This one is a word problem in algebra and tests your ability to frame equations and mathematical expressions from the information given in words. The concept tested is that of elementary algebraic factorization. Sheep in farm A are made to stand in a square formation; sheep in farm B are made to stand in another square formation. The difference between the number of sheep in the first row of the farms is 4 and the difference in the number of sheep in the two farms is 152. What is the sum of the number of sheep in the two farms? A) 620 B) 730 C) 441 D) 289 E) 480

1. GMAT QUANTITATIVE REASONING
WORD PROBLEMS
ALGEBRA: FACTORIZING
PROBLEM SOLVING
Diagnostic Test

2. Question
Sheep in farm A are made to stand in a square formation; sheep in farm
B are made to stand in another square formation. The difference between
the number of sheep in the first row of the farms is 4 and the difference
in the number of sheep in the two farms is 152. What is the sum of the
number of sheep in the two farms?
A. 620
B. 730
C. 441
D. 289
E. 480

3. Step 1
Express what is stated in words into
equations or expressions

4. Sheep in farm A are made to stand in a square formation; sheep in farm B are made to stand in another square formation.
What is the sum of the number of sheep in the two farms?

5. Step 1: Words to equations
Sheep in farm A are made to stand in a square formation; sheep in farm B are made to stand in another square formation.
What is the sum of the number of sheep in the two farms?

6. Step 1: Words to equations
Sheep in farm A are made to stand in a square formation; sheep in farm B are made to stand in another square formation.
What is the sum of the number of sheep in the two farms?
01 Sheep made to stand in square formation

7. Step 1: Words to equations
Sheep in farm A are made to stand in a square formation; sheep in farm B are made to stand in another square formation.
What is the sum of the number of sheep in the two farms?
01 Sheep made to stand in square formation
In a square formation, the number of sheep in a row is the same as the number of rows.

8. Step 1: Words to equations
Sheep in farm A are made to stand in a square formation; sheep in farm B are made to stand in another square formation.
What is the sum of the number of sheep in the two farms?
01 Sheep made to stand in square formation
In a square formation, the number of sheep in a row is the same as the number of rows.
Let there be ‘a’ rows in farm A. So, there will ‘a’ sheep in each row of farm A.

9. Step 1: Words to equations
Sheep in farm A are made to stand in a square formation; sheep in farm B are made to stand in another square formation.
What is the sum of the number of sheep in the two farms?
01 Sheep made to stand in square formation
In a square formation, the number of sheep in a row is the same as the number of rows.
Let there be ‘a’ rows in farm A. So, there will ‘a’ sheep in each row of farm A.
Therefore, total number of sheep in farm A = a * a = a2

10. Step 1: Words to equations
Sheep in farm A are made to stand in a square formation; sheep in farm B are made to stand in another square formation.
What is the sum of the number of sheep in the two farms?
01 Sheep made to stand in square formation
In a square formation, the number of sheep in a row is the same as the number of rows.
Let there be ‘a’ rows in farm A. So, there will ‘a’ sheep in each row of farm A.
Therefore, total number of sheep in farm A = a * a = a2
Let there be ‘b’ rows in farm B. So, there will ‘b’ sheep in each row of farm B.

11. Step 1: Words to equations
Sheep in farm A are made to stand in a square formation; sheep in farm B are made to stand in another square formation.
What is the sum of the number of sheep in the two farms?
01 Sheep made to stand in square formation
In a square formation, the number of sheep in a row is the same as the number of rows.
Let there be ‘a’ rows in farm A. So, there will ‘a’ sheep in each row of farm A.
Therefore, total number of sheep in farm A = a * a = a2
Let there be ‘b’ rows in farm B. So, there will ‘b’ sheep in each row of farm B.
Therefore, total number of sheep in farm B = b * b = b2

12. The difference between the number of sheep in the first row of the farms is 4 the difference in the number of sheep in the two
farms is 152
What is the sum of the number of sheep in the two farms?

13. The difference between the number of sheep in the first row of the farms is 4 the difference in the number of sheep in the two
farms is 152
What is the sum of the number of sheep in the two farms?
Step 1: Words to equations continued…

14. The difference between the number of sheep in the first row of the farms is 4 the difference in the number of sheep in the two
farms is 152
What is the sum of the number of sheep in the two farms?
Step 1: Words to equations continued…
02 First row difference is 4

15. The difference between the number of sheep in the first row of the farms is 4 the difference in the number of sheep in the two
farms is 152
What is the sum of the number of sheep in the two farms?
Step 1: Words to equations continued…
02 First row difference is 4
Number of sheep in first row of farm A
= a

16. The difference between the number of sheep in the first row of the farms is 4 the difference in the number of sheep in the two
farms is 152
What is the sum of the number of sheep in the two farms?
Step 1: Words to equations continued…
02 First row difference is 4
Number of sheep in first row of farm A
= a
Number of sheep in first row of farm B
= b

17. The difference between the number of sheep in the first row of the farms is 4 the difference in the number of sheep in the two
farms is 152
What is the sum of the number of sheep in the two farms?
Step 1: Words to equations continued…
02 First row difference is 4
Number of sheep in first row of farm A
= a
Number of sheep in first row of farm B
= b
Difference = (a – b) = 4

18. The difference between the number of sheep in the first row of the farms is 4 the difference in the number of sheep in the two
farms is 152
What is the sum of the number of sheep in the two farms?
Step 1: Words to equations continued…
02 First row difference is 4 03 Difference in number of sheep is 152
Number of sheep in first row of farm A
= a
Number of sheep in first row of farm B
= b
Difference = (a – b) = 4

19. The difference between the number of sheep in the first row of the farms is 4 the difference in the number of sheep in the two
farms is 152
What is the sum of the number of sheep in the two farms?
Step 1: Words to equations continued…
02 First row difference is 4 03 Difference in number of sheep is 152
Number of sheep in first row of farm A
= a
Number of sheep in first row of farm B
= b
Difference = (a – b) = 4
Number of sheep in farm A = a2

20. The difference between the number of sheep in the first row of the farms is 4 the difference in the number of sheep in the two
farms is 152
What is the sum of the number of sheep in the two farms?
Step 1: Words to equations continued…
02 First row difference is 4 03 Difference in number of sheep is 152
Number of sheep in first row of farm A
= a
Number of sheep in first row of farm B
= b
Difference = (a – b) = 4
Number of sheep in farm A = a2
Number of sheep in farm B = b2

21. The difference between the number of sheep in the first row of the farms is 4 the difference in the number of sheep in the two
farms is 152
What is the sum of the number of sheep in the two farms?
Step 1: Words to equations continued…
02 First row difference is 4 03 Difference in number of sheep is 152
Number of sheep in first row of farm A
= a
Number of sheep in first row of farm B
= b
Difference = (a – b) = 4
Number of sheep in farm A = a2
Number of sheep in farm B = b2
Difference in number of sheep is
(a2 – b2) = 152

22. Step 2
Algebraic Factorization to determine
a and b

23. What is the sum of the number of sheep in the two farms?
a – b =4 and a2 – b2 = 152

24. What is the sum of the number of sheep in the two farms?
a – b =4 and a2 – b2 = 152
Step 2: Factorize to determine ‘a’ and ‘b’

25. What is the sum of the number of sheep in the two farms?
a – b =4 and a2 – b2 = 152
Step 2: Factorize to determine ‘a’ and ‘b’
a2 – b2 = (a + b)(a – b)

26. What is the sum of the number of sheep in the two farms?
a – b =4 and a2 – b2 = 152
Step 2: Factorize to determine ‘a’ and ‘b’
a2 – b2 = (a + b)(a – b)
a2 – b2 = 152 … eqn (1)

27. What is the sum of the number of sheep in the two farms?
a – b =4 and a2 – b2 = 152
Step 2: Factorize to determine ‘a’ and ‘b’
a2 – b2 = (a + b)(a – b)
a2 – b2 = 152 … eqn (1)
a – b = 4 … eqn (2)

28. What is the sum of the number of sheep in the two farms?
a – b =4 and a2 – b2 = 152
Step 2: Factorize to determine ‘a’ and ‘b’
a2 – b2 = (a + b)(a – b)
a2 – b2 = 152 … eqn (1)
a – b = 4 … eqn (2)
i.e., 152 = (a + b) * 4

29. What is the sum of the number of sheep in the two farms?
a – b =4 and a2 – b2 = 152
Step 2: Factorize to determine ‘a’ and ‘b’
a2 – b2 = (a + b)(a – b)
a2 – b2 = 152 … eqn (1)
a – b = 4 … eqn (2)
i.e., 152 = (a + b) * 4
Or (a + b) =
152
4
= 38 … eqn (3)

30. What is the sum of the number of sheep in the two farms?
a – b =4 and a2 – b2 = 152
Step 2: Factorize to determine ‘a’ and ‘b’
a2 – b2 = (a + b)(a – b)
a2 – b2 = 152 … eqn (1)
a – b = 4 … eqn (2)
i.e., 152 = (a + b) * 4
Or (a + b) =
152
4
= 38 … eqn (3)
Solve equations (2) and (3) to get ‘a’ and ‘b’

31. What is the sum of the number of sheep in the two farms?
a – b =4 and a2 – b2 = 152
Step 2: Factorize to determine ‘a’ and ‘b’
a2 – b2 = (a + b)(a – b)
a2 – b2 = 152 … eqn (1)
a – b = 4 … eqn (2)
i.e., 152 = (a + b) * 4
Or (a + b) =
152
4
= 38 … eqn (3)
Solve equations (2) and (3) to get ‘a’ and ‘b’
Adding (2) and (3), (a – b + a + b) = 4 + 38
Or 2a = 42. a = 21

32. What is the sum of the number of sheep in the two farms?
a – b =4 and a2 – b2 = 152
Step 2: Factorize to determine ‘a’ and ‘b’
a2 – b2 = (a + b)(a – b)
a2 – b2 = 152 … eqn (1)
a – b = 4 … eqn (2)
i.e., 152 = (a + b) * 4
Or (a + b) =
152
4
= 38 … eqn (3)
Solve equations (2) and (3) to get ‘a’ and ‘b’
Adding (2) and (3), (a – b + a + b) = 4 + 38
Or 2a = 42. a = 21
If a = 21, in eqn (2) 21 – b = 4. So, b = 17

33. Finding a and b is not the last step. There is one more to go.
What is the sum of the number of sheep in the two farms?
a – b =4 and a2 – b2 = 152
Step 2: Factorize to determine ‘a’ and ‘b’
a2 – b2 = (a + b)(a – b)
a2 – b2 = 152 … eqn (1)
a – b = 4 … eqn (2)
i.e., 152 = (a + b) * 4
Or (a + b) =
152
4
= 38 … eqn (3)
Solve equations (2) and (3) to get ‘a’ and ‘b’
Adding (2) and (3), (a – b + a + b) = 4 + 38
Or 2a = 42. a = 21
If a = 21, in eqn (2) 21 – b = 4. So, b = 17

34. Step 3
Compute the total number of sheep in
the two farms

35. What is the sum of the number of sheep in the two farms?
a =21 and b = 17

36. What is the sum of the number of sheep in the two farms?
a =21 and b = 17
Step 3: Compute the total number of sheep

37. What is the sum of the number of sheep in the two farms?
a =21 and b = 17
Step 3: Compute the total number of sheep
Total number of sheep in farm A = a2 = 212 = 441

38. What is the sum of the number of sheep in the two farms?
a =21 and b = 17
Step 3: Compute the total number of sheep
Total number of sheep in farm A = a2 = 212 = 441
Total number of sheep in farm B = b2 = 172 = 289

39. What is the sum of the number of sheep in the two farms?
a =21 and b = 17
Step 3: Compute the total number of sheep
Total number of sheep in farm A = a2 = 212 = 441
Total number of sheep in farm B = b2 = 172 = 289
Total number of sheep in both the farms together = 441 + 289 = 730

40. What is the sum of the number of sheep in the two farms?
a =21 and b = 17
Step 3: Compute the total number of sheep
Total number of sheep in farm A = a2 = 212 = 441
Total number of sheep in farm B = b2 = 172 = 289
Total number of sheep in both the farms together = 441 + 289 = 730
730 sheep in both farms

41. What is the sum of the number of sheep in the two farms?
a =21 and b = 17
Step 3: Compute the total number of sheep
Total number of sheep in farm A = a2 = 212 = 441
Total number of sheep in farm B = b2 = 172 = 289
Total number of sheep in both the farms together = 441 + 289 = 730
Choices B is the answer.
730 sheep in both farms

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