The number of people who purchased book A is twice the number of peopl

Set up a mutual exclusive grid and input the numbers.

_______A_____Not A______Total
B______\(500\)___\(250\)______\(\frac{1}{2}x=750\)
NB_____\(y\)
Total___\(x=1500\)

\(500+y=1500\)
IMO D.

Information given :
A=2B ----- (1)
Both A and B = 500 which is twice of only B, therefore only B = 250

Overall B = Only B + Both A and B ----(2)

Solution --
From equation (2), we can find Overall B which is 750 = 250 (Only B) + 500 (Both A and B)

Putting Overall B in equation (1), we can find Overall A = 1500

Overall A = Only A + Both A and B
Hence, 1500 = Only A + 500
and Only A = 1500 - 500 i.e. 1000

The number of people who purchased book A is twice the number of people who purchased book B. The number of people who purchased both books A and B is 500, which is twice the number of people who purchased only book B. What is the number of people who purchased only book A?

A. 250
B. 500
C. 750
D. 1000
E. 1500

a = book A, b = book B, c = books A and B

a + c = 2 (b+c)
a+c =2b + 2c
a = 2b + c

c = 500 = 2b => b=250

a = 2(250) + 500 = 1000

[quote="Bunuel"]The number of people who purchased book A is twice the number of people who purchased book B. The number of people who purchased both books A and B is 500, which is twice the number of people who purchased only book B. What is the number of people who purchased only book A?

A. 250
B. 500
C. 750
D. 1000
E. 1500



This is best solved using overlapping sets or a Venn Diagram. We know that A = 2B, and that 500 people purchased both A and B. Further, those purchasing both was double those purchasing B only. This gives us 250 people purchasing B only. With the 500 that pruchased both, we have a total of 750 that purchased B and this is 1/2 of those that purchased A. So, 1500 purchased A. Less the 500 that purchased both, 1000 purchased A only. (This is much simpler to solve using the Venn diagram).

Correct answer is D. 1000

800score Official Solution:

The original Venn diagram is the following:


(D) First, let's define some values:
B = number of people who purchased book B
2B = number of people who purchased book A
500 = number of people who purchased both A and B
500/2 = 250 = number of people who purchased only B.
We are looking for the number of people who purchased ONLY book A. The number of people who purchased ONLY A is the total number of people that purchased A minus the number of people that purchased both A and B.
Since 250 people purchased ONLY B, and 500 people purchased both books, 750 people in all must have purchased B.
Thus, 1500 people in all purchased book A.
Since 500 of these people purchased both books, the number who have purchased ONLY A is:
1500 – 500 = 1000.
The correct answer is choice (D).

The solved Venn diagram is the following:

i find it much easier to use the table when total number of items is not given directly in the questions stem.
the venn diagram is often quite interesting when lots of calculations are not required.

A=2B
Purchased A+B = 500 each
Purchased B = 250
B Total 500 + 250 = 750
A = 2B, then A = 1500
1500 - 500 (from the 500 that purchased both) = 1000
Correct Answer is 1000

Let the no. of people who purchased only book A = a
Let the no. of people who purchased only book B = b
Let the no. of people who purchased both book A&B = c
According to given problem,
a+c=2(b+c)------1
c=500=2b-------2
solving 2
b=250
solving 1
a+c=2b+2c
a=2b+c
a=500+500=1000
Hence D