Rate Concept (Labour)

Additive Rate
James takes 6 hours to paint a house. If Diana helps him, they would take 4 hours. How long will Diana take to paint the house herself?


James (Rate) = 1 house / 6 hours = 1/6 house/hour
Diana (Rate) = 1/4 - 1/6 = 1/12 house/hour (Total Concept)
James + Diana (Rate) = 1 house / 4 hours = 1/4 house/hour
Diana (Time) = 1 house / (1/12 house/hour) = 12 hours
Algebra Method:
1 J = 1/6 house
1/12 house = 1 D
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Single Combo + Gap Concept
Desmond (Rate) = 1/9 kitchen/day
If ALL 13 days are done by Philip:
Gap Amount = 1 kitchen - 13/21 kitchen = 8/21 kitchen
Philip (Time) = (8/21 kitchen) / (4/63 kitchen/day)
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Double Combos + Elimination
Double Combos + Elimination Method
4 C + 2 E = 44
4 E = 84 - 44 = 40 computers
3 E = 30 computers
2 C = 42 - 30 = 12 computers
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Given: Paired Rate
Find :

First
Ahmad and Halim together took 5 days to paint their house. If Ahmad and Halim work together for 2 days, followed by Ahmad working alone for 8 days, Halim will take 1 more day to complete the remaining work. How long will Ahmad take to paint the house all by himself?

Ahmad + Halim (Rate) = 1/5 house /day
(First 2 days) (Together)
2 A + 2 H = 2/5 house

Remaining Amount of House = 3/5 house
8 A + 1 H = 3/5 house
1 A + 1 H = 1/5 house
7 A = 2/5


1 house = 7 A /2 x 5 = 17.5 days


Second
Amin and Haiqing together took 5 days to paint their house. If Amin and Haiqing work together for 2 days, followed by Amin working alone for 8 days, Haiqing would take 1 more day to complete the remaining work. How long will Amin take to paint the house all by himself?

5 A + 5 H = 5/5 house
1 A + 1 H = 1/5 house
2 A + 2 H = 2/5 house

Remainder = 1 - 2/5 = 3/5 house
8 A + 1 H = 3/5 house
1 A + 1 H = 1/5 house
7 A = 2/5 house

1 house = 5/5 house = 7 A / 2 x 5 = 17.5 A

Amin will take  10.5 days to paint the house by himself


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If Allan and Bob work together, they can complete a job in 12 days. If they work together for 3 days, followed by Bob who worked alone for the next 5 days, they can finish 5/12 of the job. How many days will each of them need to complete the job if they were to work alone?
(Ans: Bob 30 days, Alan 20 days)
12 A + 12 B = 1 job
3 A + 3 B + 5 B = 5/12 job
32 B - 12 B = 5/3 job - 1 job
Bob takes 30 days to finish the job alone
30 B = 1 job
12 A = 1 job - 2/5 job = 3/5 job
Ahmad takes 20 days to complete the job alone 
Triple Combos
John and Rauf take 4 days to build a model train. Rauf and Sean take 6 days to build the same train while John and Sean take only 3 days. How long would the three of them take to build the same model train?
1 J + 1 R = 1/4 train
1 J + 1 R + 1 R + 1 S + 1 J + 1 S = 1/4 + 1/6 + 1/3
1 / 3/8 = 8/3 = 2 2/3 days
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1 A + 1 B = 1/4 aircraft






1 J + 1 D = 1/4 house
1 D = 1/4 - 1/6 = 1/12 house

1 house = 12 D


Desmond takes 9 days to renovate 1 kitchen while Philip takes 21 days to renovate the same kitchen. If Philip starts renovating the kitchen first and leaves the rest to be completed by Desmond, they will take 13 days in total to complete renovating the kitchen. How many days does Desmond spend on the job?

Philip (Rate) = 1/21 kitchen/day
Gap Rate = 1/9 - 1/21 = 4/63 kitchen/day

Amount of Kitchen by Philip
= (1/21 kitchen/day) x 13 days
= 13/21 kitchen = 13/21 kitchen


= 6 days


When Caleb spent 2 days and Ethan spent 3 days assembling computers, they produced 42 sets of PCs. When Caleb spent 4 days and Ethan spent 2 days assembling computers, they produced 44 sets of PCs. How long will each boy require to assemble 30 sets of computers?

2 C + 3 E = 42 computers
4 C + 6 E = 84 computers


1 E = 10 computers
Ethan can assemble 10 computers in 1 day

Ethan can assemble 30 computers in 3 day

1 C = 6 computers
Caleb can assemble 6 computers in 1 day
5 C = 30 computers
Caleb can assemble 30 computers in 5 days




Amount of House = 1/5 x 2 = 2/5 house


16 A + 2 H = 6/5 house








3 A + 8 B = 5/12 job
12 A + 32 B = 5/3 job

20 B = 2/3 job
1 job = 30 B


3 B = 1/10 job
12 B = 4/10 = 2/5 job

1 job = 20 A


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1 R + 1 S = 1/6 train
1 J + 1 S = 1/3 train

2 J + 2 R + 2 S = 3/4 train
1 J + 1 R + 1 S = 3/8 train

Abel and Ben can build a model aircraft together in 4 days. Ben and Calvin can do the same job together in 1 1/2 times as many days. Abel and Calvin can complete the same job together in twice the number of days Ben and Calvin can do together. If the three boys decide to work together, how long will they take to complete their job?

4 days x 1 1/2 = 6 days
1 B + 1 C = 1/6 aircraft
6 days x 2 = 12 days
1 A + 1 C = 1/12 aircraft

2 A + 2 B + 2 C = 1/4 + 1/6 + 1/12 = 1/2 aircraft
4 A + 4 B + 4 C = 1 aircraft

They would take 4 days