Question

The ratio of the totals of the numbers in box A and box B is 2:3.

Swap two numbers, one from each box, so that the ratio of the totals of the numbers is now 9:11. Show your working.
Please answer asap!! :)

Answer 1

Swap 10 from box A and 15 from box B

Explanation:

For calculate the ratio between the boxes we find the mean of the box A and the mean of the box B and divide this two results.

For find the mean of a set you use the next formula
`\bar{x} = \frac{\text{sum numbers}}{\text{total numbers}}`

`\bar{A} = (4+5+9+10+12)/(5) = (40)/(5)= 8\\\bar{B} = (6+7+13+15+19)/(5) = (60)/(5)= 12`

Then the ratio is equal to
`\frac{\bar{A}}{\bar{B}} = (8)/(12) = (2)/(3)`

Know the ask us for a ratio of 9:11 then we know two things about the boxes.

If you swap the numbers between them the total of numbers in the mean not change is always 5 and the sum of all the elements in both boxes is always 100. With this we can state the next equation:

The new mean of
`\bar{A}` is 9 then:

`\frac{\text{sum of elements in box A}}{5} = 9`

`\text{sum of elements in box A} = 45`

The new mean of
`\bar{B}` is 11 then:

`\frac{\text{sum of elements in box B}}{5} = 11`

`\text{sum of elements in box B} = 55`

Now we know what is the sum of the elements in both boxes, but how we find the elements for swap? Well with the next relation.

When you take out a element from the box A this elements is substracted to the total sum of the elements in the box A and get a new sum of total of elements named
`c` (where
`a` is the element taken):

`40 - a = c`

And when you sum an element to the box A the current sum of total elements in the box change to (where
`b` is the added element):

`b + c = 45`

So replace
`c` for the first relation and get:

`b + 40 - a = 45\\b - a = 5`

From the last equation you can deduct that
`b` is greater than
`a` and the different between both values is 5. So the last step is find a number in the box B that substracted to a number from the box A give us 5. These both numbers are 15 and 10 because 15 - 10 = 5.