A coach for a school trip costs \(\pounds(150 + 8n)\) to hire, where \(n\) is the number of students, shared equally among them. Write an expression for the...

This question tests writing an expression for a cost per person, solving a resulting equation with the unknown in a denominator, rearranging the formula to make the number of students the subject, and proving algebraically that a stated value is impossible.

(a) Sharing the total cost \(\pounds(150+8n)\) equally among \(n\) students gives a cost per student of:

[1]

(b) Set the cost per student equal to \(\pounds13\) and multiply both sides by \(n\) to clear the fraction:

Collect the \(n\)-terms:

[3] (1 for clearing the fraction correctly, 1 for collecting the \(n\)-terms, 1 for \(n=30\))

(c) Starting from \(C=150+8n\), subtract \(150\) from both sides and divide by \(8\):

[2]

(d) Substitute \(C=342\) into the rearranged formula:

[2]

(e) If the cost per student could equal \(\pounds8\), then \(\frac{150+8n}{n}=8\), so \(150+8n=8n\). Subtracting \(8n\) from both sides gives \(150=0\), which is false for every value of \(n\). Since this leads to a contradiction regardless of \(n\), the cost per student can never equal \(\pounds8\). [3] (1 for setting up the equation, 1 for correctly cancelling to \(150=0\), 1 for the concluding statement)

Part (e) is a proof by contradiction: rather than solving for \(n\), the argument shows that assuming the cost per student is \(\pounds8\) forces an equation that is never true, so that assumption itself must be impossible for any number of students.