square, if it is the square of some natural number.

Ex: 1,4,9, …etc.

Square Root: the square root of a number x is that number when multiplied by itself gives x as the product. Square root of x is denoted bySquare root of x

Ex: Square root 1

Square root of a number in decimal form

Make the no. of decimal places even, by affixing a zero, if necessary. Now periods and find out the square root by the long division method.

Put the decimal point in the square root of as soon as the integral part is exhausted.

Ex: – To find square root of 79.21

square root of a number 1

Square root of a decimal number which is not perfect square:

if the square root is required correct up to two places of decimal, we shall find it up 3 places of decimal and then round it off up to two decimal places.

if the square root is required correct up to three places of decimal, we shall find it up 4 places of decimal and then round it off up to three decimal places.

Ex: – To find square root of 0.8 up to 2 decimal places

square root of a number 2

Pythagorean triplet:

Three natural numbers a, b and c are said to form a Pythagorean triplet if, c2 = a2 + b2

For every natural number a > 1, (2a, a2 – 1, a2 + 1).

Ex: – if we put a = 3 in (2a, a2 – 1, a2 + 1), then we get Pythagorean triplet (6, 8, 10).

Cube of a number:

The cube of a number is that number raised to the power 3.

Ex: – cube of 0.3 = 0.33 = 0.027

Cube of 2 = 23 = 8

Perfect cube:

If a number is perfect cube, then it can be written as the cube of some natural numbers.

Ex: – 1, 8, 27 and so on.

Cube root:

The cube root of a number x is that number which when multiplied by itself three times gives x as the product.

Cube root of x is denoted bycube root of x

Ex: cube root of a numbers 1  and so on.


Surds    

If ‘a’ is a positive rational number which can be expressed as the nth power of some rational number, then the irrational number  is called a Surd

n th root of n n is called order of surd, a is called ‘radicand’

  is called radical sign

Mixed surd:  A surd which has rational factor other than 1, the other factor being irrational is called mixed surd.

Ex:

Pure surd: A surd which unity has its rational factor, other factor being rational factor is called Pure surd.

Ex:  etc.

Rationalising Factor (R.F):  If the product of two surds is a rational number, then each of the two surds is a rationalising factor of the other. It is not unique.

Laws of Radicals (Surds):

Logarithms

For any two positive integers a and x if aN = x then Logarithms 1

Laws of logarithms:

Logarithms 2


  Ratio

The comparison of two quantities of same kind by using division is called ‘Ratio’

Ratio of two quantities ‘a’ and ‘b’ is denoted by a : b , read as a is to b

In a ratio a : b , a is called first term or antecedent and b is called second term or consequent.

Compound Ratio:  The compound ratio of a : b and c : d is denoted by a : b : : c : d

a : b : : c : d = a × c : b × d

the compound ratio of ( a : b), (c : d) and( x : y) is acx : bdy

Duplicate ratio of a : b is a2 : b2

Sub Duplicate ratio of a : b is sub duplicate ratio

Triplicate ratio of a : b is a3 : b3

Sub Triplicate ratio of a : b is sub triplicate of ratio


 Alligation or Mixture

Alligation: Alligation is a rule, to find the ratio in which two or more quantities at the given price must be mixed to produce a mixture of a desired price.

Mixture: Mixture means mixing two or more quantities. It can be expressed in the form of percentage or ratio.

Mean price: The cost price of a unity quantity of the mixture is called the mean price.

Alligation Rule: If two quantities are mixed, then

Alligation and Mixture 1

Suppose a container contains ‘x’ units of liquid from which ‘y’ units are taken out and replaced by water. After n operations the quantity of pure liquid = Alligation and Mixture 2units.


Proportion

If two ratios are equal, then they are in proportion.

If a : b = c : d , then a, b, c and d are in proportion

a and d are extremes

b and c are means

product of means = product of extremes

Componendo and Dividendo: Ifproportion 1, then componendo dividendo

Unitary method:  The method in which we first find the value of one unit and then the value of the   required no. of units is known as unitary method.

Direct proportion: In two quantities, when one quantity increase(decreases) the other quantity also increases(decreases) then two quantities are in direct proportion.

Inverse proportion: In two quantities, when one quantity increase(decreases) the other quantity also decreases (increases) then two quantities are in direct proportion.


Percentage

Percent means per 100, ‘x’ percent means ‘x’ hundredth.

x% = Percentage 1

Ex: 30% =Percentage 2

Converting a fraction into percentage:

Ex: Convert Percentage 4 into percentage

Sol: Percentage 4  = %  =  20 %

∎ If the price of commodity decreased by R%, then the increase in consumption so as not to

decrease the expenditure is   Percentage 6

∎ If the price of commodity increased by R%, then the increase in consumption so as not to

increase the expenditure is   Percentage 7

Results on machines: If the present value of a machine is P and it deprecates at the rate of R% per year, then

  • Value of the Machine After ‘n’ years = Percentage 8
  • Value of the Machine ‘n’ years ago =Percentage 9

Results on Population: If the present Population is P and it increases at the rate of R% per year, then

  • Population after ‘n’ years = Percentage 10
  • Population ‘n’ years ago =Percentage 11

∎ If A is R% more than B, then B is less than A by Percentage 7

∎ If A is R% less than B, then B is more than A byPercentage 6


Profit and loss

Cost price (C.P.): – Cost price is the price for which an article is bought or the price paid by a customer to by an article.

Selling price (S.P.): – Selling price is the price for which an article is sold.

Profit: – If Selling price is greater than the cost price, then we get the profit.

                  Profit = S.P – C.P.

Loss: – If Selling price is less than the Cost price then, we get loss. 

                 Loss = C.P – S.P.

Some formulae in profit and loss:

profit and loss %

For Profit:

SP and CP for profit

For Loss:

SP and CP for loss 

∎ When a person sells two similar items, one at gain of x%, and another one at loss of x%, then loss percentageProfit and Loss 1

∎If selling price is equal to cost price but uses false weight then

Gain% = Profit and Loss 2  %

∎ if the selling price of an item is at a profit of x% but uses false weight which is y% less than the original weight then

Gain % = Profit and Loss 3%

∎ if the selling price of an item is at a loss of x% but uses false weight which is y% less than the original weight then

Loss or Gain % = Profit and Loss 4%


Simple interest

Principal: – The money which is borrowed is called ‘principal’(P).

Rate of interest: – percentage of interest per year is called rate of interest(R).

Time: – The period for which money is called time(T).

Interest: – The money which is paid for the use of the principal is called interest (S.I).

Amount: – The total money which is paid after the expiry of the time is called amount.

simple interest

Amount = Principal + S.I

Amount for SI


Compound Interest

Compound interest allows us to earn interest on interest.

Amount = amount for compound interest 1 ( interest compounded annually)

P = principal, R = rate of interest and n = no. of terms or time period.

Amount = amount for compound interest half yearly ( interest compounded half yearly)

∎When the rates are different for different years, like X%, Y% and Z% for first, second and third year respectively, then

Amount = compound interest for differt rates of different years

∎ present worth of ₹ p due n years hence is given then present worth = compound interest for due after n years

∎The time period after which interest is added to principal is called conversion period. When interest is compounded h yearly, there are two conversion periods in a year. In such case, half year rate will be half of the annual rate.


Partnership

A business which is under taken by two or more persons jointly is called ‘Partnership. Those persons are called as Partners.

Types of Partnership:

Partnership is of two types (i) General or Simple partnership  (ii) Compound partnership

(i) Simple partnership or General partnership:

In this type, the period of the investment is the same. Partners divide either profit and loss in the ratio of their investments.

If X and Y invests ₹x and ₹y respectively in a business for one year, then the end of the year

X’s share of profit : Share of s profit  = x : y

(ii) Compound Partnership:

In this type, the investment and period of time are different. Their investments have to be reduced to invests per month or year. The profit or loss will be divided in the ratio of these converted investments.

If X invests ₹x per p months and Y invests ₹y per q months, then

X’s share of profit : Share of s profit  = x : y


Time and work

∎ If A can do a piece of work in ‘x’ days, then A’s one day’s work =

∎ If A’s one day work =  , then A can finish the work in ‘x’ days.

∎ If the work done by A is twice as the work done by B, then

The ratio of work done by A and B is 2 : 1

The ratio of time taken by A and B to finish the work is  1 : 2

∎ If the work done by A is thrice as the work done by B, then

The ratio of work done by A and B is 3 : 1

The ratio of time taken by A and B to finish the work is  1 : 3


Time and Distance

If the distance travelled = d, Speed = s and Time taken to travel = t, then

Distance = Speed × Time

⇒ d = s × t

Speed = Distance and Time 1 ; Time =Distance and Time 2

∎ Relation between the speed in kilometre per hour and speed in meter per seconds is

18 km/hr = 5 m/sec

x km/ hr = Distance and Time 3 m/sec

x m/sec =Distance and Time 4 km/sec

Relative Speed:

∎ In the same direction relative speed = Difference of the Speeds

∎ In the opposite direction relative speed = sum of the Speeds

∎ If the two-object moving in the same direction with speeds x km/hr and y km/hr respectively, then their relative speed = (x – y) km/hr.

∎ If the two-object moving in the opposite direction with speeds x km/hr and y km/hr respectively, then their relative speed = (x + y) km/hr.

∎ Time taken to meet the object =Distance and Time 5


Pipes and Cisterns

Inlet: A pipe connected with a tank or cistern, that fills it is called Inlet.

Outlet:  A pipe connected with a tank or cistern, that emptying is called out let.

∎If a pipe can fill a tank in x hrs. then in one hour it can fil lDistance and Time 7 part of the tank.

∎If a pipe can empty a tank in y hrs. then in one hour it can empty Distance and Time 8part of the tank.

∎ If one pipe can fill a tank in x hrs. and another pipe can empty the full tank in y hrs. then on opening both the pipes, then the tank can fill in 1           hour = Distance and Time 9

∎ If one pipe can fill a tank in x hrs. and another pipe can empty the full tank in y hrs. then on opening both the pipes, then the tank can empty in      1 hour =Distance and Time 10


Boats and Streams

Downstream:  In water the direction along the water flow (stream) is called Down stream

Upstream:  In water the direction against the stream is called Upstream.

∎If the speed of object moving in a water is x km/hr and speed of the water flow (stream) is y km/hr then

For upstream relative speed = (x – y) km/hr.

For downstream relative speed = (x + y) km/hr.

Speed in still water = ½ (x + y) km/hr

    Rate of stream = ½ (x – y) km/hr


For Train Problems

∎ Time taken by a train of length x meters to cross (pass) a pole (or a standing man) is equal to the time taken by the train to cover x meters.

∎ Time taken by a train of length x meters to cross a stationary object (like a bridge) of length y meters is the time taken by train to cover (x + y) meters.

∎ If the two trains moving in the same direction with speeds x km/hr and y km/hr respectively, then their relative speed = (x – y) km/hr.

∎ If the two trains moving in the opposite direction with speeds x km/hr and y km/hr respectively, then their relative speed = (x + y) km/hr.

∎ If two trains of length a meters and b meters are moving in a opposite directions at x km/hr and y km/hr then time taken by the trains to cross        each other = Distance and Time 6  hrs.

∎ If two trains of length a meters and b meters are moving in a same direction at x km/hr and y km/hr then time taken by the trains to cross each        other = Distance and Time 11  hrs.


Triangle

A closed figure with three-line segments is called a triangle.

triangle 1

A triangle has 3 sides, 3 angles and 3 vertices.

Types of triangle:

According to the sides:

Scalene triangle: – If no two sides of the triangle are equal then it is called scalene triangle.

Isosceles triangle- If any two sides of a triangle are equal, then it is called isosceles triangle.

Equilateral triangle: – If all sides of a triangle are equal, then it is called equilateral triangle

According to the angles:

Acute angled triangle: – If all angles of a triangle are acute, then it is called acute angled triangle.

Right angled triangle:– If one of the angles of a triangle is right angle(900), then it is called right angled triangle.

Obtuse angled triangle: If one of the angles of a triangle is obtuse, then it is called obtuse angled triangle.

Relationship between the sides of a triangle:

∎In a triangle, the sum of the lengths of two sides is greater than the third side.

∎ In a triangle, the difference of the lengths of two sides is less than the third side.

Altitude: In a triangle, a perpendicular line drawn form vertex to its opposite side is called Altitude.

∎We can draw three altitudes in a triangle.

∎ In an Isosceles triangle, the altitude from the vertex bisects the base.

Median: In a triangle, a line drawn from vertex to midpoint of its opposite side is called median.

∎We can draw three medians in a triangle.

∎ The median of a triangle divides it in to two triangles of the equal area.

Circumcentre:

The point of concurrency of medians in a triangle is called circum centre.

∎ Circumcentre divides median in the ratio 2 : 1.

Angle sum property of triangles:

Sum of the angles in a triangle is 1800

An Exterior angle: The angle formed in the exterior of a triangle when one of the sides is produced is called an exterior of the triangle.

∎ An exterior angle of a triangle is equal to sum of its interior opposite angles.

Vertical angle bisector theorem: The bisector of the vertical angle of a triangle divides the base in the ratio of the other two sides.

Similar triangles:

Two triangles are said to be similar if (i) their corresponding angles are equal (ii) their corresponding side are in proportion.

∎ The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

∎ The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding Altitudes.

Pythagorean theorem:

In a right-angled triangle the square of the hypotenuse is equal to sum of the squares of the other two sides.

In ∆ABC, ∠B = 900, then AC2 = AB2 + BC2

FORMULAE:

∎ Area of the triangle with base ‘b’ and height ‘h’ =Triangles 3 b × h sq. Units

∎ Area of the triangle whose lengths of the sides a, b and c = Triangles 4 sq. Units

Where s =Triangles 5

∎ If a and b are the lengths of any two sides of a triangle and 𝛉 is the angle between them, then area of the triangle =   ab sin 𝛉 sq. Units

∎ Area of an equilateral triangle with side a = Triangles 11   a2 sq. Units

∎Radius of circumcircle of a triangle =Triangles 7

∎ Radius of circumcircle of an equilateral triangle of side a =Triangles 8

∎ Radius of incircle of a triangle of area ∆ and semi perimeter s =Triangles 9

∎ Radius of incircle of an equilateral triangle of side a =Triangles 10

 Quadrilaterals

Quadrilateral: A simple closed figure with four-line segments is called a quadrilateral.

Diagonal: The line joining opposite vertices of a quadrilateral is called diagonal.

Trapezium: In a quadrilateral one pair of opposite sides are parallel, then it is called Trapezium.

∎ The line joining the mid points of non-parallel sides of a trapezium is parallel to each of the parallel sides and equal to half of their sum.

Parallelogram: In a quadrilateral two pair of opposite sides are parallel, then it is called Parallelogram.

∎ The diagonals of a parallelogram are bisect each other and not equal.

∎ Diagonal of a parallelogram divides it into two triangles of the equal area.

∎ In a parallelogram opposite sides are equal and opposite angles are equal.

Rectangle: In a parallelogram one of the angles is 900, then it is called rectangle.

∎ In a rectangle diagonal are bisect each other and equal.

Rhombus: In a parallel gram adjacent sides are equal, then it is called Rhombus.

∎ In a Rhombus diagonal are bisect each other perpendicularly and unequal.

A parallelogram and rectangle on the same base and lie between the same parallel lines are equal in area.

FORMULAE:

∎ Area of the Trapezium = Triangles 3 (sum of the parallel sides) × (distance between them)

∎ Area of the parallelogram = base × height

∎ Perimeter of rectangle = 2 (length + breadth)

Area if the rectangle = length × breadth

∎ Area of rhombus = Triangles 3 × product of lengths of diagonals

∎ Perimeter of Square = 4 × side

Area of square = (side)2

∎Area of four walls of a room = 2 (length + breadth) × height

Circles:

Circle: Set of points which are at a constant distance from a fixed point is called a circle.

             Fixed point is called centre of the circle and constant distance is called radius of the circle.

∎ Circumference of the circle = 2πr, where r is the radius of the circle.

Area of the circle = πr2

∎Length of an Arc = l = rθ, where θ is the angle subtended by arc at the centre.

∎Area of sector =Circles 1  πr2

∎Area of a regular polygon of side a = 1.72 a2


Mensuration

Cuboid:

QA Cuboid

Lateral surface area (L.S.A) = 2h (l + b) square units.

Total surface area (T.S.A) = 2 (lb + bh + hl) square units.

Volume = lbh cubic units.

Diagonal =diagonal of cuboid

Cube:

QA Cube

Lateral surface area (L.S.A) = 4 a2 square units.

Total surface area (T.S.A) = 6 a2 square units.

Volume = a3 cubic units.

Diagonal =diagonal of cube

Cylinder:

QA Cylinder

Curved surface area (C.S.A) = 2πr h square units.

Total surface area (T.S.A) = 2πr (r +h) square units.

Volume = πr2 h cubic units.

Cone:

QA Cone

Slant height =Slant height

Curved surface area (C.S.A) = πrl square units.

Total surface area (T.S.A) = πr (r +l) square units.

Volume = TS IX Maths Surface area and Volumes 5πr2 h cubic units.

Volume =  (R2 + r2 + Rr) πh cubic units.

Frustum of a cone:

QA Frustum of cone

When a cone is cut by a plane parallel to the base of the cone then the portion between the plane and the base is called the frustum of the cone.

Slant height = Slant height of Frustum units

L.S.A = πl (r + R) square units.

T.S.A = πl [r2 + R2 + l (R + r)] square units.

Sphere:

QA Sphere

Total surface area (T.S.A) = 4πr2 square units.

Volume = TS IX Maths Surface area and Volumes 6πr3 cubic units.

Hemisphere:

QA Hemi Sphere

Curved surface area (C.S.A) = 2πr2 square units.

Total surface area (T.S.A) = 3πr2 square units.

Volume = TS IX Maths Surface area and Volumes 7 πr3 cubic units.

Pyramid:

Surface area = Area of the base + Area of the each of the lateral faces

Volume = TS IX Maths Surface area and Volumes 5 × area of the base × height cubic units.

Average:

Average =  Average

Average of n natural numbers = ½ (n + 1).

suppose A person covers a certain distance at x km/hr and an equal distance at y km/hr, then the average speed during the whole journey is Average 1 km/hr

Probability:

Random Experiment: If in an experiment all the possible outcomes are known in advance and none of the outcomes can be predicted with certainty, then such an experiment is called a random experiment.

Ex: Throwing a die, Tossing A coin.

Events: The possible outcomes of a trail are called events.

To measure the chance of it happening numerically we classify them as follows:

Certain: Something that must happen

Equally likely: something that have the same chance of occurring.

Less likely: Something that would occur with less chance.

More likely: Something that would occur with more chance.

Impossible: Something that cannot be happen.

Probability of an event = probability of an event

Complimentary event: – Let E denote the event, ‘not E’ is called complimentary event of E. It is denoted by  . P ( ) = 1 – P(E) ⟹ P ( ) +P(E) = 1.

∎0 ≤ P(E) ≤ 1

Deck of cards: – A deck of playing cards consists of 52 cards which are divided into 4 suites of 13 cards each. They are black spade, black clubs , red heart and red diamond. The cards in each suit are: 2, 3, 4, 5, 6, 7, 8, 9 ,10, Ace, Jack, Queen and King. Jack, Queen and King are called face (picture) cards.


Calendar

Ordinary year: A year having 365 days is called ordinary year.  In ordinary years we have 52 complete weeks and 1 extra day.

7 months ⟶ 31 days.

4 months ⟶ 3 days.

1 month (February) ⟶ 28 days.

Leap year: A year having 36 days is called leap year.  In leap years we have 52 complete weeks and 2 extra days.

7 months ⟶ 31 days.

4 months ⟶ 3 days.

1 month (February) ⟶ 29 days.

∎ A leap year is divisible by 4.

∎ A century to be leap year if it must be divided by 400.

∎ In a century 76 ordinary years and 24 leap years.

Odd day: Extra days apart from the complete weeks in a given period is called odd day.

∎ An ordinary year has 1 odd day.

∎ A leap year has two odd days.

∎ A century has 5 odd days. (76 × 1 + 24 × 2 = 124 odd days = 17 weeks = 5 odd days).

∎In 200 years ⟶ 3 odd days.

In 300 years ⟶ 1 odd day.

In 400 years ⟶ 0 odd day.

In 800 years ⟶ 0 odd day.

In 1200 years ⟶ 0 odd day.

Century − Code:

Calendar Century - code


Month − odd day − Code:

Calendar Month - odd day - code

Day – Code:

Calendar Days - Code
Formula:

Calendar Formula


Clocks

A clock has two hand (i) Shorter hand or hour hand (ii) Longer hand or Minute hand

∎For every hour, both the hands are coincide once.

Hour hand:

Angle made by Hour hand in 12 hours = 3600

In one hour, hour hand can make the angle = 300.

In one minute, hour hand can make the angle =Clocks 1

Minute hand:

Angle made by Minute hand in 60 minutes = 3600

In one minute, Minute hand can make the angle = 60

Relative speed:

Relative speed of Minute hand and Hour hand = Clocks 3= Clocks 2

∎ When two hands are rotated in clockwise direction

  1. They may be opposite to each other.
  2. They may be coincide to each other.
  3. They may be perpendicular to each other.

Two hands – Coincide:

Angle between two hands = 00

In the span of 1 hour, they coincide only 1 time.

12 hours, they coincide 11 times.

1 day, they coincide 22 times.

Two hands – opposite direction:

Angle between two hands =1800

In the span of 1 hour, they coincide only 1 time.

12 hours, they coincide 11 times.

1 day, they coincide 22 times.

Two hands – perpendicular:

Angle between two hands = 900

In the span of 1 hour, they coincide 2 times.

12 hours, they coincide 22 times.

1 day, they coincide 44 times.

∎ When two hands are opposite to each other, then that two hands are 30 minutes spaces apart.

∎ When two hands are perpendicular to each other, then that two hands are 15 minutes spaces apart.

Formula:

M = Clocks 4(H × 300± θ)

Where M = Minutes

H = Hours (starting hour)

θ = angle between two hands

If starting hour is at 12’O clock, then take H = 0


 

 

 

ICSE MATHS CONCEPT FEATURE IMAGE

ICSE X Class Maths Concept

 This note is designed by the ‘Basics in Maths’ team. These notes to do help the ICSE 10th class Maths students fall in love with mathematics and overcome the fear.

These notes cover all the topics covered in the ICSE 10th class Maths syllabus and include plenty of formulae and concept to help you solve all the types of 10th class Mathematics problems asked in the ICSE board and entrance examinations.


 1. Goods and Service Tax

Two types of taxes in Indian Government:

1.Direct taxes: –

These are the taxes paid by an organisation or individual directly to the government. These include Income tax, Capital gain tax and Corporate tax.

2.Indirect taxes: –

These are the taxes on goods and services paid by the customer, collected by an individual or an organisation and deposited with the Government. Earlier there were a number of indirect taxes levied by the central and state Governments.

Goods and Service Tax (GST):

GST is a comprehensive indirect tax for the whole nation. It makes India one unified common market.

 Registration under GST:

Any individual or organisation having an annual turnover of more than ₹ 20 lakh is to be registered under GST.

Input and Output GST:

For any individual or organisation, the GST paid on purchases is called the ‘Input GST’ and the GST collection on sale of goods is called the ‘Output GST’. The input GST is set of against the output GST and the difference between two is payable in the Government account.

One currency one tax:

There is a uniform GST rate on any particular goods or services across all states and Union Territories of India. This is called ‘One currency one tax’.

Note: Assam was the first state to implement GST and Jammu & Kashmir was the last.

GST rate slabs:

ICSE X class Maths GST rate slabs

However, the tax on gold is kept at 3% and on rough precious and semi-precious is kept at 0.25%.

The multitier GST tax rate system in India has been developed keeping on mind that essential commodities should be taxed less than the luxury goods.

Benefits of GST for Traders:

• Simple tax system.

• Elimination of multiplicity of taxes.

• Development of a common market nation-wide.

• Reduction of cascading effect.

• Lower taxes result in reduction of costs making in the domestic market.

Benefits of GST for Consumers:

• Single and transparent System.

• Elimination of cascading effect has resulted in the reduction in the costs of goods and services.

• Increase in purchasing power and savings.

Benefits of GST for Traders:

• Single tax system, simple and easy to administer.

• Higher revenue efficiency.

• Better control on leakage and tax evasion.

Types of GST in India

Central GST (CGST): For any intrastate supply half of the GST collected as the output GST is deposited with the Central Governments as CGST.

State GST or Union Territory GST (SGST/UGST): For any local supply (supply with in the same state or Union Territory) half of the GST is deposited with respective state or Union Territory Government as the beneficiary. This is called SGST/UGST.

Integrated GST (IGST): The GST levied on the supply of goods or services in case of interstate trade with in India or in case of exports/imports is known as IGST.

Reverse charge Mechanism:

There are cases where the chargeability gets reversed, that is the receiver becomes liable to pay the tax and deposit it to the Government Account.

Composition shame:

The composition is meant for small dealers and service providers with an annual turnover less than ₹ 1.5 crores and also for Restaurant service providers. Under this scheme the rates of GST are:

ICSE X class Maths Composition Scheme
Input Tax Credit (ITC)
    

When a dealer sells his goods, he charges the output GST from his customer which he has to deposit in the government account, but in running his business he had paid input GST on the goods he had availed. This input GST, he utilizes as Input Tax credit and deposits the exes amount of output GST with the Government.

Input Tax credit is a provision of reducing the GST already paid on inputs in order to avoid the cascading of taxes.

GST payable = Output GST – ITC

Claiming ITC: A dealer registered under GST can claim ITC only if:

  • He possesses the tax invoice.
  • He has received the said goods/services
  • He has filed the returns.
  • The tax paid by him has been paid to the government by his supplier.

Utilization of ITC:

The Amount of ITC available to any registered dealer shall be utilized to reduce the out put tax liability in the sequence shown in the table.

ICSE X class Maths Utilization of ITC

E – ledgers under GST:
An E – ledger is an electronic form of a pass book available to all GST registrants on the GST portal. These are of three types:

(i) Electric cash ledger (ii) Electric credit ledger and (iii) Electric Liability Register

(i) Electric cash ledger: It contains the amounts of GST deposited in each to the government.

(ii) Electric credit ledger: It contains the balance of ITC available to the dealer.

(iii) Electric credit ledger: It contains all the Tax liability of the dealer.

GST Returns:

These are the information provided from time to time by the dealer to the Government regarding the ITC, output Tax liability and the amounts of GST deposited.

A GST registered person has to submit the following returns:

ICSE X class Maths GST returns

E – Way bill:
E – Way bill is an electronic way bill that can be generated on the E – Way bill portal. A registered person can not transport goods whose value exceeds ₹ 50,000 in a vehicle without an e – way bill. When an E – way bill is generated, a unique e – way bill number (EBN) is allocated and is available to the supplier, the transporter and recipient. A dealer must generate an E – way bill if he has to transport them for returning to the supplier.
 


2.Banking

 To encourage the habit of saving income groups, banks and post offices provide recurring deposit schemes.

Maturity period: An investor deposits a fixed amount every month for a fixed time period is called the maturity period,

Maturity value:  At the end of the maturity period, the investor gets the amount deposited with the interest. The total amount received by the investor is called Maturity value.

Interest =    p times frac{n(n+1))}{2times 12}times frac{r}{100}

Where p is the principle

n is no. of months

r is the rate of interest

Maturity value = (p × n) + I


3.Shares and Dividend

Capital: The total amount of money needed to run the company is called Capital.

Nominal value (N.V): – The original value of a share is called the nominal value. It is also called as face value (F.V), printed value (P.V) or registered Value (R.V).

Market value: – The price of a share at a particular time is called market value (M.V). This value changes from time to time.

Shares: The whole capital is divided in to small units is called shares.

Share at par: – If the market value of a share is equal to face value of a share, then that share is called a share at par.

Share at a premium or Above par: – If the market value of a share is greater than the face value of the share then, the share is called share at a premium or above par.

Share at discount: – If the market value of a share is lesser than the face value of the share then, the share is called share at discount.

Dividend: – The profit distributed to the shareholders from a company at the end of the year is called a dividend.

The dividend is always calculated as the percentage of face value of the share.

Some formulae:

shares formulae

Note:

  • The face value of a share always remains the same
  • The market value of a share changes from time to time.
  • Dividend is always paid on the face value of a share

 4. Linear In equations

Linear inequations: A statement of inequality between two expressions involving a single variable x with highest power one is called linear inequation.

Ex: 3x – 3 < 3x + 5; 2x + 10 ≥ x – 2 etc.

General forms of Inequations: The general forms of the linear inequations are: (i) ax + b < c   (ii) ax + by ≤ c    (iii)  ax + by ≥ c    (iv) ax + by > c, where a, b and c are real numbers and a ≠ 0.

Domain of the variable or Replacement Set: The set form which the value of the variable x is replaced in an inequation is called the Domain of the variable.

Solution set: The set of all whole values of x from the replacement set which satisfy the given inequation is called the solution set.

Ex: Solution set of x < 6, x ∈ N is {1, 2, 3, 4, 5}

Solution set of x ≤ 6, x ∈ W is {0, 1, 2, 3, 4, 5, 6}

Inequations – Properties:

• Adding the same number or expression to each side of an inequation does not change the inequality.

Ex: 3 < 5

Add 2 on both sides

3 + 2< 5 + 2

5 < 7 (no change in inequality)

• Subtracting the same number or expression to each side of an inequation does not change the inequality.

Ex: 3 < 5

subtract 2 on both sides

3 – 2 < 5 – 2

1 < 3 (no change in inequality)

• Multiplying or Dividing the same positive number or expression to each side of an inequation does not change the inequality.

Ex: 3 < 5

Multiply 2 on both sides

3 × 2< 5 × 2

6 < 10 (no change in inequality)

6 < 8

Divide 2 on both sides

6 ÷ 2< 8 ÷ 2

3< 4 (no change in inequality)

•Multiplying or Dividing the same negative number or expression to each side of an inequation can change(reverse) the inequality.

Ex: 3 < 5

Multiply 2 on both sides

3 × –2< 5 × –2

–6 > –10 (change in inequality)

6 < 8

Divide 2 on both sides

6 ÷ –2< 8 ÷ –2

–3 > –4 (change in inequality)

Note:

  • a < b iff b > a
  • a > b iff b < a

Ex: x < 4 ⇔ 4 > x

x > 3 ⇔ 3 < x

Method of solving Liner Inequations:

  • Simplify both sides by removing group symbols and collecting like terms.
  • Remove fractions by multiplying both sides by an appropriate factor.
  • Collect all variable terms on one side and all constants on the other side of the inequality sign.
  • Make the coefficient of the variable 1.
  • Choose the solution set from the replacement set.

Ex: Solve the inequation 3x – 2 < 2 + x, x ∈ W

Sol: given in equation is

3x – 2 < 2 + x

Add 2 on both sides

3x – 2 + 2< 2 + x + 2

3x < 4 + x

3x – x < 4

2x < 4

Dividing both sides by 2

x < 2

∴ Solution set = { 0, 1}


5.Quadratic Equations

Quadratic Equation: An equation of the form ax2 + bx + c = 0, where a, b, and c are real and a ≠ 0 is called a Quadratic equation in a variable ‘x’.

Ex: x 2 – 3x + 4 = 0 is a quadratic equation in a variable ‘x’

t2 + 5t = 6 is a quadratic equation in a variable ’t’

Roots of a quadratic equation: A number α is called a root of the quadratic equation ax2 + bx + c = 0, if aα2 + bα + c = 0.

Solution set:  The set of elements representing the roots of a quadratic equation is called solution set of the give quadratic equation.

Solving Quadratic equation by using Factorization method:

Step – 1: Make the given equation into the standard form of ax2 + bx + c = 0.

Step – 2: Factorise ax2 + bx + c into two linear factors.

Step – 3: Put each linear factor equal to zero.

Step – 4: Solve these linear equations and get two roots of the given quadratic equation.

Ex: Solve x2 – 3x – 4 = 0

x2 – 4x + x – 4 = 0

x (x – 4) + 1 (x – 4) = 0

(x – 4) (x + 1) = 0

x – 4 = 0 or x + 1 = 0

x = 4 or x =– 1

∴ Solution set = {– 1, 4}

Solving Quadratic equation by using Formula:

The roots of the quadratic equation ax2 + bx + c = 0 are:

ICSE X maths Quadratic Formula

Ex: Solve x2 – 3x – 4 = 0

Sol: Given equation is x2 – 3x – 4 = 0

Compare with ax2 + bx + c = 0

a = 1, b = – 3, c = – 4

  ICSE X maths Quadratic Formula example

x = 4 or x = – 1

∴ Solution set = {– 1, 4}

Nature of the roots:

Discriminant: – For a quadratic equation ax2 + bx + c = 0, b2 – 4ac is called discriminant.

(i) If b2 – 4ac > 0, then roots are real and un equal.

     Case – 1: b2 – 4ac > 0 and it is a perfect square, then roots are rational and unequal.  

    Case – 2: b2 – 4ac > 0 and it is not a perfect square, then roots are irrational and unequal.     

(ii) If b2 – 4ac = 0, then roots are equal and real.

(iii) b2 – 4ac < 0, then roots are imaginary and un equal.


6.Problems on Quadratic equations

To solve word problems and determine unknown values, by forming quadratic equations from the information given and solving them by using methods of solving Quadratic equation.

The problems may be based on numbers, ages, time and work, time and distances, mensuration etc.

Method of Solving word problems in Quadratic equation:

Step – 1: Read the given problem carefully and assume the unknown be x.

Step – 2: Translate the given statement and form a quadratic equation in x.

Step – 3: Solve for x.


7.Ratio and Proportion

Ratio: Comparing two quantities of same kind by using division is called a ratio.

The ratio between two quantities ‘a’ and ‘b’ is written as a : b and read as ‘a is to b’

In the ratio a : b, ‘a’  is called ‘first term’ or ‘antecedent’ and ‘b’ is called ‘second term’ or ‘consequent’.

Note:  The value of a ratio remains un changed if both of its terms are multiplied or divided by the same number, which is not a zero.

Lowest terms of a Ratio:

In the ratio a : b, if a, b have no common factor except 1, then we say that a : b is in lowest terms.

Ex: 4 : 12 = 1 : 3 ( lowest terms)

Comparison of Ratios:  

  • (a : b) > (c : d) ⇔ ad > bc
  • (a : b) = (c : d) ⇔ ad = bc
  • (a : b) < (c : d) ⇔ ad < bc

Proportion:

An equality of ratios is called a proportion.

a, b, c and d are said to be in proportion if a : b = c : d and we write as a : b : : c : d.

a and d are ‘extremes’, b and c are ‘means’

product of extremes = product of means

Continued proportion: If a, b, c, d, e and f are in continued proportion, then ICSE X maths Continued Proportion

Mean proportion:  If ICSE X maths Mean Proportion 1then b2 = ac or b =