## Prologue

Speed Time work questions routinely appears in almost every aptitude exam

A can complete a piece of work in 10 days, B can do it in 12 days, C can to it in 15 days. Now A starts the work and continues for 2 days, then B joins and then C joins ….etc.etc. how much time will it take?

• Sometimes, Instead of character names, they’ll give your three pipes. Pipe A can fill the tank in X minutes, Pipe B can do in ….But the approach remains the same. (if pipe is emptying then use minus sign in the speed.)
• Such questions take very little time. Just LCM, fill in the table, and use STD formula and done.

# Case: three men start the work simultaneously.

Aiyyar can paint Gokuldham society in 10 days, Bhide Master can do it in 12 days and Champaklal can do it in 15 days. If all three of them start working simultaneously from day 1, how many days will it take for them to finish the painting work?

It is similar to the pipes and cisterns problem.so make a table

 Aiyyar Bhide Champak Together Speed Time 10 12 15 Distance

The “time” for each individual character, is already given in the problem.

Now take the LCM of Time, means that factorize number into its prime numbers (2,3,5,7,11,13,17,19 etc)

10=2×5

12=22×3

15=3×5

Now take each prime number in its highest power and multiply them together.

LCM=22×31×51=60

• So LCM (10,12,15)=60. This is the total distance covered, and in this case it means total work done. i.e. entire surface area of walls of Gokuldham is 60 square meters.
• Whether these characters paint the society individually or together, the area of the walls will remain one and same 60.
 Aiyyar Bhide Champak Together Speed Time 10 12 15 Distance 60 60 60 60

Not only empty box is of speed.

Simply write the number required to get 60 by multiplication.

Example Aiyyer’s time is 10 days. So to get 60 from ten, you’ve to multiply 10 with 6.

## (Alternatively) for Aiyyer’s case

Speed x time = distance

Speed x 10 days = 60 ; because it is already given in the problem that Aiyyar takes 10 days.

Speed = 60/10 = 6.

Write 6, in Aiyyar’s speed box.

Similarly for Bhide and Champak you’ll get 5 and 4 respectively.

 Aiyyar Bhide Champak Together Speed 6 5 4 6+5+4=15 Time 10 12 15 Distance 60 60 60 60

When these three characters work together, their speed will increase: 6+5+4=15.

Now we have to find out the time required,

Again STD formula

Speed  x time = distance

15 x time = 60

Time = 60/15= 4 days.

Final answer: if these three characters work together, they can paint the society in only four days.

# Case: Three Men Start Working One After Another

Same problem, but with little modification

Aiyyar can paint Gokuldham society in 10 days, Bhide Master can do it in 12 days and Champaklal can do it in 15 days. Aiyyar starts painting the society and Bhide joins him after 2 days. Then Both of paint the society together for one day and then Champaklal joins the gang. In this manner, how many total days will be required to finish the paint work?

## Part I: Aiyaar starts the work

The table remains seem up to this part

 Characters Work Aiyyar Bhide Champak Aiyyar starts Speed 6 5 4 6 Time 10 12 15 x2 Distance 60 60 60 =12

It is given that For the first two days, Aiyyar works alone.

So in the last column, fill up Aiyyar’s speed and time accordingly.

Speed x time = distance

6 x 2 =12 ; because Aiyyar’s speed is 6 and he works for 2 days.

Meaning, in the first two days Aiyyar has painted 12 square meters of walls.

How much work remains?

Total 60 minus 12 = 48.

## Part II: Bhide joins

it is given that, after two days Bhide master joins and together Aiyyar and Bhide work for one day. (and then Champak joins)

So make a new column in a table, Aiyyar +Bhide

since they’re working together, this bill will increase and we will do addition : 6 + 5 =11

 Characters Work Aiyyar Bhide Champak Aiyyar starts Bhide Joins Speed 6 5 4 6 6+5=11 x Time 10 12 15 x2 x 1 =Distance 60 60 60 12 =11

Again

Speed x time = distance

11 x 1 =11

Meaning, in one-day Aiyyar and Bhide painted another 11 square meters.

How much work is left??

48 minus 11 =37

## Part III : Champaklal joins

Now Champaklal also joins, so combine speed is 6+5+4=15

and remaining work is 37 We have to find out the time

 Characters Work Progress report Aiyyar Bhide Champak Aiyyarstarts BhideJoins ChampakJoins Speed 6 5 4 6 6+5=11 6+5+4=15 xTime 10 12 15 2 1 ?? Distance 60 60 60 60 11 37 remains

Again STD

Speed x Time =distance

15 x Time = 37

Time = 37/15= approx. 2.5 days

Caution : This “2.5” is not the final answer. Because they’ve asked you to find total number of days required to finish the paint work. So total time is right from the day one when Aiyyar started.

2+1+2.5=5.5

Final Answer: The work will be finished in 5.5 days.

# Case: Share in Daily Wage

Abdul can do a piece of work in 10 days; Bhide in 15 days. They work for 5 days. The rest of the work was finished by Champak in 2 days. If they get Rs.1500 for the whole work. Find the daily wages of Bhide and Champak.

They got Rs.1500 for the whole work, meaning the work is worth Rs.1500 so take it as our distance

Repharse: Abdul can do road-construction of 1500 kms in 10 days!.. and so on

 Abdul Bhide Speed Time 10 15 distance 1500 1500

Apply STD formula and find out speed value of each column

 A B Speed 150 100 xTime 10 15 =distance 1500 1500

Given: they (A and B) worked for five days

 A B A+B Speed 150 100 150+100=250 Time 10 15 5 distance 1500 1500 250 x 5=1250

Combined speed of A+B=250 so in two days they constructed road of 1250 kms.

Remaining work = 1500 minus 1250 = 250kms

This was done by C alone, in two days.

 A B A+B C Speed 150 100 250 Time 10 15 5 2 distance 1500 1500 1250 250

Run STD formula on C’s column and find his speed

 A B A+B C Speed 150 100 250 125 Time 10 15 5 2 distance 1500 1500 1250 250

Their daily wage depends on how much work can they do per day?

Which is actually their speed. E.g. B can construct 100 km roads per day.

So daily wage of B=Rs.100

And daily wage of C=Rs.125

Total daily wage of B and C=100+125=225

# Case: Share in money after work finished?

Q. Abdul alone can do a piece of work in 6 days and Bhide alone in 8 days. They undertook to do it for Rs.3200. With the help of Champak, they completed the work in 3 days. How much is to be paid to Champak?

Total road construction is 3200 kms this time.

 A B Speed Time 6 8 distance 3200 3200

Run STD formula to find speeds

 A B Speed 3200/6 400 Time 6 8 distance 3200 3200

Given: With the help of C, they completed the work in 3 days.

We don’t know the speed of C, assume it is “c” but construct his column and fill remaining data.

 A B C A+B+C Speed 3200/6 400 c (3200/6)+400+c Time 6 8 ?? 3 distance 3200 3200 3200

Run STD formula on third column

[ (3200/6)+400+c ] x 3 = 3200

Solve this equation you get c=400/3

Meaning speed of c=400/3

He works for 3 days.

Speed x time = distance

(400/3) x 3 =distance

400= distance

Meaning C covered 400kms

Which means he was paid Rs.400

# Case: Time equations: set of two people working

Q. A & B can do a piece of work in 12 days , B and C in 15 , C & A in 20 days. How long would each take separately to do the same work?

In the STD table, when people come and go, we can do addition and subtraction in the “SPEED” boxes only and now in the Time boxes.

Now fill up the table.

 A+B B+C C+A SPEED TIME 12 15 20 DISTANCE

TAKE THE LCM (12,  15, 20) =60. Based on that we can find individual speeds using STD formula.

 A+B B+C C+A SPEED 5 4 3 TIME 12 15 20 DISTANCE 60 60 60

Assume that these three entities work together

 A+B B+C C+A (A+B)+(B+C)+(C+A) SPEED 5 4 3 5+4+3=12 TIME 12 15 20 ?? DISTANCE 60 60 60 60

Speed of A+B+C together can be derived using

(A+B)+(B+C)+(C+A)

= 2(A+B+C)=5+4+3=12

Hence speed of A+B+C together =12/2=6

Now STD formula

Speed x time = distance

6 x time = 60

Time = 10

Meaning if A+B+C work together, they can finish work in 10 days. [Although Time of A+B+C, is not asked in the question, but I’ve given it only for explanation]

Now back to the question, we’ve to find individual time. So far our table looks like this-

 A+B B+C C+A A+B+C SPEED 5 4 3 6 TIME 12 15 20 10 DISTANCE 60 60 60 60

Now you can solve it three unknown equation.

Because this is ‘speed’, we can do addition, subtraction and solve it as three unknown equations

Speed of A+B+C=6

Speed of A+B=5

Hence speed of C= 6 –(A+B)= 6-5=1

 A+B B+C C+A A+B+C (A+B+C)-(A+B)=C SPEED 5 4 3 6 6-5=1 TIME 12 15 20 10 ?? DISTANCE 60 60 60 60 60

Speed of C is 1 and he has to cover 60 kilometers alone,

STD formula,

1 x Time = 60

Hence it’ll take him 60 days.

 A+B B+C C+A A+B+C (A+B+C)-(A+B)=C SPEED 5 4 3 6 6-5=1 TIME 12 15 20 10 60 DISTANCE 60 60 60 60 60

Do similar procedure for B and A.

 A+B B+C C+A A+B+C C B A SPEED 5 4 3 6 6-5=1 6-3=3 6-4=2 TIME 12 15 20 10 60 20 30 DISTANCE 60 60 60 60 60 60 60

Final answer, time taken by A,B,C if they work individually= 30,20,60

Run a cross verification to see the answer is correct. Plug the answer values.

 A B C A+B B+C C+A Speed 2 3 1 5 4 3 Time 30 20 60 12 15 20 Dist 60 60 60 60 60 60

Yes it satisfies the statements given in the Question.

# Case: Time Ratios are given

Q. A takes twice as much time as B and thrice as much as C to finish a piece of work. They together finish the work in one day. Find the time taken by each of them to finish the work.

A takes twice as much time as B,

meaning if B takes 1 day, A takes 2 days

So the ratio of time is A:B=2:1

Similarly A:C=3:1

We can write the same thing as

B:A=1:2

A:C=3:1

If we want the three terms ratio, we want mid term (A) to be same in both equation.

In the first eq. A is 2 and in second eq. A is 3. LCM (2,3)=6

Multiply something in both equations so that we get 6 in both equation.

Meaning, Multiply first eq with (2) and second equation with (3)

B:A=3:6

A:C=6:2

Now we can make a three terms ratio

B:A:C=3:6:2

Keep in mind, this is ratio and not ‘absolute number’

So we fill “Time” boxes as

A’s time = 6x

B’s time = 3x

C’s Time = 2x

Run the STD table, fill the data

 A B C A+B+C Speed 1 2 3 6 Time 6x 3x 2x 1 day Distance 6x 6x 6x 6x

LCM of (6x,3x,2x)= 6x

From the STD fomula, we can get the individual speeds of A,B,C,= 1,2,3

Together their speed A+B+C=1+2+3=6

For the last column

Speed x time = distance

6 multiplied with 1 =6x

Hence x= 1

Put the value of x=1 in the table and you get individual time

A will take 6 days individually, B will take 3 days and C will take 2 days.

Cross verification (plug the answers we got)

 A B C A+B+C Speed 1 2 3 6 Time 6 3 2 1 day Distance 6 6 6 6

From the table,

A takes twice as much time as B and thrice as much as C to finish a piece of work

Yes it satisfies the statement given in the question itself.

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