A rectangular floor area can be completely tiled with 200 square tiles. If the side length of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor.
Generated by claude-sonnet-4-6 · 2026-06-15 10:33 · grounding stimulus
Model Answer
(i) Original area of each tile = $x^2$ sq. units.
Total floor area = $200x^2$
When side is increased by 1 unit, new tile area = $(x+1)^2$
Number of tiles needed = 128
So: $128(x+1)^2 = 200x^2$
(ii) Expanding:
$128(x^2 + 2x + 1) = 200x^2$
$128x^2 + 256x + 128 = 200x^2$
$72x^2 - 256x - 128 = 0$
Dividing by 8:
$$9x^2 - 32x - 16 = 0$$
This is the standard form.
(iii) Factorising $9x^2 - 32x - 16 = 0$:
$9x^2 - 36x + 4x - 16 = 0$
$9x(x - 4) + 4(x - 4) = 0$
$(9x + 4)(x - 4) = 0$
$x = 4$ or $x = -\dfrac{4}{9}$
Since side length cannot be negative, $x = 4$ units.
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Explanation
- (i) The key step is equating total floor area: both tile arrangements cover the same floor, so $200x^2 = 128(x+1)^2$.
- (ii) Always rearrange to $ax^2 + bx + c = 0$ and simplify (divide by HCF = 8) to get clean numbers.
- (iii) For factorisation, split the middle term using product = $9×(-16) = -144$ and sum = $-32$ → factors $-36$ and $+4$. Reject negative value since length must be positive. Examiners award 1 mark for correct factors and 1 mark for correct value of $x$.