work&rate problems&answers

8/9/2019 Work&Rate Problems&Answers

1/15

1.Working, independently X takes 12 hours to finish a certain work. he finishes 2/3 of thework . The rest of the work is finished by Y whose rate is 1/1 th of X. !n how "uch ti"edoes Y finsh his work#

$% 1/12 & t $ 1/3 $% t$' h

2.(unning at their respecti)e constant rates, "achine X takes 2 days longer to produce wwidgets than "achine Y. *t these rates, if the two "achines together produce +/' w widgets in3 days, how "any days would it take "achine X alone to produce 2w widgets#

(w/t w/-t 2 & 3 $ +w/' t$ 0 1w 0 days

2w .# days ?= 12 days

*. '. 0. 4

5. 1E. 12

3.Working together, printer * and printer would finish the task in 2' "inutes. 6rinter * alonewould finish the task in 0 "inutes. 7ow "any pages does the task contain if printer prints +

pages a "inute "ore than printer * #180284381'812+81+

.

(ate ti"e work

X 1/12 12 2/3

9 1/12 t 1/3

(ate ti"e work

X w/t t w

Y w/-t 2 t 2 w

X Y w/t w/-t 2 3 +w/'

(ate ti"e work

* X pg/"in 0 1 job

-X + pg/"in t $ ' 1 job

* -2X + pg/"in 2' 1 :ob


2/15

;tandard Work


3/15

* $ +

*>;W?( *. +

6.when a certain tree was first planted, it was ' feet tall, and the height of the tree increased

by a constant a"ount each year for the neBt 0 years. *t the end of the 0th year, the tree was1/+ taller than it was at the end of the 'th year. y how "any feet did the height of the treeincrease each year#3/12/+1/22/30/+

let ( $ the growth of the tree each year

height in four years $ ' '&( height in siB years $ ' 0&(

therefore,

' 0&($-0/+ &-' '&(

( $ 2/3

7.!f 9i" earns B dollars per hour, it will take hi" ' hours to earn eBactly enough "oney to purchase a particular :acket. !f To" earns y dollars per hour, it will take hi" eBactly + hoursto earn enough "oney to purchase the sa"e :acket. 7ow "uch does the :acket cost#-1 To" "akes 2 C less per hour than 9i" does.-2 B y $ D'3.@+

'&B $ +&y

-1 y $ .4&B y $ 4/1 &B 1 &y $ 4&B$% +&y $ '&B >ET ;F

-2 B y $ D'3.@+ ;F

(ate ti"e work

9 B ' 1 :ob

T y + 1 :ob


4/15

pool was filled by 118 p" and not earlier, when was the drain opened#& at 28 p"& at 283 p"& at 38 p"& at 383 p"

& at '8 p"

Answer1:

let * $ the nu"ber of hours the filling )al)e is open by itself let $ the nu"ber of hours the filling and draining )al)e are open together

;W?(8 ?. 383 6I

Answer 2:!"agine both the drain and the )al)e are on at the sa"e ti"e. You can use a for"ula hereH !

:ust con)ert each to the sa"e a"ount of ti"e -2 hours 8

)al)e will fill + pools in 2 hoursdrain will e"pty ' pools in 2 hours)al)e drain will fill 1 pool in 2 hours

;o we really ha)e two JworkersJ hereH the )al)e, which fills one pool in ' hours, is on for thefirst t hours, and the )al)e drain co"bo, which fills one pool e)ery 2 hours, which was on

for the re"aining 1 t hours. While the )al)e was on, t/' of the pool was filled, and whilethe )al)e drain co"bo were on, -1 t /2 of the pool was filled. *dding these, we "ust get 1 pool8


5/15

t/' -1 t /2 $ 1+t 1 t $ 2't $ 1t $ 2.+

;o the drain was turned on 2.+ hours after 1p", or at 383 p".

Answer3:;o in one hour K1 will fill 1/'th of the poolK2 will e"pty 1/+th of the pool.

!f the )al)es were opened at the sa"e ti"e8 1/2 th of the pool will be full in 1 hour- 5ifference of 1/' and 1/+

;o if both the )al)es were opened at 1 6I it would take 2 hours to fill the pool. ThatJs notthe case. Work fro" the answer choices.

!f K2 was opened at 3 6I. !n 2 hours K1 would ha)e filled 8 1/2 of the tank. To fill there"aining half it would take 1 hours. 3 1 $ 13. ;o the ti"e should be after 3.

!f K2 was opened at 3.3 8 in 2.+ hours, K1 would ha)e filled 1/'&2.+ 8 +/4th of the tank.


6/15

This Guestion "ost rese"bles the one fro" the post abo)e. * will co"plete 1.+ :obs in thefirst 1+ hours.


7/15

& 14& 2& 2'& 3Answer:

(ate ti"e work

* ' /B B '

' /-B ' B ' '

* + /0 0 "in +

We need to find B first, before going for the printer * 4 pages

-' /-B ' -' /B $ -+ /0 $% B$12 $% it takes * 2' "inutes

11. 0 "achines each working at the sa"e constant rate together can co"plete a :ob in 12 days.7ow "any additional "achines, each working at the sa"e constant rate, will be needed toco"plete the :ob in 4 days#Answer 8

(ate Ti"e Work 0 "achines 1/12 12 1

1 "achine has a rate of 1/@2X "achines X & 1/@2 4 1

X/@2 & 4 $1 $% B$LL 0$ 3 additional "achines are needed

12 .*t their respecti)e rates, pu"p *, , and can fulfill an e"pty tank, or pu"p out the fulltank in 2, 3, and 0 hours. !f * and are used to pu"p out water fro" the half full tank, while

is used to fill water into the tank, in how "any hours, the tank will be e"pty#*. 2/3

. 1". 3#45. 3/2?. 2

(ate ti"e work

* 1/2 2 1


8/15

1/3 3 1

1/0 0 1

let X $ M of hours for the tank to e"pty

X&(ate * X&(ate X&(ate $ 1/2 tank

X&-1/2 X&-1/3 X&-1/0 $ 1/2X $ 3/'

!t takes 3/' hrs to e"pty the tank

13 .Working together at their respecti)e rates, "achine *, , and can finish a certain work in4/3 hours. 7ow "any hours will it take * to finish the work independently#

-1 Working together, * and can finish the work in ' hours.-2 Working together, and can finish the work in '4/@ hours.(ate ti"e work

* -original 3w/4 4/3 w

* -state"ent1 w/' ' w

-state"ent2 @w/'4 '4/@ w

* - $3w/4 $% * @w/'4 $ 3w/4 $% *$11w/'4

Answer 2 8let * $ ti"e it takes "achine * to finish the :ob by itself let $ ti"e it takes "achine to finish the :ob by itself let $ ti"e it takes "achine to finish the :ob by itself

;F

*>;W?(8 . ;tate"ent 2 alone is sufficient

14 .9ohn can co"plete a gi)en task in 2 days. 9ane will take only 12 days to co"plete the sa"etask. 9ohn and 9ane set out to co"plete the task by beginning to work together. 7owe)er, 9ane


9/15

was indisposed ' days before the work got o)er. !n how "any days did the work get o)er fro"the ti"e 9ohn and 9ane started to work on it together#

(ate ti"e work

9ohn 1/2 2 1

9ane 1/12 12 1

9ohn 9ane 1/12 1/2 $4/0 *s they donOt worktogether all the ti"ewe cannot write here3@.+

1

Answer1:

let X $ days they need to finish the :ob

4/0 &- X ' 1/2 &' $1 $% X$1

Answer2 8

let X $ days working together

X&-1/@.+ '&-1/2 $ 1 :ob $%X $ 0

therefore, 0 ' $ 1 days1$ .working together at their constant rates , * and can fill an e"pty tank to capacity in1/2hr.what is the constant rate of pu"p #1 *Js constant rate is 2+PT; / "in2 the tanks capacity is 12 lts.

Answer:(ate for * $ 1/3 tank per "inute(ate for *8

1/2+ & 12 $ '4 $% 1/'4 tank per "inute

16 .Pindsay can paint 1/B of a certain roo" in 2 "inutes. What fraction of the sa"e roo" can9oseph paint in 2 "inutes if the two of the" can paint the roo" in an hour, working together attheir respecti)e rates#

* 1/3B3B/B 3B 3/3B

5 B/B 3? B 3/B

(ate ti"e work


10/15

P 1/2 B 2 B 1

9

P 9 1/9 1/2 B 0 "in 1

-1/9 1/2 B &0 $ 1 $% 9$ 0 X/X 3 $% (ate of 9 is8 -B 3 /0 B roo" per "inute!n 2 "inutes8 -B 3 /0 B & 2 $ -B 3 /3B

17 . Iachines X and Y run at different constant rates, and "achine X can co"plete a certain :ob inL hours. Iachine X worked on the :ob alone for the first 3 hours and the two "achines, workingtogether, then co"pleted the :ob in ' "ore hours. 7ow "any hours would it ha)e taken "achineY, working alone, to co"plete the entire :ob#* 14

13 1/2 @ 1/+

5 ' 1/2? 3 2/3

Y$ ti"e it takes for "achineY to co"plete the :ob

1/L &3 -1/L 1/Y & ' $ 1Y$ 14

1% .;iB "achines, each workingat the sa"e constant rate,together can co"plete a certain :ob in 12 days. 7ow "any additional "achines, each working atthe sa"e constant rate, will be needed to co"plete the :ob in 4 days#

* 2 3 '

5 0? 4

Answer 8(ate Ti"e Work

0 "achines 1/12 12 1

1 "achine has a rate of 1/@2X "achines X & 1/@2 4 1

X/@2 & 4 $1 $% B$LL 0$ 3 additional "achines are needed

1 .Iachince * and are each used to "anufacture 00 sprockets. !t takes * 1 hours longer to produce 00 sprockets than "achine . produces 1 percent "ore sprockets per hour than *.

(ate ti"e work

X 1/L L 1

Y 1/Y Y 1

X Y 1/L 1/Y 1


11/15

7ow "any sprockets per hours does "achine * produce#*. 0

. 0.0

. 05. 1

? 11

Answer1 8*$ nr of spokers * produces

per hour * & t $ 00 1,1&*&t N 11 * $ 00

$% * $ 0

Answer2: $ a"ount of ti"e takes to

produce 00 sprockets inhours

produces 1 C "oresprockets per hour than *8

-00 / / -00 / -1 $ 1.1 $% $ 1 therefore, 00 /-1 1 $ 0

2!. * co"pany has two types of "achines, type ( and type ;. Eperating at a constant rate, a"achine of type ( does a certain :ob in 30 hours and a "achine of type ; does the sa"e :ob in14 hours. !f the co"pany used the sa"e nu"ber of each type of "achine to do the :ob in 2hours, how "any "achines of type ( were used#

a 3 b 'c 0d Le 12

X $ nr of "achines-1/30 1/14 &X&2$1

$% X $ 0

21. Ene hour after Yolanda started walking fro" X to Y, a distance of '+ "iles, ob startedwalking along the sa"e road fro" Y to X. !f YolandaJs walking rate was 3 "iles /hour and

obJs was ' "iles / hour, how "any "iles had ob walked when they "et#

* 2' 23

(ate ti"e work

* * T 00

1,1 & * t 1 00

(ate ti"e work

* 00 / 1 1 00

00 / 00

(ate ti"e work ( 1/30 30 1

; 1/14 14 1

( ; 1/30 1/14 2 1


12/15

225 21? 1L.+

Answer1: &ISS A''roa h

;peed of both 83 '$@Y already walked 3 "/h &1h $ 3 "ilesre"aining distance8 '+ 3$'2 "iles

obQs ti"e 8 '2/@ $0 hob walked 8 '"/h & 0h $2' "iles

Answer2:

d1$ )1& t

d2$ )2 & -t 1)1$ ' H )2$3)1 & t )2& -t 1 $ '+ $% t-)1 )2 )2 $ '+

t$ -'+ 3 /@ $% t$0 hob walked 8 '"/h & 0h $2' "iles

)ote that *+ they wo,-d ha e been wa-/*n0 towards ea hother the a''roa h wo,-d ha ebeen the sa e

22. Working alone at its own constant rate, a "achine seals k cartons in 4 hours,and working alone at its own constant rate, a second "achine seals k cartons in' hours. !f the two "achines, each working at its own constant rate and for thesa"e period of ti"e, together sealed a certain nu"ber of cartons, what percentof the cartons were sealed by the "achine working at the faster rate#

* 2+C 33 1/3C + C 66 2#35

? @+C

X $ nr of hours in which they both worked

Work I18 -4/k & XWork I28 -'/k & X

--4/k &X / --4/k &X -'/k &X $ 2/R

23. Iicheal and *da" can do together a piece of work in 2 days. *fter they ha)e worked

together for 12 days Iicheal stops and *da" co"pletes the re"aining work in 1 days. !n how"any days Iicheal co"pletes the work separately.

(ate ti"e work

' "/h t d1

Y 3"/h t 1 d2

Y @"/h '+

(ate ti"e work

I1

4/k 4 k

I2 '/k ' k


13/15

* 4 days 1!! days 12 days

5 11 days? L days

1/2 & 12 1/* &1 $ 1*$2+ $% (ate for * is 1/2+1/2+ 1/I $ 1/2 $% 1/I $1/1I$ 1 days

24. Iatt and 6eter can do together a piece of work in 2 days. *fter they ha)e worked togetherfor 12 days Iatt stops and 6eter co"pletes the re"aining work in 1 days. !n how "any days6eter co"plete the work separately. * 20days

2@days 23days 2$days

? 2' days

1/2 & 12 1/* &1 $ 16$2+ days

2$. * certain car a)erages 2+ "iles per gallon of gasoline when dri)en in the city

and ' "iles per gallon when dri)en on the highway. *ccording to these rates,which of the following is closest to the nu"ber of "iles per gallon that the car a)erages when it is dri)en 1 "iles in the city and then + "iles on thehighway#

24 3 33 30 34Total "iles dri)en8 + 1 $ 0

Total gallons used dri)ing in the city82+ "iles .1gallon1 "iles .B gallonX$1 /2+ $ 2/+ gallon

Total gallons used dri)ing on the highway8' "iles .1gallon+ "iles .B gallonX$+/' gallon

0 "iles / -2/+ +/' gallon $ 30 "iles/ gallon

26 .Working together, 9ohn and 9ack can type 2 pages in one hour. They will be able to type 22 pages in one hour if 9ack increases his typing speed by 2+C. What is the ratio of 9ackJs nor"altyping speed to that of 9ohn#1/3

(ate ti"e work

I 1/I I 1

* 1/* * 1

I * 1/2 2 1

(ate ti"e work

I 1/I I 1

6 1/6 6 1

I * 1/2 2 1


14/15

2/+1/22/33/+

urrently8 9o 9 $ 12!f 9ackQs speed increases by 2+C 8

9o 1,2+ 9$229o$129$4

o 9o/9$ 2/3

27. To", working alone, can paint a roo" in 0 hours. 6eter and 9ohn, working independently,

can paint the sa"e roo" in 3 hours and 2 hours, respecti)ely. To" starts painting the roo" andworks on his own for one hour. 7e is then :oined by 6eter and they work together for an hour.ET ;FT

E a '-e :PetQs say that X produced * bottles in 'h $% Y produced */2 products in 3h

X produced */' bottles in 1h;o, for X to produce all the a"ount alone8 */' & W hours $ * */2 $% W$0 hours

2 .Ene s"urf and one elf can build a treehouse together in two hours, but the s"urf wouldneed the help of two fairies in order to co"plete the sa"e :ob in the sa"e a"ount of ti"e. !fone elf and one fairy worked together, it would take the" four hours to build the treehouse.*ssu"ing that work rates for s"urfs, el)es, and fairies re"ain constant, how "any hourswould it take one s"urf, one elf, and one fairy, working together, to build the treehouse#

(ate ti"e work 9ohn

9ack

9o 9 2 pg/h 1 h 2 pg

(ate ti"e work

T 1/0 0 1

6 1/3 3 1

9 1/2 2 1


15/15

-* +/@- 1- 1 /@-5 12/@

-? 22/@let ; $ ti"e it takes a s"urf to finish a treehouselet ? $ ti"e it takes an elf to finish a treehouselet < $ ti"e it takes a fairy to finish a treehouse

1/; 1/? $ 1/21/; 2/< $ 1/21/? 1/< $ 1/'

3 eGuations and three unkowns

sol)e for X8

1/; 1/< 1/? $ 1/X

X $ 12/@

*>;W?(8 5. 12/@

3!. o"pany ; produces two kinds of stereos8 basic and deluBe. Ef the stereos produced byo"pany ; last "onth, 2/3 were basic and the rest were deluBe. !f it takes @/+ as "any

hours to produce a deluBe stereo as it does to produce a basic stereo, then the nu"ber of hours it took to produce the deluBe stereos last "onth was what fraction of the totalnu"ber of hours it took to produce all the stereos#*.@/1@

.1'/31

. @/1+5.1@/3+?.1/2

@/+ &t & 1/3 & X @/1+ &t&B $ $% @/1@

@/1+&t&B 2/3&t&X 1@/1+&t&X

(ate ti"e work

b t 2/3 & X

d @/+ &t 1/3 & X

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