Question

7/10 of a city animal population are cats. 2/3 of the remainder are dogs and the rest are goats. If there are 60,000 goats, how many cats and dogs are there?

Answer 1

`420,000 " cats and " 120,000 " dogs"`

Explanation:

Let the total number of animals be `x`

If `7/10` of them are cats, it means that the dog and goats together make up `3/10`

The fraction of dogs will be `2/3 of 3/10. = 2/10 = 1/5`

The number of goats, which is 60,000 is 1`/3 of 3/10` of the total number of animals.

`1/3 xx 3/10 xx x = 60,000`

`1/10 x= 60,000`

`x = 60,000xx10`

`x =600,000`
If there are 60,000 animals, then:

There are `7/10 xx 600,000 = 420,000` cats.

There are `1/5 xx 600,000 = 120,000` dogs
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
By a ratio/ proportion method:

Cats make up `7/10` of the population.

Dogs are `2/3 xx 3/10 = 2/10 = 1/5`

Goats are the remaining `1/3 xx 3/10 = 1/10`

For the total, increase 60,000 in the ratio `10:1`

`60,000 xx 10/1 = 600,000`

By proportion:

`1/10 = x/60000`

`x = 10xx60000`

`x = 600000`

Once we know the total, we can find the fractions the same as before.

Be careful not to write:
`1/10 = 60,000 " " OR" " 1/3 = 60000`
These values are clearly NOT equal!

Answer 2

`"There are "540,000" cats and dogs"`

Splitting into

`420,000" cats"`
`120,000" dogs"`

Explanation:

For cats: `7/10` of all of the animals

For dogs: `2/3` of remainder `->2/3xx(1-7/10) = 1/5`

Thus dogs and cats`-> 7/10+1/5" " ->" "7/10+2/10" " =" " 9/10`

So the remainder (goats) is `1-9/10=1/10` of all the animals

But goat count is 60,000`-> 1/10`of the total animal count

So by using ratio `("animal count")/("proportion of all animals") -> (60000)/(1/10)`

To change this to all the animals we need to change the `1/10` into 1.

Multiply top and bottom by 10

`("animal count")/("proportion of all animals") -> (60000)/(1/10) ->(60000xx10)/(1/10xx10)`

`=(600000)/1`

So the total count of animals is 600,000

As there are 60,00 goats that means the count of cats and doge is:

`600,000`
`ul(color(white)(6)60,000)" "larr" Subtract"`
`540,000" "larr" Cats and dogs combined"`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Total cats `-> 7/10xx600000 = 420,000`

Total dogs

`540,000`
`ul(420,000)" "larr" Subtract"`
`120,000`