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Word Problem – RatesWork TimeProblem Solving – Question 6
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.469
days
B.509
days
C.5011
days
D.367
days
E.21041
days
◴What is the approach?
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
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
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
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.
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
◴Understanding relation between rate of work
and time taken
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 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
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 1a
of the task in a day
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 1a
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.
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 1a
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
120
th of the task and
B will complete 130
th of the task.
Together they will complete 1
20+
130
=3+260
=1
12
Or they will complete the task in 12 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?
01 Understanding relation between time taken and rate. Continued….
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.
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 1a
+1b
=1
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. 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 1a
+1b
=1
10
◴Words to Mathematical Expressions and
Equations
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
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
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 1a
of the task in a day
02 Words to mathematical expressions and equations
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 1a
of the task in a day
B will complete 1b
of the task in a day
02 Words to mathematical expressions and equations
1 A can complete a task in ‘a’ days and B in ‘b’days
2 They take turns in doing the task with eachworking 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 1a
of the task in a day
B will complete 1b
of the task in a day
02 Words to mathematical expressions and equations
1 A can complete a task in ‘a’ days and B in ‘b’days
2 They take turns in doing the task with eachworking 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 1a
of the task in a day
B will complete 1b
of the task in a day
If A works 2 days at a time, A will
complete 2a
of the task.
02 Words to mathematical expressions and equations
1 A can complete a task in ‘a’ days and B in ‘b’days
2 They take turns in doing the task with eachworking 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 1a
of the task in a day
B will complete 1b
of the task in a day
If A works 2 days at a time, A will
complete 2a
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 5a
of the task.
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…
3 If A starts they finish the task in exactly 10days.
(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…
3 If A starts they finish the task in exactly 10days.
(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…
3 If A starts they finish the task in exactly 10days.
(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…
3 If A starts they finish the task in exactly 10days.
(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…
3 If A starts they finish the task in exactly 10days.
(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…
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…
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…
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…
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…
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…
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…
◴Solve the two equations
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?
6a
+4b
=1 …… eqn (1) and 92a
+6b
=1 …… eqn (2)
03 Solve the two equations
·
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?
6a
+4b
=1 …… eqn (1) and 92a
+6b
=1 …… eqn (2)
03 Solve the two equations
·
Multiply eqn (1) by 3: 18a
+12b
=3 … eqn (3)·
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?
6a
+4b
=1 …… eqn (1) and 92a
+6b
=1 …… eqn (2)
03 Solve the two equations
·
Multiply eqn (1) by 3: 18a
+12b
=3 … eqn (3)·
Multiply eqn (2) by 2: 182a
+12b
=2 … eqn (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?
6a
+4b
=1 …… eqn (1) and 92a
+6b
=1 …… eqn (2)
03 Solve the two equations
·
Multiply eqn (1) by 3: 18a
+12b
=3 … eqn (3)·
Multiply eqn (2) by 2: 182a
+12b
=2 … eqn (4)·
Subtract eqn (4) from eqn (3), 9a
=1 or a = 9·
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?
6a
+4b
=1 …… eqn (1) and 92a
+6b
=1 …… eqn (2)
03 Solve the two equations
·
Multiply eqn (1) by 3: 18a
+12b
=3 … eqn (3)·
Multiply eqn (2) by 2: 182a
+12b
=2 … eqn (4)·
Subtract eqn (4) from eqn (3), 9a
=1 or a = 9·
Substitute a = 9 in eqn (3): 189
+12b
=3 or 12b
= 1 or b = 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?
6a
+4b
=1 …… eqn (1) and 92a
+6b
=1 …… eqn (2)
03 Solve the two equations
·
Multiply eqn (1) by 3: 18a
+12b
=3 … eqn (3)·
Multiply eqn (2) by 2: 182a
+12b
=2 … eqn (4)·
Subtract eqn (4) from eqn (3), 9a
=1 or a = 9·
Substitute a = 9 in eqn (3): 189
+12b
=3 or 12b
= 1 or b = 12.·
a = 9; b = 12
◴Last step: Finding time taken when they work
together
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.
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 19
th of the task and B will complete 112
th of the task.
04 Time taken when they work together. A takes 9 days and B takes 12 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?
In a day A will complete 19
th of the task and B will complete 112
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 19
+112
=4 + 3
36=736
th of the task in a day.·
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 19
th of the task and B will complete 112
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 19
+112
=4 + 3
36=736
th of the task in a day.·
Therefore, they will complete the task in 367
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?
In a day A will complete 19
th of the task and B will complete 112
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 19
+112
=4 + 3
36=736
th of the task in a day.·
Therefore, they will complete the task in 367
days.·
Choice D is the answer
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