A paint shop mixes red, blue and yellow bases to make shades A, B and C. Litres needed per can are s

Question

A paint shop mixes red, blue and yellow bases to make shades A, B and C. Litres needed per can are shown.

	$A$	$B$	$C$
Red	$2$	$1$	$3$
Blue	$3$	$2$	$1$
Yellow	$1$	$4$	$2$

Let $M=\begin{pmatrix}2&1&3\\3&2&1\\1&4&2\end{pmatrix}$, with rows Red, Blue and Yellow and columns A, B and C.

The shop makes 4 cans of A, 2 cans of B and $x$ cans of C. Represent this as a $3\times1$ matrix $N$.[1]
Find $L=MN$ in terms of $x$.[2]
State what the elements of $L$ represent.[1]
The shop has 25 litres of each base colour. Find the greatest integer $x$ such that no colour runs out.[2]

Accepted Answer

(a) $N=\begin{pmatrix}4\cr 2\cr x\end{pmatrix}$; (b) $L=\begin{pmatrix}10+3x\cr 16+x\cr 12+2x\end{pmatrix}$; (c) total litres of red, blue and yellow base respectively; (d) $x=5$.

	\(A\)	\(B\)	\(C\)
Red	\(2\)	\(1\)	\(3\)
Blue	\(3\)	\(2\)	\(1\)
Yellow	\(1\)	\(4\)	\(2\)

2025 SJI PRELIMS P1 Q23

2025 SJI PRELIMS P1 Q23

Solution:

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