The last blog post I wrote showed how modifiers can fool people on quant problems “ here’s the link.

Several of my students who got the baseball problem from that post correct dismissed the issue entirely and scoffed at me for showing them such an easy problem, then inevitably missed a variant of the problem I’m about to show you. Try it for yourself, and watch out for the modifiers!

The town of Malmo, Sweden has only two late-night food options: Pizza and Kebab. All sellers of late-night food have either a street permit or a permanent store permit. 60% of all the late-night food sellers in Malmo are street vendors that serve Kebab; 20% of all the late-night food sellers who have a permanent store serve Pizza. If Malmo’s ratio of total street permits to total permanent store permits is exactly 7 to 3, what percentage of all late-night food sellers in Malmo serve pizza?

(A)  10%

(B)  16%

(C)  24%

(D) 30%

(E)  70%

(If you’re not sure how to approach this problem, try brushing up on overlapping sets, covered in the Manhattan GMAT Word Problems strategy guide. Then come back and give it a shot.)

( *** SPOILER *** ) —————————

Did you choose (D)? Sorry, that’s not the right answer! But if you picked it, don’t beat yourself up: it’s a trap.

60% of all the late-night food sellers in Malmo are street vendors that serve Kebab means:

1 “ Out of all food sellers (assume there are 100 food sellers in Malmo)

2 “ 60% of those

3 “ are street vendors AND serve Kebab

Therefore, 60 vendors are street vendors that serve Kebab.

However, 20% of all the late-night food sellers who have a permanent store serve Pizza means something DIFFERENT:

1 “ Out of ONLY the permanent store vendors (of which there are 30 “ we know that from the 7:3 ratio and our 100-vendor total assumption)

2 “ 20% of those

3 “ serve Pizza

Therefore, only 6 (20% of 30) vendors are permanent vendors that serve pizza, NOT 20! This is the critical step to understand. Whenever the GMAT mentions a percentage, there are two quantities involved: the part and the whole. In the first case, the whole is all vendors. But in the second case, the whole is restricted by the MODIFIER who have a permanent store. The problem is difficult largely because humans have a tendency to ignore modifiers that come at the end of long sentences. Be extra diligent on these types of word problems, especially ones involving many different percentages! Often the GMAT will trap you by showing you percentages of different wholes. When the different wholes are distinguished only by a modifier, the change can be very difficult to spot. Be sure you always know what you are taking a percentage of!

The correct answer is (B). If you missed it, try and work out the correct version on your own (I’m purposefully not giving a complete solution here), and use the comment section if you need help!

There’s one final thing I’d like you to take away from the Malmo problem above and the baseball problem in my previous post: when you see a hard problem, try and relate it to an easy one! The best GMAT students see the connection between the MalmÃ¶ problem and the baseball problem, because they didn’t dismiss the baseball problem just because it was easy.

#### Ryan Jacobs

Ryan Jacobs is the son of an English teacher and a math teacher, which probably means he was fated from birth to teach the GMAT. He scored a 760 on the GMAT and a perfect 1600 on the SAT, both on his first attempt. He has a bachelor's degree in Music Composition as well as an MBA, both from the University of California, San Diego.

### 7 responses to Return of the Hardest Easy Math Problem in the World

1. Love this question. Brilliant wording and again stressed the fact for me that no question is simple if we are not attentive.

2. I did pick the incorrect answer (30%) but only because I couldn’t find a way to employ the modifier when I read the sentence “20% of all the late-night food sellers who have a permanent store serve Pizza”. This is because one cannot work out 20% of 30 until after one has processed the following sentence – ” If Malmoâ€™s ratio of total street permits to total permanent store permits is exactly 7 to 3″

So we have to process those two sentence in reverse order to be able to get it right. My approach to such problems has been
1. Draw a double set matrix
2. Start putting in numbers using sentence that follow in order.
3. Calculate missing information.

I guess I have to modify step 2 in my approach.

3. could someone say how to set-up the matrix ??

thanks

4. Mukesh, great point. You have two choices in that situation: (1) ignore information that you don’t know how to use yet and come back to it later, or (2) use a variable like “x” to represent the whole, and “0.2x” to represent the part. I use strategy (1) most often, but many of my students prefer (2). Try both and see what works for you.

5. A matrix to help resolve such a question would be something like the following:

……….. Pizza Kebab Total
Street 10 60 70

Permanent 6 24 30

Total 16 (Ans) 84 100

I assumed the total vendors = 100.

Since the ratio of the licenses is 7 : 3, that means of the 100 total, 70 must have street, and 30 have permanent licenses. Since 60% of the TOTAL (100) are street vendors selling kebabs, 60 goes directly under Kebabs column and in front of the Street row. Further since 20% of all the PERMANENT license holders sell pizzas, that’s 20% of 30, which is 6 – therefore 6 goes under the Pizza column and in front of the Permanent row, and the remaining 80% of the 30 permanent license holders (which equals 24) goes right next to it. The remaining numbers can now be found through simple arithmetic. Hope this helped.

6. A matrix to help resolve such a question would be something like the following:

………………….Pizza…….. Kebab…… Total
Street …………..10 …………. 60 ……… 70

Permanent ……..6 ………….. 24………. 30

Total …………….16 (Ans) ……84 ………100

I assumed the total vendors = 100.

Since the ratio of the licenses is 7 : 3, that means of the 100 total, 70 must have street, and 30 have permanent licenses. Since 60% of the TOTAL (100) are street vendors selling kebabs, 60 goes directly under Kebabs column and in front of the Street row. Further since 20% of all the PERMANENT license holders sell pizzas, that’s 20% of 30, which is 6 – therefore 6 goes under the Pizza column and in front of the Permanent row, and the remaining 80% of the 30 permanent license holders (which equals 24) goes right next to it. The remaining numbers can now be found through simple arithmetic. Hope this helped.