Can someone help with this GCSE maths question?

[Schmoana]

I did A level maths 30 years ago and can’t do it ...

There are only blue and green pens in a box.

There are three more blue pens than green pens in the box. There are more than 12 pens in the box.

Simon is going to take at random two pens from the box.

The probability that Simon will take two pens of the same colour is 27/55.

Work out the number of green pens in the box.


Can anyone help? TIA

[Thimbleberries]

It might be that trial and error is the easiest, but if you want an algebraic way, you could try this:

x = number of green pens
x + 3 = number of blue pens
2x + 3 = total number of pens.

To find probability of two events, you multiply individual probabilities, so for p(green, green), you get

prob of picking a green pen to start = x/(2x+3)

to find prob of picking a second green pen, you have to subtract 1 from both the number of green pens and number of total pens, so
(x-1)/(2x+2)

Multiply those: x(x-1)/(2x+3)(2x+2)

Do the same for the blue pens:
probability of picking a first blue pen is (x+3)/(2x+3)
subtract 1 from number of blue pens and number of total pens to get the probability of picking a second blue pen: (x+2)/(2x+2)

Multiply those two probabilities to get the probability of picking two blue pens: (x+3)(x+2)/(2x+3)(2x+2).

Then to find the probability of two pens the same colour, you have to add both these possibilities. Luckily you already have a common denominator for the fractions:

[x(x-1) + (x+3)(x+2)]/(2x+3)(2x+2)

Expand the top and bottom

(x2 - x + x2 + 5x +6)/(4x^2 + 10x + 6)

Simplify the top:

(2x2 + 4x +6)/(4x2 + 10x +6)

Further simplify by dividing everything through by 2 (or, factor a 2 out of the top and the bottom, and then cancel, if that is easier to understand)

(x2 + 2x + 3)/(2x2 + 5x + 3)

This is now your expression for picking two pens of the same colour. You can make it into an equation by setting it equal to 27/55

(x2 + 2x + 3)/(2x2 + 5x + 3) = 27/55

Cross multiply (other ways of simplifying are available!)

55 (x2 + 2x + 3) = 27 (2x2 + 5x + 3)

expand

55x2 + 110x + 165 = 54 x2 + 135x + 81

rearrange so that one side = 0

x^2 -25x + 84 = 0

factor

(x-21)(x-4) = 0

x = 21 or x = 4.

Because there are more than 12 pens, x must be 21.

So 21 green, 24 blue, 45 total.

Hope the brackets, division, powers etc makes sense - easier when it's written out as fractions on top of each other.

Other ways of simplifying and solving will also work fine, once you have the basic equation set up.

[noblegiraffe]

If it’s for GCSE maths (and this is a classic last question on Higher), they won’t allow trial and error and you’ll only get full marks for the algebraic method above.

[BenoneBeauty]

Wow @Thimbleberries - very impressed!

[Schmoana]

Amazing thimbleberries, thank you. I did know the logic but completely fumbled the execution. That’s really helpful, thanks

[Thimbleberries]

You're welcome. It looks more complicated than it is, when you try to write it out on lines like that!

[Schmoana]

Your explanations were excellent!