QUANTITATIVE REASONING-SAT MATH WORD PROBLEMS

1. Word Problem –
Rates WorkTime
Problem Solving –

2. Questio
nA 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?
9
A. 46 days
9
B. 50 days
11
C. 50 days
7
D. 36 days
41
E. 210 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 andTime
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 andTime
Understand relation
between time taken
and rate
2. Words toequations
Translate what is given in
words to mathematical
expressions and
equations

6. How long does it take to complete the task if they both
work together?
1. Rate andTime
Understand relation
between time taken
and rate
2. Words toequations
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.
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 andTime
Understand relation
between time taken
and rate
2. Words toequations
Translate what is given in
words to mathematical
expressions and
equations
3. Solve theequations
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 andTime
Understand relation
between time taken
and rate
2. Words toequations
Translate what is given in
words to mathematical
expressions and
equations
3. Solve theequations
Solve the equations to
find the value of the
unknowns.
4. It’s not overyet
Using the values
computed for the
unknowndetermine
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
A will complete
1
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
A will complete
1
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
A will complete
1
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.
  20
In a day A will complete 1 th of the task
and 1
B will complete 30th of the
task.
Together they will complete 1 + 1 = 3+2 = 1
20 30 60 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
1 1
a+ b = 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
1 1
a+ b = 10

19. ◴Words to Mathematical Expressionsand
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
A will complete
1
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
A will complete
1
of the task in a
day
b
B will complete
1
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
A will complete
1
of the task in a
day
b
B will complete
1
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
A will complete
1
of the task in a
day
b
B will complete
1
of the task in a
day
2
If A works 2 days at a time, A
will complete 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
A will complete
1
of the task in a
day
b
B will complete
1
of the task in a
day
2
If A works 2 days at a time, A
will complete a of the task.
02 Words to mathematical expressions and equations
a
For e.g., if A works for a total of 5 days, A will complete
5
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 4
𝑎 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 6
𝑎 𝑏
+ = 1 Or
9 6
2𝑎 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?
03 Solve the two equations
· 6 4 9 6
a + b =1 …… eqn (1) and 2a + b =1 …… eqn (2)

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?
03 Solve the two equations
· 6 4 9 6
a + b =1 …… eqn (1) and 2a + b =1 …… eqn (2)
a b
· Multiply eqn (1) by 3:
18
+
12
=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?
03 Solve the two equations
· 6 4 9 6
a + b =1 …… eqn (1) and 2a + b =1 …… eqn (2)
a b
· Multiply eqn (1) by 3:
18
+
12
=3 … eqn
(3)
2a b
· Multiply eqn (2) by 2:
18
+
12
=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 4 9 6
a + b =1 …… eqn (1) and 2a + b =1 …… eqn (2)
03 Solve the two equations
·
a b
Multiply eqn (1) by 3:
18
+
12
=3 … eqn
(3)
·
2a b
Multiply eqn (2) by 2:
18
+
12
=2 … eqn
(4)
·
9
· Subtract eqn (4) from eqn (3), 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 4 9 6
a + b =1 …… eqn (1) and 2a + b =1 …… eqn (2)
03 Solve the two equations
·
a b
Multiply eqn (1) by 3:
18
+
12
=3 … eqn
(3)
·
2a b
Multiply eqn (2) by 2:
18
+
12
=2 … eqn
(4)
·
9
· Subtract eqn (4) from eqn (3), a =1 or a
=9
Substitute a = 9 in eqn (3):
18
+
12
=3 or
12
= 1 or b =
12.
9 b b
·

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 4 9 6
a + b =1 …… eqn (1) and 2a + b =1 …… eqn (2)
03 Solve the two equations
·
a b
Multiply eqn (1) by 3:
18
+
12
=3 … eqn
(3)
·
2a b
Multiply eqn (2) by 2:
18
+
12
=2 … eqn
(4)
·
9
· Subtract eqn (4) from eqn (3), a =1 or a
=9
Substitute a = 9 in eqn (3):
18
+
12
=3 or
12
= 1 or b =
12.
9 b b
·
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?
04 Time taken when they work together. A takes 9 days and B takes 12 days.
· In a day A will complete
1
th of the task and B will complete
1
th of the
task.
9 12

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?
04 Time taken when they work together. A takes 9 days and B takes 12 days.
· In a day A will complete
1
th of the task and B will complete
1
th of the
task.
9 12
So, together they will complete 9 +
12 =
1 1 4 + 3 7
36 =36th 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?
04 Time taken when they work together. A takes 9 days and B takes 12 days.
· In a day A will complete
1
th of the task and B will complete
1
th of the
task.
9 12
So, together they will complete 9 +
12 =
1 1 4 + 3 7
36 =36th of the task in a
day.
·
7
Therefore, they will complete the task in
36
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?
04 Time taken when they work together. A takes 9 days and B takes 12 days.
· In a day A will complete
1
th of the task and B will complete
1
th of the
task.
9 12
So, together they will complete 9 +
12 =
1 1 4 + 3 7
36 =36th of the task in a
day.
·
7
Therefore, they will complete the task in
36
days.
·
Choice D is the answer

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55. Question---just have a try
The arithmetic mean of the 5 consecutive integers starting with 's' is 'a'. What is the arithmetic mean of 9
consecutive integers that start with
s + 2 ?
A. 2 + s + a
B. 22 + a
C. 2s
D. 2a + 2
E. 4 + a

56. • Consecutive integers are equally and symmetrically
distributed with respect to the middle number.
• i.e., the middle number is the average of a set of
consecutive integers.
• Determine which term is 'a' and then work out the
average of the second set relative to that term.
Hints to solve this GMAT Statistics question

57. • The first sequence of 5 consecutive numbers
starts with 's' and its mean is 'a'
• The mean of 5 consecutive numbers is the
3rd term - the middle term.
• Hence, 'a' the mean is the middle (3rd) term.
Understanding the given data

58. • The first sequence starts with 's'
• Hence, the terms are s, s + 1, s + 2, s + 3, and
s + 4
• The middle term is s + 2
• Therefore, a = s + 2
Relating 'a' to 's'

59. • The second series of 9 consecutive numbers starts from s
+ 2
• The terms will therefore, be s + 2, s + 3, s + 4, s + 5, s + 6,
s + 7, s + 8, s + 9, and s + 10
• The average of these 9 numbers is the middle term of the
second series. i.e., the 5th term = s + 6
• If a = s + 2, then s + 6 will be a + 4
• The average of the second sequence is a + 4
The second series and its mean

60. Alternative Approach
The fastest way to solve

61. Alternative Approach
The fastest way to solve such questions is to assume a value for 's'.
Let sbe 1
Therefore, the 5 consecutive integers that start with 1 are 1, 2, 3, 4, and 5
The average of these 5 numbers is the middle term, which is 3. Hence, a = 3
9 consecutive integers that start with s + 2 will start from 1 + 2 = 3
The second sequence is therefore, 3, 4, 5, 6, 7, 8, 9, 10, and 11
The average of these 9 number is the middle term, which is 7
If the average of the first sequence 3 = a, the average of the second sequence 7 = 4 + a
Choice E is the correct answer.

62. It’s for your try only!
The recommended daily calcium intake for a 20-year-old is 1,000
milligrams (mg). One cup of milk contains 299 mg of calcium and one
cup of juice contains 261 mg of calcium. Which of the following
inequalities represents the possible number of cups of milk m and
cups of juice j a 20-year-old could drink in a day to meet or exceed
the recommended daily calcium intake from these drinks alone?

63. Let’s have a correct choice!
A. 299 m plus 261 j greater or equal than 1 comma 000
B. 299 m plus 261 j greater than 1 comma 000
C. 299 over m plus 261 over j greater or equal than 1 comma 000