50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS


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1 0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS BASIC KNOWLEDGE () RATIOS: Rtios re use to ompre two or more numers For n two numers n ( 0), the rtio is written s : = / Emple : If 4 stuents in lss of 0 stuents were present, wht perent of the stuents were sent? (Mthounts Hnooks) Solution: 0% The numer of stuent who were sent: 0 4 = 6 6/ 0 = 0 = 0% Emple : There re 6 girls in lss of 0 stuents Fin the rtio of girls to os Epress our nswer s ommon frtion (Mthounts Hnooks) Solution: 8/7 The numer of os in the lss is 0 6 = 4 6/4 = 8/7 Properties of rtios: The first term of rtio n e n numer The seon term n lso e n numer eept ero If the two terms re multiplie the sme numer, the rtio oes not hnge : ( ) : ( ) Emple: : 7 ( ) : (7) : If the two terms re ivie the sme numer ( 0), the rtio oes not hnge : ( ) : ( ) Emple : 0 : (0 ) : ( ) : 40
2 0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS 0 Emple : : ( 6) : () : 6 If the totl numer of prts is m = A + B, n A : B = :, then the frtionl prt of is the frtionl prt of is, n A m, n B m If the totl numer of prts is m = A + B +C, n A : B : C = : :, then the frtionl prt of is the frtionl prt of is the frtionl prt of is,, n A m, B m, n C m Emple : A ertin pint olor is rete miing prts of re with ever prts of lue How mn gllons of re pint re neee to mi 40 gllons of this olor? (Mthounts Hnooks) Solution: A m = 40 4
3 0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS Emple : In group of 7 stuents if the rtio of os to girls is :, how mn os re in the group? Solution: 4 A m = 7 4 Emple : Keith ought pper for mking origmi figure He ought pkges of ornge pper, pkges of ellow pper, n pkges of lue pper Wht frtion of the ppers ws lue? Solution: = 0 Emple 4: Keith ought 0 pkges of pper for mking origmi figure The rtio of ornge pper, ellow pper, n lue pper is : : How mn pkges of lue pper i he u? Solution: A m = 0 () RATES: A rte is rtio use to ompre two numers of ifferent units If the seon term of the rtio is, the rte is lle unit rte Emple : Sm rove 00 miles in hours Wht re his rte n the unit rte? Solution: The rte is 00 miles/ hours n the unit rte is 0/ or 0 miles per hour Emple : Mihel tpes 0 wors in 0 minutes How mn hours will it tke him to tpe 700 wor pper? (Mthounts Hnooks) Solution: 0 4
4 0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS The unit rte is 0 0 = wors per minute The time to tpe 700 wors is 700 = 600 minutes = 0 hours Emple : A r gets 7 miles per gllon How mn miles will it go on 9 gllons of gs? (Mthounts Hnooks) Solution: 4 The numer of miles will the r go is 7 9 = 4 Emple 4: A sketll pler mkes 80% of the shots he ttempts in eh gme In ertin gme, he me 0 of his shots How mn shots i he ttempt in the gme? (Mthounts Hnooks) Solution: Let e the totl numer of shots he me 08 = 0 = Emple : A pork rost shoul e ooke 0 minutes per poun How mn hours shoul 6poun rost e ooke? (Mthounts Hnooks) Solution: The numer of hours it tkes is 0 6 = 00 minutes = 6 hours () PROPORTIONS: A proportion is n eqution of two rtios For emple,,, n or we know n the vlue of / Properties of Of Proportion: We n fin if we know Propert :, is equivlent to :,, 4
5 0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS Emple : Emple : The rtio of with to length of retngulr room is 4 n the with is 7 8 Wht is the length? 0 Solution: L W L W Propert : If, then n Emple: (= 4) n (= ) Propert : If, then Emple: ( =) 4 0 Propert 4: If n n Then n n Emple : Fin if 0 Solution: 0 ( ) ( ) = 44
6 0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS 4 Emple : If for three positive numers,, n, ll ifferent, then wht is the vlue of? (99 AMC) Solution: ) ( ) ( ) ( Proof of properties () n () Sine, we hve () We lso hve () () (), (note n ) () (4) CONTINUED RATIO The rtio of three or more quntities is lle the ontinue rtio For emple, :: is omintions of three seprte rtios :, :, n : () If : : = : : 4, then : = :, : = : 4, n : = 4 : () If : = :, : = : 4, n : = 4 :, then : : = : : 4,
7 0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS () If : = :, n : = : 4 (note ), then : : = ( ) : ( ) : ( 4) = 0 : : Emple : Three numers,, n in the rtios of : = : 4 n : = : 6 hve sum of 8 Wht re the vlues of,, n? Solution: = 0, = 40, n = 48 Metho : : : 4 : 0 n : : 6 0 : 4 B the propert of the ontinue rtio, we get: : : : 0 : 4 We lso know tht 8, so 8 0, 0 4 Similrl, = 40, n = 48 Metho : : : 4 n : : 6 B the propert of the ontinue rtio, we get: : : = : 0 : 4 so 8 0, n = 40, n =
8 0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS Emple : Mhine A n fill o of nils in 6 minutes Mhine B n fill o of nils in 9 minutes The strte to work t the sme time n the stoppe lso t the sme time Totl the fille 00 oes How mn were fille mhine A? Solution: 60 Metho : Mhine A woul fill oes of nils in 8 minutes Mhine B woul fill oes of nils in 8 minutes So the rtio of their work is : The numer of oes fille mhine A is: Metho : Sine the rtio of their work is :, let the numer of oes fille mhines A e, n the numer of oes fille mhines B e + = 00 = 0 = 60 Emple : Ale pi $94 to trnsport his nimls ferr The osts re $, $ n $ for eh ts, og, n squirrel, respetivel The rtios of ts to ogs is : 9, n og to squirrel : 7 How mn ts were there? Solution: The rtio of the numer of nimls n e otine s follows: : = : 9 n : s = : 7 : : s = 6 : 7 : 6 = : 9 : Then the rtio of the ost is then: ( ) : ( 9) : ( 7) = : 6 : 7 So the ost for ts is lulte s follows: The numer of ts is 6 = 4 47
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