Question

Alex has a bag of marbles. Each marble is red, blue, or yellow , and the ratio of the red to blue to yellow marbles is 6:10:15. After Alex adds 10 red marbles and removes 25 yellow marbles form the bag, the ratio of red to blue to yellow is 2:3:4. How many blue marbles are in Alex's bag

Answer 1

There are 150 blue marbles in Alex's bag.

Explanation:

Let
`x`,
`y`,
`z` be the quantities of red, blue and yellow marbles, respectively. From statement we derive three rational equations for initial and final ratios:

Initial ratio

`(x)/(y) = (6)/(10)` (1)

`(y)/(z) = (10)/(15)` (2)

Final ratio

`(x + 10)/(z - 25) = (1)/(2)` (3)

After some algebraic handling we get the following system of linear equations:

`x - (3)/(5)\cdot y = 0` (1)

`y - (2)/(3)\cdot z = 0` (2)

`2\cdot (x + 10) = z - 25`

`2\cdot x +20 = z - 25`

`2\cdot x - z = -45` (3)

The solution of the system of linear equations is:

`x = 90`,
`y = 150`,
`z = 225`

There are 150 blue marbles in Alex's bag.