WORKING AT THE SAME RATES

1. a. In work problems, there are three items involved: the number of people working, the time spent working, and the amount of work done.

b. The number of people working is directly proportional to the amount of work done; that is, the more people on the job, the more of the work will be done, and vice versa.

c. The number of people working is inversely proportional to the time needed to finish the job; that is, the more people on the job, the less time it will take to finish it, and vice versa.

d. The time expended on a job is directly proportional to the amount of work done; that is, the more time expended on a job, the more work that is done, and vice versa.

WORKING AT THE SAME RATES

2. a. When given the time required by a number of people working at same rates to complete a job, multiply the number of people by the time required to find the time required by one person to do the complete job.

Example: If it takes 4 people working at equal rates 30 days to finish a job, then 1 person will take 30 3 4 = 120 days.

b. When given the time required by 1 person to complete a job, to find the time required by a number of people working at equal rates to complete the same job, divide the time by the number of people.

Example: If 1 person can do a job in 20 days, it will take 4 people working at the same rate 20 4 4 = 5 days to finish.

3. To solve problems involving people who work at the same rates:

a. Multiply the number of people by the time to find the time required by 1 person.

b. Divide this time by the number of people actually working.

Problem: Four workers can do a job in 48 days. How long will it take 3 workers to finish the job?

SOLUTION: One worker can do the job in 48 3 4 = 192 days.

Three workers can do the job in 192 4 3 5 64 days.

Answer: It would take 3 workers 64 days.

4. In some work problems, the rates, though unequal, can be equalized by comparison.

To solve such problems:

a. Determine from the facts given how many equal rates there are.

b. Multiply the number of equal rates by the time given.

c. Divide this by the number of equal rates.

Problem: Three workers can do a job in 12 days. Two of the workers work twice as fast as the third. How long would it take one of the faster workers to do the job himself?

SOLUTION: There are 2 fast workers and 1 slow worker. The amount of time required, therefore, would be the same as the amount of time required by 5 slow workers working at equal rates.

One slow worker will take 12 3 5 = 60 days.

One fast worker 5 2 slow workers; therefore he will take 60 4 2 = 30 days to complete the job.

Answer: It will take 1 fast worker 30 days to complete the job.

5. Unit time is time expressed in terms of 1 minute, 1 hour, 1 day, etc.

6. The rate at which a person works is the amount of work he can do in unit time.

7. If given the time it will take one person to do a job, then the reciprocal of the time is the part done in unit time.

Example: If a worker can do a job in 6 days, then he can do1

6of the work in 1 day.

8. The reciprocal of the work done in unit time is the time it will take to do the complete job.

Example: If a worker can do 3

7of the work in 1 day, then he can do the whole job in7

3, or 21 3, days.

9. If given the various times in which each of a number of people can complete a job, to find the time it will take to do the job if all work together:

a. Find the reciprocal of the time of each to find how much each can do in unit time.

b. Add these reciprocals to find what part of the job all working together can do in unit time.

c. Find the reciprocal of this sum to find the time it will take all of them together to do the whole job.

Problem: If it takes A 3 days to dig a certain ditch, whereas B can dig it in 6 days, and C in 12, how long would it take all three to do the job together?

SOLUTION: A can do it in 3 days; therefore, he can do1

3of the job in 1 day.

B can do it in 6 days; therefore he can do1

6of the job in 1 day.

C can do it in 12 days; therefore, he can do 1

12 of the job in 1 day.

1 311

61 1 125 7

12 A, B, and C can do 7

12of the work in 1 day; therefore, it will take them12

7, or 15

7, days to complete the job.

Answer: A, B, and C, working together, can complete the job in 15 7days.

10. If given the total time it requires a number of people working together to complete a job, and the times of all but 1 are known, to find the missing time:

a. Find the reciprocal of the given times to find how much each can do in unit time.

b. Add the reciprocals to find how much is done in unit time by those whose rates are known.

c. Subtract this sum from the reciprocal of the total time to find the missing rate.

d. Find the reciprocal of this rate to find the unknown time.

Problem: A, B, and C can do a job, working together, in 2 days. B can do it in 5 days, and C can do it in 4 days. How long would it take A to do it himself?

SOLUTION: B can do it in 5 days; therefore he can do1

5 in 1 day. C can do it in 4 days; therefore, he can do1

4in 1 day. The part that can be done by B and C together in 1 day is:

1 511

45 9 20

The total time is 2 days; therefore, all together can do1

2in 1 day.

1 22 9

205 1 20 A can do 1

20 in 1 day; therefore, he can do the whole job in 20 days.

Answer: It would take A 20 days to complete the job himself.

11. In some work problems, certain values are given for the three factors—number of workers, the amount of work done, and the time. It is then usually required to find the changes that occur when one or two of the factors are given different values.

One of the best methods of solving such problems is by directly making the necessary divisions and multiplications.

In the following problem, it is easily seen that more workers will be required since more houses are to be built in a shorter time.

Problem: If 60 workers can build four houses in 12 months, how many workers would be required to build six houses in 4 months?

SOLUTION: To build six houses instead of four in the same amount of time, we would need6

4 of the number of workers.

6

4360 5 90

Since we now have 4 months where previously we needed 12, we must triple the number of workers.

90 3 3 5 270

Answer: 270 workers will be needed to build six houses in 4 months.