A would takes 6 hours to paint a wall working alone. B takes 4 hours. C 3 hours
A starts painting at noon, at 1.00 pm B joins him, C at 1.30 pm. At what time do they finish the painting ?
Let t = the time in hours after 1:30PM until the painting is finished.
Make this chart and fill in t for the time after 1:30PM until the painting
is done.
Walls painted | Time | Rate in Walls/hour
A working alone | | |
B working alone | | |
C working alone | | |
A from 12 to 1PM | | |
A&B from 1 to 1:30PM| | |
A&B&C after 1:30PM | | t |
>>...A would takes 6 hours to paint a wall working alone.
B takes 4 hours. C 3 hours...<<
Fill in the times for A, B, and C and 1 for the number of walls painted
in those times.
Walls painted | Time | Rate in Walls/hour
A working alone | 1 | 6 |
B working alone | 1 | 4 |
C working alone | 1 | 3 |
A from 12 to 1PM | | |
A&B from 1 to 1:30PM| | |
A&B&C after 1:30PM | | t |
Next fill in the Walls/hour rates for A, B and C individually, by
dividing their number of walls painted, which is 1, by their times
in hours to paint 1 wall working alone:
Walls painted | Time | Rate in Walls/hour
A working alone | 1 | 6 | 1/6
B working alone | 1 | 4 | 1/4
C working alone | 1 | 3 | 1/3
A from 12 to 1PM | | |
A&B from 1 to 1:30PM| | |
A&B&C after 1:30PM | | t |
Fill in A's rate of 1/6 walls/hour also when he paints from 12 noon to 1PM
and 1 hour for the time:
Walls painted | Time | Rate in Walls/hour
A working alone | 1 | 6 | 1/6
B working alone | 1 | 4 | 1/4
C working alone | 1 | 3 | 1/3
A from 12 to 1PM | | 1 | 1/6
A&B from 1 to 1:30PM| | |
A&B&C after 1:30PM | | t |
Add A's rate and B's rate to find their combined rate:
1/6 + 1/4 = 2/12 + 3/12 = 5/12 and 1/2 hour for their time working
together from 1PM to 1:30PM
Walls painted | Time | Rate in Walls/hour
A working alone | 1 | 6 | 1/6
B working alone | 1 | 4 | 1/4
C working alone | 1 | 3 | 1/3
A from 12 to 1PM | | 1 | 1/6
A&B from 1 to 1:30PM| | 1/2 | 5/12
A&B&C after 1:30PM | | t |
Add A's, B's and C's rate to find the combined rate for all three
after 1:30PM
1/6 + 1/4 + 1/3 = 2/12 + 3/12 + 4/12 = 9/12 = 3/4
Walls painted | Time | Rate in Walls/hour
A working alone | 1 | 6 | 1/6
B working alone | 1 | 4 | 1/4
C working alone | 1 | 3 | 1/3
A from 12 to 1PM | | 1 | 1/6
A&B from 1 to 1:30PM| | 1/2 | 5/12
A&B&C after 1:30PM | | t | 3/4
Fill in the fraction of a wall painted by A alone from 12 noon to 1PM
by multiplying his rate times his time: (1/6)(1) = 1/6
Walls painted | Time | Rate in Walls/hour
A working alone | 1 | 6 | 1/6
B working alone | 1 | 4 | 1/4
C working alone | 1 | 3 | 1/3
A from 12 to 1PM | 1/6 | 1 | 1/6
A&B from 1 to 1:30PM| | 1/2 | 5/12
A&B&C after 1:30PM | | t | 3/4
Fill in the fraction of a wall painted by A&B from 1PM to 1:30PM
by multiplying their combined rate times their time: (1/2)(5/12) = 5/24
Walls painted | Time | Rate in Walls/hour
A working alone | 1 | 6 | 1/6
B working alone | 1 | 4 | 1/4
C working alone | 1 | 3 | 1/3
A from 12 to 1PM | 1/6 | 1 | 1/6
A&B from 1 to 1:30PM| 5/24 | 1/2 | 5/12
A&B&C after 1:30PM | (3/4)t | t | 3/4
Fill in the fraction of a wall painted by A&B&C after 1:30PM
by multiplying their combined rate times their time: (3/4)t
The equation is obtained by adding the three fractions
of a wall painted beginning at 12 noon, and setting that
sum equal to 1 wall:
1/6 + 5/24 + (3/4)t = 1
Clear of fractions by multiplying through by 24:
4 + 5 + 18t = 24
9 + 18t = 24
18t = 15
t = 15/18
t = 5/6 of an hour
5/6 of an hour is (5/6)(60) = 50 minutes. So the painting was
finished at 50 minutes after 1:30PM or 2:20PM.
Edwin
