The answer
(a) shown below
(b) \(\approx 12.3\) cm
O-Level E-Math 2017 Paper 1 Question 23 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 23 of the O-Level E-Math 2017 Paper 1. It tests test for direct proportion via constant unit rate, in the Direct proportion / similar solids area. It is worth 5 marks: 2 + 3. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a) If cost were directly proportional to quantity, the cost per millilitre (cost ÷ quantity) would be the same for every bottle. Computing it: \[\frac{30.80}{400} = $0.0770/\text{ml}, \qquad \frac{19.25}{250} = $0.0770/\text{ml}, \qquad \frac{5.95}{75} = $0.0793/\text{ml}.\] The 400 ml and 250 ml bottles give $0.0770 per ml, but the 75 ml bottle gives $0.0793 per ml, which is different. Since the cost per millilitre is not constant, the cost is not directly proportional to the quantity.
(b) For geometrically similar bottles, volume scales as the cube of the height. The height scale factor from the 250 ml bottle to the 75 ml bottle is the cube root of the volume ratio: \[\text{scale factor} = \sqrt[3]{\frac{75}{250}} = \sqrt[3]{0.3} = 0.6694.\] Height of the 75 ml bottle \(= 18.4 \times 0.6694 = 12.3\) cm (3 s.f.).
Answer: (a) shown below
(b) \(\approx 12.3\) cm
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a test for direct proportion via constant unit rate question from Direct proportion / similar solids, worth 5 marks: 2 + 3.
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