Pets And Proportions

Algebra Level 3

Each person in a group of friends has a cat and/or a dog.

$\dfrac{1}{5}$ of the people with cats have a dog, and
$\dfrac{1}{4}$ of the people with dogs have a cat.

What percent of people in the group have both?

Note: If your answer is $X\%,$ submit your answer as $X.$

The answer is 12.5.

2 solutions

Pi Han Goh
Mar 3, 2016

It wouldn't hurt to draw out a Venn diagram as a visual aid.

Let $x$ denote the number of people in the group that have both a cat and a dog. Then,

Then from the first statement: $\dfrac15$ of the people with cats have a dog . This also means that $1 - \dfrac15 = \dfrac45$ of the people with cats don't have a dog. Thus, the number of people with cats that don't have a dog is 4 times the number of people in the group has both a cat and a dog. In other words, the number of people with cats that don't have a dog is $4x$ .

Similarly, from the second statement: $\dfrac14$ of the people with dogs have a cat . This also means that $1 - \dfrac14 = \dfrac34$ of the people with dogs don't have a cat. Thus, the number of people with dogs that don't have cats is 3 times the number of people in the group has both a cat and a dog. In other words, the number of people with dogs that don't have a cat is $3x$ .

Since each person in this group of friends has at least a cat or a dog, then the total number of people in group must be $4x+x+3x= 8x$ , hence the fraction of people in the group that have both cats and dogs is $\dfrac{x}{8x} = \dfrac18 = \boxed{12.5\%}$ .

Moderator note:

Good explanation of how to interpret the statements to create the solution.

What number is a multiple of 4 and 5 ?

So there are 20 pet owners.

1/4 = 25%

So 25% of 20 is 5.

This leaves 15 pet owners.

Now, 1/ 5 = 20%

So 20% of 15 is 5.

5 + 3 = 8

Now here's the problem with your answer. You cannot assume 100 just because you want to.

There are 20 pet owners.. So the percent that has both is 8 / 20 = 2/ 5 = 40 %. Not 12,5 % !!

So when I see that 12.5 percent as the answer, I have to ask you dude , what are you talking about. ?

DarkMind S. - 4 years, 6 months ago

I have no idea what you're talking about. We are not given the number of pet owners in the first place.

Pi Han Goh - 4 years, 6 months ago

There is nothing wrong with the percentage being a decimal. You are assuming that there are 20 people, which need not be the case. Note that it's "1/5 of cat owners", which doesn't imply that the total number must be divisible by 5. It only implies that the number of people who own a cat is a multiple of 5.

The conditions in the question can be satisfied with a group of 8 people, of which 1 has both cats and dogs, 3 have dogs only, 4 have cats only.

I have marked this report as resolved.

Calvin Lin Staff - 4 years, 5 months ago

I meant.20 % of. 15. = 3

DarkMind S. - 4 years, 6 months ago

Pi Han Goh. Thanks for pointing out the obvious fact that we are not given the number of pet owners. That's where something called deduction comes in handy

DarkMind S. - 4 years, 6 months ago

Cristi Ursuţ
Mar 10, 2016

A - Number of people with dogs; B - Number of people with cats; C - Number of people with dogs and cats;

A = 5 C; B= 4 C; A + B = 9C - C (because 9C means is actually the total number of pets, but there are 1C people with both dogs and cats) = 8C;

The result is: 100/8 = 12.5%

Very well explained. No complication. Thanks. Although you can add one explanatory line why the total is 8C , because 5C and 4C both count one extra 'C', counted twice. so Total number of pet owners is 4C + 3C + C = 8C

Ajit Deshpande - 5 years, 3 months ago

Pets And Proportions

The answer is 12.5.

2 solutions

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