Group Problems

Arithmetic: Group Problems

Read the following SAT test question and then select the correct answer. 

Work all math problems the same way so that you can approach even the most difficult problems with confidence.  Start by reading the question carefully.  Many problems have several steps, so you must identify the bottom line: what is the question asking?  Assess your options and choose the most efficient method to attack the problem.  Finally, loop back to make sure that your answer matches the bottom line.

In a community of 416 people, each person owns a dog or a cat or both. If there are 316 dog owners and 280 cat owners, how many of the dog owners own no cat?

Bottom Line: just dogs = ?

When you get to the step where you look at the answers, notice that (E) comes from not reading carefully.  Yes, there are 316 total dog owners, but some of them also own cats.  You must find how many own only dogs.

Assess your Options:  You could try to work backwards using the answer choices, but trying to think about the steps of a problem backwards often leads to mistakes.  You could also realize that this is a problem involving two overlapping groups and draw a Venn Diagram.  Forget those methods because the fastest method is to use the Group Formula.  Take a moment now to memorize this formula if you have not already done so: Total = Group 1 + Group 2 + Neither – Both.

Attack the Problem:  Plug all the information that you know into the formula.  How many total people are there? 416.  Then there are your two groups: Those who own dogs and those who own cats.  Plug in the numbers 316 and 280 to represent these groups.   Now, the problem tells you that “each person owns a dog or a cat or both,” so how many people own neither animal?  Zero.  The only thing that you are not given in the problem is how many people own both a dog and a cat.  Your formula should now look like this:

Total = Group 1 + Group 2 + Neither – Both

416 = 316 + 280 + 0 – B  

That B represents the unknown Both, but you can now solve for it because it is the only variable left in your equation.  Start by simplifying the problem.

416 = 316 + 280 + 0 – B  

416 = 596 – B   (add B to each side to make it positive)

416 + B = 596   (subtract 416 from each side)

B = 180

You have finished one step, but you have not yet reached your bottom line!  Do not look at the answer choices yet or you will be tempted to pick a wrong answer!

You just solved for the number of people who own both a cat and a dog.  How do you find the number of people who own just a dog?  Take the number who own both and subtract it from the total number of dog owners.  Remember that the total number of dog owners was given in the problem as 316.

316 – 180 = 136

Loop Back:  Your second step solved for those who just own dogs, so you are ready to look at your answer choices.

(A) 36

(B) 100

(C) 136

(D) 180

(E) 316

The correct answer is (C). 

On sat.collegeboard.org, 46% of the responses were correct.

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