Question

There are some books on three shelves. On the bottom shelf there are half as many books as on the other two shelves. On the middle shelf there are three times less than on the other two shelves. On the top shelf there are 30 books. How many books are on all three shelves?

Answer 1

There are 72 books on all the three shelves.

Explanation:

Given:

On the bottom shelf there are half as many books as on the other two shelves.

On the middle shelf there are three times less than on the other two shelves.

On the top shelf there are 30 books.

Now, to find the books on all three shelves.

Let the books on the middle shelf be
`x.`

The books on the top shelf are 30.

The books on the bottom shelf =
`(1)/(2) (x+30)`

So, the books on the middle shelf:

`3x=30+(1)/(2)(x+30)`

`3x=30+(x+30)/(2)`

`3x=(60+x+30)/(2)`

Multiplying both sides by 2 we get:

`6x=60+x+30`

`6x=x+90`

Subtracting both sides by
`x` we get:

`5x=90`

Dividing both sides by 5 we get:

`x=18.`

Thus, the books on middle shelf are 18.

Now, to get the books on the bottom shelf by substituting the value of
`x`:

`(1)/(2) (x+30)`

`=(1)/(2) (18+30)\\\\=(1)/(2) * 48\\\\=24.`

Hence, the books on the bottom shelf are 24.

Now, to get the total books on all the three shelves we add the books of top, middle and bottom shelf:

`30+18+24\\\\=72.`

Therefore, there are 72 books on all the three shelves.