A household's energy supplier writes to say it will estimate next year's bill, \(\pounds b\), from a smart-meter reading, \(m\), and a tariff constant, \(k\...
Question 1 Report
A household's energy supplier writes to say it will estimate next year's bill, \(\pounds b\), from a smart-meter reading, \(m\), and a tariff constant, \(k\), using \(b=\dfrac{5m-k}{m+k}\).
- Show that this formula can be rearranged to give \(k(b+1)=m(5-b)\). (3)
- Hence make \(k\) the subject of the formula. (3)
Answer Details
(a) Multiplying both sides of \(b=\dfrac{5m-k}{m+k}\) by \((m+k)\) to clear the fraction gives \(b(m+k)=5m-k\). Expanding the left-hand side, \(bm+bk=5m-k\). [1 mark] Gathering every term with \(k\) onto the left and every other term onto the right, adding \(k\) to both sides and subtracting \(bm\) from both sides, gives \(bk+k=5m-bm\). [1 mark] Factorising each side, \(k\) out of the left and \(m\) out of the right, gives \(k(b+1)=m(5-b)\), as required. [1 mark]
(b) Dividing both sides by \((b+1)\) makes \(k\) the subject: \(k=\dfrac{m(5-b)}{b+1}\). [3 marks]
Gathering every occurrence of \(k\) onto one side before factorising is essential here, exactly as with any subject that appears more than once in the original formula; \(k\) could not simply be divided out while it still appeared on both sides.
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