A football clubs memorabilia shop sells only shirts and caps.

Question

A football club’s memorabilia shop sells only shirts and caps. On a weekend, the shop sold 260 memorabilia items, with each fan buying either one shirt, one cap, or one of each. If 90 fans bought both a shirt and a cap, how many shirts did the shop sell?

(1) 50 fans did not buy a shirt.
(2) 140 fans bought a cap.

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Bunuel · Accepted Answer

Official Solution:

A football club’s memorabilia shop sells only shirts and caps. On a weekend, the shop sold 260 memorabilia items, with each fan buying either one shirt, one cap, or one of each. If 90 fans bought both a shirt and a cap, how many shirts did the shop sell? 
 
90 fans buying both a shirt and a cap accounts for 90 * 2 = 180 items sold. The remaining 80 items were bought by fans who purchased exactly one item. This makes the total number of fans equal to 90 + 80 = 170.
 
(1) 50 fans did not buy a shirt.
 
Since there are 170 fans, 50 not buying a shirt means the remaining 120 did, and since each could have bought only one shirt, the total number of shirts sold is also 120. Sufficient.
 
(2) 140 fans bought a cap.
 
Since each fan could have bought only one cap, these 140 fans must have bought exactly 140 caps. Given that 260 items were sold in total, the number of shirts sold must have been 260 - 140 = 120. Sufficient.
 
P.S. To give a complete breakdown: 90 fans bought both shirts and caps, 30 bought only shirts, and 50 bought only caps, making a total of 170 fans. Therefore, 90 + 30 = 120 shirts were sold, and 90 + 50 = 140 caps were sold.

Answer: D

Missinga · Answer

.............S..............NS............T
C..........90............. ...........
NC....... ............. ..........
T.......... ............ ......... 260
We need to find S+NC=?

1) 50 fans did not buy a shirt.

.............S..............NS................T
C..........90............. 50 ...........140
NC....... ............. ........... 120
T.......... ............ ......... 260
Insufficient

(2) 140 fans bought a cap.
.............S..............NS............T
C..........90.............( 50)........140
NC....... ............. ..........(120)
T.......... ............ ......... 260
Insufficient

Both (1)&(2) give same information, so clearly insufficient

E

ManifestDreamMBA · Answer

Let number of shirt only purchases be s and cap only purchases be c

We know T = group of only 1 item + both + none
T = 260 and none = 0
Both = 90
We intend to double count the shirt and cap purchases as 2 articles were bought by the 90 fans
s+c+90+90=260
s+c=80

Total shirts = s+90 =?

Need to know s or c to calculate the total shirts sold

Statement 1
If 50 didn't buy a shirt, they bought a cap
c=50
Sufficient

Statement 2
c+90=140
c=50
Sufficient

Answer D

Bunuel · Answer

The answer is correct. However, there is a problem with the solution.

ManifestDreamMBA · Answer

Thanks for pointing it out Bunuel. Can you please share what is the problem? Is it because I haven't used sets strategy to solve this, like t = s+c+b+n, (given s and c are shirt only and cap only purchases)?

Bunuel · Answer

If I understand your solution correctly, you concluded that 50 caps were sold. But we're told that 90 fans bought both a shirt and a cap, so at least 90 caps must have been sold. How is it possible, then, for only 50 caps to be sold?

I'll post the official solution shortly. Let's also see what others come up with.

ManifestDreamMBA · Answer

You are absolutely right, and s and c are single purchase numbers. I had mentioned that while assuming the variables but it might not have been that clear with the wording I used. Let me include more commentary

Dereno · Answer

Given, each fan buys a shirt , or a cap or both. So, none = 0 Lets consider the two set venn diagram https://gmatclub.com/forum/download/file.php?id=83936 let the common be x Given x = 90 Set 2, which denotes shirt alone = s-x set1, which denotes cap alone = c-x s-x + x + c-x = 260 s + c = 260+x = 260+ 90 = 350 s+c = 350. We need to find: Shirts sold statement 1: (1) 50 fans did not buy a shirt. 50 did not buy shirt, so 50 must have bought caps. As none = 0 x denotes both cap and shirt. So, 50 will not come for x. c-x = 50. Thus, c = 50+x = 50+ 90 = 140. we can find the number of shirts sold. Shirt alone will be 260-140 = 120 Total shirts = 120+90 = 210. Hence, Sufficient (2) 140 fans bought a cap. This statement, is same as what we have deduced from statement 1. Shirt alone will be 260-140 = 120 Total shirts = 120+90 = 210. Hence, Sufficient Hence, Option D GMAT-Club-Forum-t5h7a7nh.png

gmatophobia · Answer

Given  :

The shop sold 260 memorabilia items.

Possible purchases 
1. Only Caps
2. Only Shirts
3. Shirts and Caps

Question  :

Only Shirts = ?

Statement 1
   
 (1) 50 fans did not buy a shirt.

The premise states that every fan bought atleast one item. Hence, if 50 fans did not buy a shirt, they must have boughts only caps.

Only Shirts = 260-(50+90) = 120

The statement alone is sufficient. We can elimiante options B, C and E

Statement 2
   
 (2) 140 fans bought a cap.

The premise states that every fan bought atleast one item, and we also know that 90 fans bought both a shirt and a cap. Therefore we can conclude that 50 fans bought only caps.

Only Shirts = 260-(50+90) = 120

The statement alone is sufficient.

Option D

A football clubs memorabilia shop sells only shirts and caps.

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A football clubs memorabilia shop sells only shirts and caps.

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