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1 2 WORD PROBLEMS STEP 1. LABEL UNKNOWNS a. DRAW A PICT URE , if po ss ib le , or b. MAK E A T ABLE , o r c. US E “LET” STATEMENTS NOTE: I n or de r to de ter mine what the unk no wns a re, s imp ly r ead the q ue stio n. Us ua lly, we le t o ne o f the unk no wns b e x, a nd la be l the othe r unk nowns in te r ms o f x. NOTE: The pr ob le m will give yo u two d is tinc t p ie ce s o f info r ma tio n abo ut the unk no wns. Us e o ne o f the m to La be l the Unk no wns a nd the o ther to Wr ite the Eq ua tio n. 2. WRI TE EQUATION 3. SOLVE EQ UATION 4. CHECK SO LUTIONS a. Ans wer all the questio ns (Go back to where you lab eled the unknowns). b. Check to see that your so lutio ns are feas ib le and do what they are s upposed to. 3 MIXTURE PR OB LEM S P ERCENT P ERCENT P ERCENT x + x = x AMOUNT AMOUNT AMOUNT THE EQUATION I S : (P ERCENT x A MOUNT) + (P ERCENT x AMOUN T) = (P ERCENT x AMOUNT) EXA MP LE : How ma ny ga llons on a 12% sa lt s olut ion mus t be combine d w ith a 42% salt s olut ion to obta in 30 ga llons of an 18% solut ion? 12% 42% 18% x + 30 – x = 30 .12x + .42( 30 – x) = .18( 30) 12x + 42( 30 – x) = 18( 30) 12x + 1260 – 42x = 540 -30x = - 720 x = -720/- 30 x = 24 24 ga llons of t he 12% sa lt solut ion must be used. INVESTMENT P ROBLEM S ARE A TYP E O F MI XTUR E P ROBLEM S. EXA MP LE : If you ha ve tw ice as muc h invested a t 8% as at 5% and if your a nnua l interest inc ome from t hese tw o inves t ments is $315, how muc h is investe d at each rate? 5% 8% x + 2x = 315 .05x + .08( 2x) = 315 5x + 8( 2x) = 31500 5x + 16x = 31500 21x = 31500 x = 31500/ 21 x = 1500 $1500 a t 5% and $3000 a t 8% 4 MOTION PR OB LEM S D R T A B THREE COLU MNS : 1. G IVEN You are given t he two distances , a nd the two rates , or t he tw o t imes. 2. UKNOWN Read t he quest ion t o deter mine t his c olumn. 3. FOR MULA D = RT R = D/T T= D/R THE EQUATION COM ES FROM TH E FOR MULA COLUMN !!! EXA MP LE : The s peed of a strea m is 4 m/ h. A boat trave ls 36 mile s downstrea m in t he same t ime it tra ve ls 1 2 miles ups trea m. F ind t he spee d of t he boat in s t ill water. THREE COLU MNS : 1. G IVEN You are given t he two D ISTANC ES , 36 miles dow nst ream a nd 12 miles upstrea m. 2. UKNOWN You are as ked to f ind the s peed (RATE) of the boat in st il l water. Let t his be x. Kee p in mind tha t a ll e ntr ies in t he ta ble apply to the boa t in t his s tream, w hic h is moving at a s peed of 4 m/ h. Theref ore , t he rate of t he boa t going dow nstrea m is x + 4 m/ h, and the rate of the boa t going ups tream is x – 4 m/ h. 3. FOR MULA The re ma ining c olumn is t he TI ME column, and the f or mula f or t ime is T = D/R ; he nce , t he t ime going dow nst ream is 36/( x+4) and the t ime going upst ream is 12/ (x- 4). D R T DOWNSTREAM 36 x+ 4 36/( x + 4) UP STREAM 12 x– 4 12/( x – 4) 5 S ince the EQUATION c ome fr om the FORMU LA COLUMN , we must rea d t he pr oblem aga in a nd f ind a rela t ion between t he two TI MES . It says t he boa t trave ls downs tream and upstrea m “in t he SAM E TI ME ,” he nce 36 12 CLEAR t he fract ions by cr oss - mult iplying. x4 x4 36(x – 4) = 12( x + 4) Now CLEAR the parent heses by us ing the dis tr ibut ive proper ty. 36x – 144 = 12x + 48 Next , I SOLATE the “x-ter ms ,” 36x – 12x = 48 + 144 …and c ombine like ter ms. 24x = 192 Now ISOLA TE t he x by dividing bot h s ides by its coef f ic ie nt , x = 192/24 …and s implif y. x= 8 The s peed of t he boat in s t ill water is 8 m/h. 6 WORK PROB LEM S ALTERNATE METHOD (Necessar y f or s ome W or k P roble ms afte r DSM 086) : TIME RATE TIME P ART ALONE WORKING COMP LETED A B THE EQUATION COM ES FROM TH E LAS T COLUMN. THE TWO P ART S COMP LETED ADD UP TO 1 CO MP LETED JOB. EXA MP LE : In a certa in pos t off ice , A lice can s ort a stac k of ma il in 30 minutes; B ob ca n sort t he same stack in 40 minutes. If t he y wor k t oget her , how fas t can t he y sort t he stac k? TIME RATE TIME P ART ALONE WORKING COMP LETED Alice 30 1/ 30 x x/ 30 Bob 40 1/ 40 x x/ 40 x/ 30 + x/ 40 = 1 40x + 30x = 120 70x = 120 x = 120/70 x = 17.1 min Wor king t oge ther , the y can s or t t he stack of ma il in about 17 min. 7 WORK PROB LEM S Let t 1 = t ime it ta kes f or t he f irst part ic ipa nt t o do t he job t 2 = t ime it takes f or the sec ond part ic ipa nt to do t he job etc… T = t ime it takes f or a ll t he par t ic ipa nts to do t he job w or king t oge ther 1 1 1 FORMULA : t t 1 2 T NOTE : If a part ic ipant is w or king a ga inst the ot hers , t he n t he fract ion 1/t for t hat part ic ipa nt is negat ive ins tead of pos it ive. EXA MP LE : In a certa in pos t off ice , A lice can s ort a stac k of ma il in 30 minutes; B ob ca n sort t he same stack in 40 minutes. If t he y wor k t oget her , how fas t can t he y sort t he stac k? 1 1 1 30 40 x 40x + 30x = 120 70x = 120 x = 120/70 x = 17.1 min Wor king t oge ther , the y can s or t t he stack of ma il in about 17 min. 8 MIXTURE PR OB LEM S 1. How ma ny quarts of crea m conta ining 36% butterfa t mus t be a dde d t o 100 quar ts of milk c onta ining 3% but terfat to pr oduce 4% milk? 2. An automobile ra diat or c onta ins 12 quarts of a 20% ant ifree ze and water solut io n. H ow much must be re placed by 80% ant ifree ze t o ma ke a 30% solut ion? 3. Al investe d $10,000 in tw o sa vings acc ounts , one t hat pa id 4% and t he ot her 5% . If his t ota l earnings in one year was $480, the n how much was investe d at each rate? 9 MOTION PR OB LEM S 1. One hour and 20 minutes af ter a hiker s tarte d dow n a mounta in tra il a ranger started a fter him on horsebac k and over took him in 40 minutes. F ind t he s peed of each if t he ranger r ode 6 miles per hour faster tha n t he man wa lked. 2. A la dy dr ove t o a c it y 30 miles away t o s hop a nd returne d home in t he eve ning. She s pent 15 minutes longer dr iving t he ret ur n t r ip t han in the going, a nd she dr over three -four ths as fast whe n ret ur ning as she did w hen going to the c it y. How long did it take her t o dr ive t o t he c ity? 10 WORK PR OB LEM S 1. A boy could s pade his gar den in 6 hours , w hile his fat her c ould s pa de it in 4 hours. H ow long w ould it take the two w or king toget her to s pade the gar den? 2. It ta kes Ash le y t hree t imes as long t o pa int a r oom as it does Aar on. Wor king toget her , t hey ca n pa int t he r oom in 6 hours. H ow long w ould it take Aar on to pa int t he r oom? 3. A la borat or y instr uct or ca n pre pare t he solut ions f or a certa in c hemica l demonstra t ion in 1 hour , w hile his ass ista nt requires 1 ½ hours t o prepare t he same solut ions. A fter the y ha d w orke d t oget her f or 30 minutes , t he inst ruc tor had to meet a c lass a nd the ass ista nt f inishe d t he w ork. H ow long did it ta ke him? 11 WORD PROBLEMS Age P r oble ms 1. R ic k is 8 year s older t ha n S te ve. T he s um of t he ir a ges is 46 year s. H ow old is each now? (R ic k - 27, Ste ve - 19) 2. C har les is 4 year s older t ha n his w ife D ia na. T he s um of t he ir a ges is 52. W ha t are t he ir pre se nt a ges? ( C har les - 28, D ia na - 24) Age P r oble ms – 2 t ime f ra mes 1. S ue a nd J oa n a re s is ter s. J oa n is 7 yea rs older t ha n S ue . I n 3 year s J oa n’s a ge w ill be t w ic e S ue’ s a ge. W ha t a re t he ir pre se nt a ges? ( S ue - 4, J oa n - 11) 2. Se lma is now 3 t imes a s old as J oyce. F our year s a go, Se lma was 4 t ime s as old as J oyce was t he n. F ind t he ir prese nt a ges . ( Se lma - 36, J oyce - 12) 3. Mr s. Bar r y is 20 yea rs older t ha n Mrs. C ook. S ixt ee n yea rs a go, Mrs . B arr y was 3 t imes a s old a Mr s. C ook wa s t he n. F ind t he ir pres e nt a ges. ( Mrs . Ba rr y - 46, M rs. C ook- 26) 4. A my is 8 yea rs older t ha n her pe t ha ms ter Ge or ge . I n 2 year s A my’s a ge w il l be 3 t imes Ge or ge ’s a ge. W ha t are t he ir pres e nt a ges? (A my - 10, Ge or ge - 2) 5. Da vid is 3 year s older t ha n his br ot he r J ohn. T hree yea rs a go J ohn’ s a ge was 6 more t ha n one -f our t h of Da vid’ s a ge. W ha t a re t he ir prese nt a ges ? ( Da vid - 15, J ohn- 12) C onsec ut ive I nte ger 1. T he s um of t hr ee c onse c ut ive inte gers is 300. W ha t are t he int e ger s? (99,100,101) 2. T he s um of tw o c onsec ut ive inte ge rs is - 34. W ha t ar e t he inte ge rs? ( - 18, - 16) 3. T he s um of tw o c onsec ut ive odd inte ge rs is 1460. W hat are t he int e ger s? ( 729, 731) 4. T he s um of t hr ee c onse c ut ive inte gers is - 54. W ha t a re t he inte ge rs? ( - 19, - 18, - 17) 12 5. F ind t hree c onsec ut ive int e ger s w it h t he pr ope rt y t ha t 3 t imes t he f ir st inte ger plus t he se c ond inte ge r minus 2 t imes t he t hir d int e ger is 9. ( 6, 7, 8) 6. F ind t hree c onsec ut ive odd inte ge rs w it h t he pr ope rt y t ha t 3 t imes t he f irs t inte ge r plus tw ice t he s ec ond inte ge r is 5 more t ha n tw ice t he t hir d inte ge r. ( 3, 5, 7) P er ime te r 1. T he pe r imete r of a re cta ngle is 21 c m. T he le ngt h is 9. 5 c m gre ate r t ha n t he w idt h. F ind t he w idt h a nd le ngt h of t he rec ta ngle. (w idt h -. 5 c m, le ngt h - 10c m) 2. T he pe r imete r of a re cta ngle is 38 m. T he w idt h is 13 m le ss t ha n t he le ngt h. F ind t he w idt h a nd t he le ngt h. (w idt h - 3m, le ngt h- 16m) 3. T he pe r imete r of a re cta ngle is 64 f t. T he le ngt h of t he r ecta ngle is 3 f t less t ha n 4 t imes t he w idt h. W ha t are t he dime ns ions of t he rec ta ngle? ( w idt h - 7 ft , le ngt h- 25 f it) 4. T he pe r imete r of a re cta ngle is 50 yd. T he w idt h is 1 yd more t ha n one -t hir d of t he le ngt h. W ha t a re t he dime ns ions of t he rec ta ngle? (w idt h - 7 yd, le ngt h- 18yd) 5. T he pe r imete r of a t r ia ngle is 17 c m. T he sec ond s ide is t w ic e t he le ngt h of t he f ir st s ide , a nd t he t hir d s ide is 1 c m less t ha n 3 t imes t he le ngt h of t he f ir st s ide. W hat a re t he le ngt hs of t he s ides of t he t r ia ngle ? ( 3c m, 6c m, 8c m) 6. T he pe r imete r of a re cta ngle is 26 in. If 5 in. are a dde d t o t he w idt h of t he recta ngle , a s qua re w it h per ime te r 36 in. is obta ine d. W hat are t he dime ns ions of t he or ig ina l rec ta ngle? ( w idt h - 4in , le ngt h- 9in) Unknow n numbe r 1. W he n 15 is s ubt rac te d f r om 11 t imes a ce rta in numbe r , t he res ult is - 37. W ha t is t he numbe r? ( - 2) 2. If you double a number a nd t he n a dd 14, you get 5/ 6 of t he or igina l number. W ha t is t he or igina l number? ( - 12) 3. If you s ubt rac t 2/ 3 of a number fr om t he number itse lf , you ge t 7. W ha t is t he numbe r? ( 21) 4. If 5 t imes a number is decr ease d by 8, t he re s ult is t hree t ime s t he number increa se d by 4. F ind t he number. ( 6) 5. T he dif fere nc e bet wee n tw o number s is 24. F ind t he numbe rs is t he ir s um is 88. ( 32, 56) 13 6. T he lar ge r of t w o numbe rs is 5 les s t ha n t w ice t he s ma l ler. If t he ir s um is 70, f ind t he number s. ( 25, 45) 7. A 450 m w ire is c ut int o 3 piec es. T he s ec ond piece is 11 t imes as long as t he f ir st. T he t hir d piece is 3 t imes as long a s t he sec ond. H ow long is eac h piece? ( 10m, 110m, 330m) 8. Jac k has $4. 80 in quar ter s a nd dimes . He has a t ota l of 30 c oins . H ow ma ny qua rte rs a nd how ma ny dimes doe s he ha ve ? ( 18 dime s , 12 qua rt ers ) 9. Ra y has $435 in f ives , te ns , a nd twe nt ie s. He has 2 mor e $10 bil ls t ha n $5 bil ls. H ow ma ny of eac h t ype of bil l does he ha ve? ( 17 f ives , 19 te ns , 8 twe nt ie s) 10. T he dif fere nc e of tw o number s is 7. T he s um of t he number s is - 1. W hat are t he numbe rs? (- 4, 3) M ot ion P r oble ms 1. He nr y lea ve s t ow n a t noon on a bic yc le t ra ve ling w est at a ra te of 15 mph. At 4:00 pm Be tt y lea ves t ow n in a c ar tra ve l ing in t he sa me direct ion a t a rate of 45 mph. H ow ma ny hours w il l it ta ke Be tt y t o ca tc h He nr y? 2. Jennie lea ve s sc hool in a ca r tra ve l ing nor t h a t a rat e of 45 mph. At t he sa me t ime A lber t lea ves sc hool in a ca r t ra ve ling s out h at a ra te of 55 mph. I n how ma ny hours w il l A lber t a nd Je nnie be 250 mi les a par t? ( 2. 5 hour s) 3. A car t ra ve ls f r om one t ow n t o a not he r in 6 hour s. O n t he re t ur n tr ip, t he spe e d is increa se d by 10 mph a nd t he tr ip t a kes 5 hours. F ind t he r ate on t he ret ur n t r ip. H ow fa r a pa rt are t he t ow ns? ( 50 mph, 300 mi) 4. Car ol has 8 hours t o s pe nd on a hike up a mounta in a nd bac k a ga in. S he ca n wa lk up t he tr a il a t a n a ve ra ge of 2 mph a nd ca n wa lk dow n a t a n a vera ge of 3 mph. H ow long s hould s he pla n t o s pe nd on t he uphill par t of t he hike? ( 4 4/ 5 hr ) 5. Ma r ia jogs t o t he c ountr y at a ra te of 10 mph. S he re t ur ns a long t he sa me route at 6 mph. I f t he t ota l tr ip t ook 1 hour 36 minute s , how fa r did s he jog? (6 mi) 6. T om’ s r ow boa t ca n tra ve l 7 mi dow ns trea m in t he sa me t ime t ha t it c a n tra ve l 3 mi upst rea m. If t he s pee d of t he c ur re nt is 1 mph, w ha t is T om’ s row ing s pee d in s t il l w ate r? 14 Invest me nt P r oble ms 1. P hil ha s $20,000, par t of w hic h he inves ts a t 8% int ere st a nd t he res t a t 6% . If his t ota l inc ome was $1,460 f r om t he tw o inves t me nts , how muc h did he inves t a t eac h ra te? ( $13,000 a t 8% , $7,000 at 6% ) 2. Lar r y ma de t w o invest me nt s t ota ling $21,000. O ne inves t me nt ma de him a 13% pr of it , but on t he ot he r inves t me nt he t ook 9% loss . I f his ne t loss was $196, how muc h was eac h inves t me nt? ( $7,700 a t 13% , $13,300 at 9% ) 3. Jane ha d $30,000, par t of w hic h s he inve sts at 9% inte res t a nd t he res t at 7% . If her inc ome fr om t he 7% inves t me nt w as $820 more t ha n t hat f r om t he 9% inves t me nt , how muc h did s he inves t at eac h rat e? ( $8,000 at 9% , $22,000 at 7% ) 4. An inve st me nt of $3,500 is ma de a t a n a nnua l s imple intere st ra te of 13. 2% . How muc h a ddit iona l mone y must be inve ste d at a n a nnua l s imp le inte res t rate of 11. 5% s o t hat t he t ota l inter est ear ne d is $1,037? ( $5,000) M ixt ure P r oble ms 1. How ma ny ounces of a gold a lloy t ha t c osts $320 a n ounce mus t be mixe d w it h 100 oz of a n a l loy t ha t c ost s $100 a n ounc e t o ma ke a mixt ur e t ha t c ost s $160 a n ounce? ( 37. 5 ounce s) 2. How ma ny pounds of pea nuts t hat c os t $2. 25 pe r pound must be mixe d w it h 40 lb of cas he ws t ha t c os t $6. 00 pe r pound t o ma ke a mixt ure t ha t c os ts $3. 50 per pound? ( 80 pounds ) 3. A c he mis t mixes a n 11% ac id s olut ion w it h a 4% ac id s olut ion. H ow ma ny mil l il it ers of ea c h s olut ion s hould t he c he mis t use t o ma ke a 700 ml s olut ion t hat is 6% ac id? ( 200ml of 11% , 500 ml of 4% ) 4. How ma ny quar ts of wa te r mus t be a dde d t o 5 qt of a n 80% a nt ifree ze solut ion t o ma ke 50% a nt ifre e ze s olut ion? ( 3 qt) 5. A tea mixt ure wa s ma de fr om 30 lbs of t ea c os t ing ^ 6. 00 per pound a nd 70 lbs of te a c os t ing $3. 20 pe r pound. F ind t he c os t per pound of t he tea mixt ure. ( $4. 04 pe r pound) W or k P r oble ms 1. Joyc e ca n mow he r yar d in 2 hr. Her s on D a nny ca n mow t he ya r d in 3 hr. W or king t oget her , how long w il l it ta ke t he m t o mow t he ya r d? ( 6/ 5 hr) 2. J immy ca n w as h his ca r in 30 min. H is 4 yea r old s on J immy Jr . ta kes 2 hour s t o wa s h t he car . W or king t oge t he r how long w ill it ta ke t he m t o w as h t he ca r? ( 24 min) 15 3. W or king t oget her , La rr y, J oe , a nd S hir le y c a n wa s h t he w indows at t he M oss C ount r y C ourt house in 1 hr . W or king a lone , Lar r y ca n was h t he w indows in 2 hr a nd J oes ca n was h t he w indows in 3 hr. H ow long does it t a ke S hir le y t o was h t he w indow s? ( 6 hrs ) 4. Gar y ta kes t w ice as long as Ba r bara t o s or t t he M oss C ountr y ma il. T oge t her t he y ca n s or t t he ma in in 8 hr . H ow long does it ta ke eac h of t he m t o s or t t he ma i l a lone? ( Ba r bara - 12 hrs , Gar y - 24 hr s) 5. A sw imm ing pool ca n be f il le d by a n inle t pipe in 12 hr s. T he dra in c a n empt y t he pool in 36 hr s. H ow long w il l it ta ke t o f ill t he pool if t he dra in is lef t ope n? ( 18 hr s)

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