This one is a problem solving question in rates and is a word problem. Note the crux of solving a word problem is translating words into mathematical expressions and equations. A can complete a task in ‘a’ days and B in ‘b’ days. They take turns in doing the task with each working 2 days at a time. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more. How long does it take to complete the task if they both work together? 46/9 days 50/9 days 50/11 days 36/7 days 210/41 days

1. Word Problem – Rates
Work Time
Problem Solving – Question 6

2. Question
A can complete a task in ‘a’ days and B in ‘b’ days. They take turns in doing the task with each
working 2 days at a time. If A starts they finish the task in exactly 10 days. If B starts, they take half
a day more. How long does it take to complete the task if they both work together?
A.
46
9 days
B.
50
9 days
C.
50
11 days
D.
36
7
days
E.
210
41
days

3. ◴What is the approach?

4. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
1. Rate and Time
Understand relation
between time taken
and rate

5. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
1. Rate and Time
Understand relation
between time taken
and rate
2. Words to equations
Translate what is given in
words to mathematical
expressions and
equations

6. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
1. Rate and Time
Understand relation
between time taken
and rate
2. Words to equations
Translate what is given in
words to mathematical
expressions and
equations
The crux of solving a
word problem is this
step

7. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
1. Rate and Time
Understand relation
between time taken
and rate
2. Words to equations
Translate what is given in
words to mathematical
expressions and
equations
3. Solve the equations
Solve the equations to
find the value of the
unknowns.

8. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
1. Rate and Time
Understand relation
between time taken
and rate
2. Words to equations
Translate what is given in
words to mathematical
expressions and
equations
3. Solve the equations
Solve the equations to
find the value of the
unknowns.
4. It’s not over yet
Using the values
computed for the
unknown determine
what is asked

9. ◴Understanding relation between rate of work
and time taken

10. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
01 Understanding relation between time taken and rate

11. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
01 Understanding relation between time taken and rate
· If A takes ‘a’ days to complete a task

12. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
01 Understanding relation between time taken and rate
· If A takes ‘a’ days to complete a task A will complete
1
a
of the task in a day

13. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
01 Understanding relation between time taken and rate
· If A takes ‘a’ days to complete a task A will complete
1
a
of the task in a day
Therefore, if A takes 20 days and B takes
30 days to complete a task, it will be
absurd to say that working together they
will take 20 + 30 = 50 days.


14. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
01 Understanding relation between time taken and rate
· If A takes ‘a’ days to complete a task A will complete
1
a
of the task in a day
Therefore, if A takes 20 days and B takes
30 days to complete a task, it will be
absurd to say that working together they
will take 20 + 30 = 50 days.
  In a day A will complete
1
20
th of the task and
B will complete
1
30
th of the task.
Together they will complete
1
20 +
1
30 =
3+2
60 =
1
12
Or they will complete the task in 12 days.

15. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
01 Understanding relation between time taken and rate. Continued….

16. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
01 Understanding relation between time taken and rate. Continued….
Similarly, if A takes ‘a’ days to complete a
task and B takes ‘b’ days to complete a
task and if working together they will
complete the task in 10 days,
it is INCORRECT to say a + b = 10.


17. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
01 Understanding relation between time taken and rate. Continued….
Similarly, if A takes ‘a’ days to complete a
task and B takes ‘b’ days to complete a
task and if working together they will
complete the task in 10 days,
it is INCORRECT to say a + b = 10.
  The correct expression is
1
a +
1
b =
1
10

18. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
01 Understanding relation between time taken and rate. Continued….
Only the rate of work done in a day by two people
working together can be added or subtracted. The time
taken individually should not be added or subtracted.
Similarly, if A takes ‘a’ days to complete a
task and B takes ‘b’ days to complete a
task and if working together they will
complete the task in 10 days,
it is INCORRECT to say a + b = 10.
  The correct expression is
1
a +
1
b =
1
10

19. ◴Words to Mathematical Expressions and
Equations

20. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
02 Words to mathematical expressions and equations

21. 1 A can complete a task in ‘a’ days and B in ‘b’
days
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
02 Words to mathematical expressions and equations

22. 1 A can complete a task in ‘a’ days and B in ‘b’
days
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
A will complete
1
a
of the task in a day
02 Words to mathematical expressions and equations

23. 1 A can complete a task in ‘a’ days and B in ‘b’
days
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
A will complete
1
a
of the task in a day
B will complete
1
b
of the task in a day
02 Words to mathematical expressions and equations

24. 1 A can complete a task in ‘a’ days and B in ‘b’
days
2 They take turns in doing the task with each
working 2 days at a time.
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
A will complete
1
a
of the task in a day
B will complete
1
b
of the task in a day
02 Words to mathematical expressions and equations

25. 1 A can complete a task in ‘a’ days and B in ‘b’
days
2 They take turns in doing the task with each
working 2 days at a time.
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
A will complete
1
a
of the task in a day
B will complete
1
b
of the task in a day
If A works 2 days at a time, A will
complete
2
a
of the task.
02 Words to mathematical expressions and equations

26. 1 A can complete a task in ‘a’ days and B in ‘b’
days
2 They take turns in doing the task with each
working 2 days at a time.
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
A will complete
1
a
of the task in a day
B will complete
1
b
of the task in a day
If A works 2 days at a time, A will
complete
2
a
of the task.
02 Words to mathematical expressions and equations
For e.g., if A works for a total of 5 days, A will complete
5
a
of the task.

27. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
02 Words to mathematical expressions and equations. Continued…

28. 3 If A starts they finish the task in exactly 10
days.
(Note each of them works 2 days at a time)
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
02 Words to mathematical expressions and equations. Continued…

29. 3 If A starts they finish the task in exactly 10
days.
(Note each of them works 2 days at a time)
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
A will work on the following days:
Day 1, day 2, day 5, day 6, day 9, and day 10
02 Words to mathematical expressions and equations. Continued…

30. 3 If A starts they finish the task in exactly 10
days.
(Note each of them works 2 days at a time)
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
A will work on the following days:
Day 1, day 2, day 5, day 6, day 9, and day 10
B will work on the following days:
Day 3, day 4, day 7, and day 8
02 Words to mathematical expressions and equations. Continued…

31. 3 If A starts they finish the task in exactly 10
days.
(Note each of them works 2 days at a time)
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
A will work on the following days:
Day 1, day 2, day 5, day 6, day 9, and day 10
B will work on the following days:
Day 3, day 4, day 7, and day 8
So, A works 6 days and B works 4 days
to complete the task.
02 Words to mathematical expressions and equations. Continued…

32. 3 If A starts they finish the task in exactly 10
days.
(Note each of them works 2 days at a time)
So, the equation is
6
𝑎
+
4
𝑏
= 1 …… eqn (1)
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
A will work on the following days:
Day 1, day 2, day 5, day 6, day 9, and day 10
B will work on the following days:
Day 3, day 4, day 7, and day 8
So, A works 6 days and B works 4 days
to complete the task.
02 Words to mathematical expressions and equations. Continued…

33. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
02 Words to mathematical expressions and equations. Continued…

34. 4 If B starts, they take half a day more.
(Note each of them works 2 days at a time)
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
02 Words to mathematical expressions and equations. Continued…

35. 4 If B starts, they take half a day more.
(Note each of them works 2 days at a time)
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
B will work on the following days:
Day 1, day 2, day 5, day 6, day 9, and day 10
02 Words to mathematical expressions and equations. Continued…

36. 4 If B starts, they take half a day more.
(Note each of them works 2 days at a time)
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
B will work on the following days:
Day 1, day 2, day 5, day 6, day 9, and day 10
A will work on the following days:
Day 3, day 4, day 7, day 8, and half day on day 11.
02 Words to mathematical expressions and equations. Continued…

37. 4 If B starts, they take half a day more.
(Note each of them works 2 days at a time)
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
B will work on the following days:
Day 1, day 2, day 5, day 6, day 9, and day 10
A will work on the following days:
Day 3, day 4, day 7, day 8, and half day on day 11.
So, A works 4.5 days and B works 6 days to complete
the task.
02 Words to mathematical expressions and equations. Continued…

38. 4 If B starts, they take half a day more.
(Note each of them works 2 days at a time)
So, the equation is
4.5
𝑎
+
6
𝑏
= 1 Or
9
2𝑎
+
6
𝑏
= 1 ….. eqn (2)
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
B will work on the following days:
Day 1, day 2, day 5, day 6, day 9, and day 10
A will work on the following days:
Day 3, day 4, day 7, day 8, and half day on day 11.
So, A works 4.5 days and B works 6 days to complete
the task.
02 Words to mathematical expressions and equations. Continued…

39. ◴Solve the two equations

40. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
6
a
+
4
b
=1 …… eqn (1) and
9
2a
+
6
b
=1 …… eqn (2)
03 Solve the two equations
·

41. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
6
a
+
4
b
=1 …… eqn (1) and
9
2a
+
6
b
=1 …… eqn (2)
03 Solve the two equations
·
Multiply eqn (1) by 3:
18
a
+
12
b
=3 … eqn (3)·

42. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
6
a
+
4
b
=1 …… eqn (1) and
9
2a
+
6
b
=1 …… eqn (2)
03 Solve the two equations
·
Multiply eqn (1) by 3:
18
a
+
12
b
=3 … eqn (3)·
Multiply eqn (2) by 2:
18
2a
+
12
b
=2 … eqn (4)·

43. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
6
a
+
4
b
=1 …… eqn (1) and
9
2a
+
6
b
=1 …… eqn (2)
03 Solve the two equations
·
Multiply eqn (1) by 3:
18
a
+
12
b
=3 … eqn (3)·
Multiply eqn (2) by 2:
18
2a
+
12
b
=2 … eqn (4)·
Subtract eqn (4) from eqn (3),
9
a
=1 or a = 9·

44. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
6
a
+
4
b
=1 …… eqn (1) and
9
2a
+
6
b
=1 …… eqn (2)
03 Solve the two equations
·
Multiply eqn (1) by 3:
18
a
+
12
b
=3 … eqn (3)·
Multiply eqn (2) by 2:
18
2a
+
12
b
=2 … eqn (4)·
Subtract eqn (4) from eqn (3),
9
a
=1 or a = 9·
Substitute a = 9 in eqn (3):
18
9
+
12
b
=3 or
12
b
= 1 or b = 12.·

45. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
6
a
+
4
b
=1 …… eqn (1) and
9
2a
+
6
b
=1 …… eqn (2)
03 Solve the two equations
·
Multiply eqn (1) by 3:
18
a
+
12
b
=3 … eqn (3)·
Multiply eqn (2) by 2:
18
2a
+
12
b
=2 … eqn (4)·
Subtract eqn (4) from eqn (3),
9
a
=1 or a = 9·
Substitute a = 9 in eqn (3):
18
9
+
12
b
=3 or
12
b
= 1 or b = 12.·
a = 9; b = 12

46. ◴Last step: Finding time taken when they work
together

47. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
04 Time taken when they work together. A takes 9 days and B takes 12 days.

48. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
In a day A will complete
1
9
th of the task and B will complete
1
12
th of the task.
04 Time taken when they work together. A takes 9 days and B takes 12 days.
·

49. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
In a day A will complete
1
9
th of the task and B will complete
1
12
th of the task.
04 Time taken when they work together. A takes 9 days and B takes 12 days.
·
So, together they will complete
1
9
+
1
12
=
4 + 3
36
=
7
36
th of the task in a day.·

50. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
In a day A will complete
1
9
th of the task and B will complete
1
12
th of the task.
04 Time taken when they work together. A takes 9 days and B takes 12 days.
·
So, together they will complete
1
9
+
1
12
=
4 + 3
36
=
7
36
th of the task in a day.·
Therefore, they will complete the task in
36
7
days.·

51. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
In a day A will complete
1
9
th of the task and B will complete
1
12
th of the task.
04 Time taken when they work together. A takes 9 days and B takes 12 days.
·
So, together they will complete
1
9
+
1
12
=
4 + 3
36
=
7
36
th of the task in a day.·
Therefore, they will complete the task in
36
7
days.·
Choice D is the answer

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