can u help me plz with an ALGEBRA II problem?

Question

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. thx

Answer 1

Total number of people T = 416
Number of dog owners D = 316
Number of cat owners C = 280

An important piece of information is that each person falls into one of the three categories:

(1) Dog owners
(2) Dog and Cat owners
(3) Cat owners

Your question says you need to find the number of dog owners who own no cat - that's group (1) above.

A useful piece of information is the number of people who own both a dog and a cat. Since the number of dog owners D = 316 and the number of cat owners C = 280, the total number of dog owners and cat owners (D + C) is greater than the total number of people in the community (T).

The number of people who have both a dog and a cat is then (D + C) - T, or (316 + 280) - 416 = 180.

Now, you can easily find out how many people fall into groups (1) and (3) above:

The number of dog owners who have no cat is (D - 180), or 316 - 180 = 136. (You can perform a similar operation to find the number of cat owners who have no dog.)

Hope this helps - good luck!

Answer 2

280 cats
+316 dogs
=596

since there are only 416 people:
596 - 416 = 180 people own both a cat and a dog

since 180 of the 316 dog owners also own a cat
316 - 180 = {[136 dog owners w/ NO cat]}

Answer 3

Let:
B = the number of people that own both cats & dogs
C = the number of people that own only cats
D = the number of people that own only dogs

Then:
The total population of people that own both, only cats, or only dogs should add up to 416:
(1) B + C + D = 416
The number of people that own dogs is the sum of the people that own both plus the people that own just dogs.
(2) B + D = 316
The number of people that own cats is the sum of the people that own both plus the people that own just cats.
(3) B + C = 280

If you substitute equation (3) into equation (1), you have:
(B + C) + D = 416
280 + D = 416

Subtract 280 from both sides of the equation:
D = 136
So the number of people that own only dogs and no cats is 136.