# GMAT Question of the week

There are 125 members in the winter sports club. Of those, 20 do not ski or snowboard. If 1/3 of the people who ski also snowboard, and if the number of people who only snowboard is equal to 1/2 of the number of people who ski and snowboard, how many people only snowboard?

A) 15

The fact that you are given two activities (skiing and snowboarding) and then given information about people who do both or do neither should be an indication to you that you can use a Venn Diagram to work on this problem. You are told that 20 members of the 125 person club neither ski nor snowboard. You are told that 1/3 of the skiers also snowboard, which can be written as:

**Both=1/3Ski**

You are told that the number of people who only snowboard is equal to one half of the number of those who do both, which can be written as:

**Snowboard Only=1/2Both**

Because you know that, of the 125 club members, 20 neither ski nor snowboard, you know that, total, there are 105 people who ski, snowboard, or do both.

Remember that there are two equations that are commonly used alongside Venn Diagrams:

* Total=A+B–Both+NeitherTotal=A+B–Both+Neither* and

**Only A+Only B+Both+Neither=Total**

Because you are given information about people who do ONLY one or the other, you should use the second equation, which can be rewritten as:

**Ski Only+Snowboard only+Both+Neither=125Ski Only+Snowboard only+Both+Neither=125**

And because both pieces of information you were given in the question stem were in terms of people who both skied and snowboarded, look to rewrite each of these terms in terms of the number of people who do both.

The first substitution you do is direct, since you are given the number of people who only snowboard in terms of the number of people who do both. You can then rewrite the equation as:

**Ski Only+1/2Both+Both=105**

You are told that **1/3Ski=Both**, which can be rewritten as **Ski=3Both**. However, since this equation expresses the total number of people who ski in terms of those who do both rather than the people who **ONLY** ski, you must rewrite it. You can think of the number of people who ski as:

**Ski=Ski Only+Both**

Substituting the information given in the question stem into this equation you get: **3Both=Ski Only+Both**

Solving for “Ski Only” by subtracting “Both” from both sides gives you:

**Ski Only=2BothSki Only=2Both**

Substituting this back into the equation you wrote gives you:

**2Both+3/2Both=105**

Collecting like terms then gives you

**7/2Both=105**

You can then solve for the “Both” value by multiplying both sides by 2/7 to get

**Both=30**

However, since you are asked for the number of people who only snowboard, you must then divide that value by 2 to get Snowboard Only=15.