[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-204151-105":59,"doc-detail-204151-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","important-formulae-for-competitive-exams-quick-reference","IMPORTANT FORMULAE FOR COMPETITIVE EXAMS - Quick reference","","A compact collection of exam-focused mathematical and commerce-reasoning formulas, designed for rapid revision. Covers key algebraic identities for simplification, counting rules (sums, squares, cubes), percentage change and profit or loss relationships, interest calculations for simple and compound interest, and population-growth style compound formulas. Includes time, speed, and distance relations with average-speed concepts, ratio and proportion properties, alligation and mixture methods, and system/number-theory tools such as LCM/HCF and divisor-factor counting. Useful as a formula checklist for competitive problem-solving.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/important-formulae-for-competitive-exams-quick-reference/204151/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/important-formulae-for-competitive-exams-quick-reference/204151.png","ImageObject",300,407,{"name":92,"@type":93},"Mali","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-08","2026-09-05",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",17,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"Which algebraic identities are included for simplification?","Question",{"text":113,"@type":114},"The document lists standard expansions such as (a+b)^2, (a-b)^2, and identities for a^2-b^2, a^3±b^3, plus a relation involving a^2 and b^2 in a difference-of-squares style.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How are simple and compound interest formulas represented?",{"text":118,"@type":114},"Simple interest is given using P, r%, and n in a direct proportion form. Compound interest provides an amount after n years using a growth factor based on r% p.a., and also includes related profit-growth population-style expressions.",{"name":120,"@type":111,"acceptedAnswer":121},"What speed–distance average concepts and boat-travel relations are provided?",{"text":122,"@type":114},"Average-speed rules are included: arithmetic mean when time is constant and harmonic mean when distance is constant. Boat-travel relations define downstream/upstream/speed in still water with equations linking D, U, B, and R, plus derived forms for B and R.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},204151,1788571096,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145},2336475104362,"https://ap-avatar.wpscdn.com/avatar/22000c4c46a41b752dd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786595829695023868","IMPORTANT FORMULAE FOR COMPETITIVE EXAMS  \nProfit 􀁵 100  \nCP  \n􀂧 P 􀂷  \n2. SP = CP + P% of CP = CP 􀂩􀂨 1 􀀎 100 􀂹􀂸  \n3. Discount = Marked Price – Selling Price  \n1. Important formulae used in simplification:  \n(1) (a + b)2 = a2 + b2 + 2ab  \n(2) (a – b)2 = a2 + b2 – 2ab  \n(3) (a + b)2 = (a – b)2 + 4ab  \n(4) a2 – b2 = (a – b) (a + b)  \n(5) a3 + b3 = (a + b) (a2 – ab + b2)  \n(6) a3 – b3 = (a – b) (a2 + ab + b2)  \n(7) a2 􀀎 b2 􀀠 12 [(a 􀀎 b)2 􀀎 (a– b)2 ]  \n2. Rules of counting numbers  \n1. Sum of first n natural numbers n 􀀋 n 􀀎 1􀀌  \n=  \n2  \n2. Sum of first n odd natural numbers = n2  \n3. Sum of first n even natural numbers = n(n + 1)  \n4. Sum of the squares of first n natural numbers = n(n~~ ~~􀀎~~ ~~1)6(2n~~ ~~􀀎~~ ~~1)  \n5. Sum of the cubes of first n  \n􀂪  n(n 􀀎 1) 􀂺 2 natural numbers = 􀂬􀂫 2 􀂼􀂻  \nPERCENTAGES  \n1. Two successive percentage changes of a% and b% is an effective change of  \n110000􀁲rr~~ ~~ % less/more than A.  \nINTEREST  \n1. P = Principal, A = Amount, I = Interest, n = no. of years, r% = rate of interest  \nP 􀁵 r 􀁵 n  \nThe Simple Interest (S.I.) = 100  \n2. If P is the principal kept at Compound Interest (C. I. )@ r% p.a. , amount after n years  \n= P 􀂧􀂨􀂩 1 􀀎 ~~ ~~1r00~~ ~~ 􀂷􀂹􀂸n  \n3. Amount = Principal + Interest  \n4. Let P = Original Population, P􀁣 = Population after n years, r% = rate of anual growth  \n􀂧  r 􀂷 n P' 􀀠 P 􀂩􀂨 1 􀀎 100 􀂹􀂸  \n5. Difference between CI and SI for 2 and 3 years respectively:  \n(CI)2 –(SI)2 = Pa2 for two years  \n(CI)3 –(SI)3 = Pa2 (a + 3) for three years where, a = 1r00  \n6. A principal amounts to X times in T years at S. I. It will become Y times in:  \nYears 􀀠 􀂧􀂨􀂩 ~~ ~~YX~~ ~~––~~ ~~11􀂷􀂹􀂸 􀁵 T  \n7. A principal amounts to X times in T years at C. I. It will become Y times in:  \nYears = T × n  \nwhere n is given by Xn = Y  \nPROFIT AND LOSS  \n1. Profit % = ~~ ~~  \n4. Discount % = MDariskceuPnrtice 􀁵 100  \n5. The selling price of two articles is same . If one is sold at X% profit and the other at loss of X%, then there is always a loss of X10~~ ~~ %  \n􀂧􀂨􀂩 a+b+ 1ab00 􀂷􀂹􀂸 % .  \n2. If A is r% more/less than B, B is  \nIMPORTANT FORMULAE FOR COMPETITIVE EXAMS Page 1  \na cb 􀀠 d then:  \na 􀀎 b c 􀀎 d  \n(a)  b  􀀠  d  – Componendo Lawa – b c – d  \n(b)  b  􀀠  d  – Dividendo Lawa 􀀎 b c 􀀎 d  \n(c) a – b 􀀠 c – d – Componendo & Dividendo Law  \n(d)  \na  \nb  \n􀀠  \nc  \nd  \n􀀠  \ne  \nf  \n= K, then:  \n􀂧  V 􀀐 x  􀂷 n K 􀀠 Ko 􀂩􀂨 V 􀂹􀂸  \nTIME, SPEED AND DISTANCE  \n1. 1 km/hr = 158 m/s and 1m/s =  \n2. Average Speed =  \n(a) a 􀀎 c 􀀎 e 􀀠 Kb 􀀎 d 􀀎 f  \n(b)  \nRATIO & PROPORTION  \n1. It a : b : : c : d, then ad = bc  \n2. If a \u003C b and x is a positive quantity, then  \na a 􀀎 x a a – x  \nb 􀀟 b 􀀎 x and b 􀀡 b – x  \n3. If a > b and x is a positive quantity, then  \na a 􀀎 x a a – x  \nb 􀀡 b 􀀎 x and b 􀀟 b – x  \n4. If  \na 􀁲 c a  \n~~ ~~ 􀀠 ~~ ~~  \nb 􀁲 d b  \n5. If ~~ ~~ ~~ ~~ ~~ ~~  \npa 􀀎 qc 􀀎 re  \n pb 􀀎 qd 􀀎 rf = K  \n(p, q and r are not all zero)  \nALLIGATION, MIXTURESAND MEAN  \n1. Alligation is a method of calculating weighted averages. The ratio of the weights of the two items mixed will be inversely proportional to the difference of each of these two items from the average attribute of the resultant mixture.  \nx1  \nx2– x : x– x1 w1 : w2  \n2. Arithmetic Mean = x1~~ ~~􀀎~~ ~~x2~~ ~~􀀎~~ ~~x~~ ~~􀀎~~ ~~......~~ ~~􀀎~~ ~~xn~~ ~~  \n3. Geometric Mean = nx1 􀁵 x2 􀁵 x3 􀁵 . ..... 􀁵 xn  \n4. Harmonic Mean =  \nn  \n􀂧 1 1 1 1 􀂷  \n􀂨 ~~ ~~ 􀀎  􀀎 ~~ ~~ 􀀎 ...... 􀀎 ~~ ~~ 􀂸  \n􀂩 x1 x 2 x3 x n 􀂹  \n5. Let Ko be the initial concentration of a solution and K is the final concentration after n dilutions. V is the original volume and x is the volume of the solution replaced each time, then  \n18  \n5 km/hr  \nTotal Distance Travelled  Total Time Taken  \n3. When the distance is constant, the average speed is the harmonic mean of the two speeds  \n2S1S2  \nS 􀀠 ~~ ~~  \navg S1 􀀎 S2  \n4. When the time is constant, the average speed is the arithmetic mean of the two speeds.  \nS1 􀀎 S2  \nS 􀀠 ~~ ~~  \navg 2  \n5. D – Speed of the boat downstream U – Speed of the boat upstream B – Speed of the boat in still","cbCaiaOtvoZMwcz8","https://ap.wps.com/l/cbCaiaOtvoZMwcz8","pdf",2418095,14,"English","# Important formulae overview\n## Profit and loss, percentages, interest\n## Counting, ratios, and means\n## Time, speed, distance, boats\n## Alligation and mixtures\n## Number theory (LCM/HCF, factors, totient)","[{\"question\":\"Which algebraic identities are included for simplification?\",\"answer\":\"The document lists standard expansions such as (a+b)^2, (a-b)^2, and identities for a^2-b^2, a^3±b^3, plus a relation involving a^2 and b^2 in a difference-of-squares style.\"},{\"question\":\"How are simple and compound interest formulas represented?\",\"answer\":\"Simple interest is given using P, r%, and n in a direct proportion form. Compound interest provides an amount after n years using a growth factor based on r% p.a., and also includes related profit-growth population-style expressions.\"},{\"question\":\"What speed–distance average concepts and boat-travel relations are provided?\",\"answer\":\"Average-speed rules are included: arithmetic mean when time is constant and harmonic mean when distance is constant. Boat-travel relations define downstream/upstream/speed in still water with equations linking D, U, B, and R, plus derived forms for B and R.\"}]","IMPORTANT FORMULAE FOR COMPETITIVE EXAMS - Quick reference | PDF",35]