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?

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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?

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Answer 1 · 2016-08-28T20:03:45

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

Explanation:

Let the total number of animals be $x$ $x$

If $\frac{7}{10}$ $7/10$ of them are cats, it means that the dog and goats together make up $\frac{3}{10}$ $3/10$

The fraction of dogs will be $\frac{2}{3} o f \frac{3}{10} . = \frac{2}{10} = \frac{1}{5}$ $2/3 of 3/10. = 2/10 = 1/5$

The number of goats, which is 60,000 is 1 $/ 3 o f \frac{3}{10}$ $/3 of 3/10$ of the total number of animals.

$\frac{1}{3} \times \frac{3}{10} \times x = 60, 000$ $1/3 xx 3/10 xx x = 60,000$

$\frac{1}{10} x = 60, 000$ $1/10 x= 60,000$

$x = 60, 000 \times 10$ $x = 60,000xx10$

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

There are $\frac{7}{10} \times 600, 000 = 420, 000$ $7/10 xx 600,000 = 420,000$ cats.

There are $\frac{1}{5} \times 600, 000 = 120, 000$ $1/5 xx 600,000 = 120,000$ dogs
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
By a ratio/ proportion method:

Cats make up $\frac{7}{10}$ $7/10$ of the population.

Dogs are $\frac{2}{3} \times \frac{3}{10} = \frac{2}{10} = \frac{1}{5}$ $2/3 xx 3/10 = 2/10 = 1/5$

Goats are the remaining $\frac{1}{3} \times \frac{3}{10} = \frac{1}{10}$ $1/3 xx 3/10 = 1/10$

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

$60, 000 \times \frac{10}{1} = 600, 000$ $60,000 xx 10/1 = 600,000$

By proportion:

$\frac{1}{10} = \frac{x}{60000}$ $1/10 = x/60000$

$x = 10 \times 60000$ $x = 10xx60000$

$x = 600000$ $x = 600000$

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

Be careful not to write:
$\frac{1}{10} = 60, 000 O R \frac{1}{3} = 60000$ $1/10 = 60,000 " " OR" " 1/3 = 60000$
These values are clearly NOT equal!

Answer 2 · 2016-08-29T11:39:06

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

Splitting into

$420, 000 cats$ $420,000" cats"$
$120, 000 dogs$ $120,000" dogs"$

Explanation:

For cats: $\frac{7}{10}$ $7/10$ of all of the animals

For dogs: $\frac{2}{3}$ $2/3$ of remainder $\to \frac{2}{3} \times (1 - \frac{7}{10}) = \frac{1}{5}$ $->2/3xx(1-7/10) = 1/5$

Thus dogs and cats $\to \frac{7}{10} + \frac{1}{5} \to \frac{7}{10} + \frac{2}{10} = \frac{9}{10}$ $-> 7/10+1/5" " ->" "7/10+2/10" " =" " 9/10$

So the remainder (goats) is $1 - \frac{9}{10} = \frac{1}{10}$ $1-9/10=1/10$ of all the animals

But goat count is 60,000 $\to \frac{1}{10}$ $-> 1/10$ of the total animal count

So by using ratio $\frac{animal count}{proportion of all animals} \to \frac{60000}{\frac{1}{10}}$ $("animal count")/("proportion of all animals") -> (60000)/(1/10)$

To change this to all the animals we need to change the $\frac{1}{10}$ $1/10$ into 1.

Multiply top and bottom by 10

$\frac{animal count}{proportion of all animals} \to \frac{60000}{\frac{1}{10}} \to \frac{60000 \times 10}{\frac{1}{10} \times 10}$ $("animal count")/("proportion of all animals") -> (60000)/(1/10) ->(60000xx10)/(1/10xx10)$

$= \frac{600000}{1}$ $=(600000)/1$

So the total count of animals is 600,000

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

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

Total dogs

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

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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?

2 Answers

Explanation:

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