There are only blue pens, green pens and red pens in a box - Edexcel - GCSE Maths - Question 4 - 2017 - Paper 3

Adding the number of each type of pen, we get the total number of pens in the box:

$Total = 2x + 5x + y = 7x + y.$

Substituting $y$ from earlier:

$Total = 7x + \frac{5x}{4} = \frac{28x}{4} + \frac{5x}{4} = \frac{33x}{4}.$

Since the total number of pens must be less than 100, we set up the inequality:

$\frac{33x}{4} < 100.$

To find the maximum permissible value of $x$, we multiply both sides of the inequality by 4:

$33x < 400$

Dividing both sides by 33 gives:

$x < \frac{400}{33} \approx 12.12.$

Since $x$ must be a whole number, the largest integer value for $x$ is 12.

Using $x = 12$, we substitute back to find the number of red pens:

$y = \frac{5(12)}{4} = 15.$

Thus, the greatest possible number of red pens in the box is 15.