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I need some help with this question, My answer to question E) is 1880 pounds per year, can anyone suggest a different answer, also does anyone know how would I revise for questions of this type.

Give answers, where appropriate, to 1 decimal place.

Julia Parsons runs a small business hiring out canoes and pedaloes (pedal boats) in a seaside holiday resort during the summer months. The canoes and pedaloes are subjected to considerable “wear and tear”.

Julia’s “fleet” consists of

  • 10 single canoes (canoes for a single person)
  • 12 double canoes (canoes for 2 persons)
  • 12 pedaloes (pedal boats)
  • 2 large canoes (canoes for 4 persons)

The hiring costs per hour for these are,
Single £4;
Double £6;
Pedaloes £3;
Large £8

Julia has kept a record of usage over several years and assesses the total usage according to the following table.

Table 1 Number of occasions when canoe/pedalo is hired for the period shown

Due to the heavy “wear and tear” each pedalo has to be replaced every 3 years and each canoe every 5 years. Julia maintains a depreciation fund for this purpose and puts the same amount of money each year in to this fund. The costs for new are,
Single £200;
Double £230;
Pedaloes £150;
Large £400

(e) How much does Julia put in to the fund each year?

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  • $\begingroup$ By my calculations the replacement cost averages £1712 a year: in a 15-year span she must replace 30 single canoes, 36 double canoes, 60 pedaloes, and 6 large canoes, at a total cost of £25680. Putting in £1712 a year is therefore a steady-state solution. However, it’s not enough starting from scratch, because at the end of 6 years she’s had to pay £12920 and has deposited only £10272. $\endgroup$
    – Brian M. Scott
    Feb 18, 2012 at 23:23
  • $\begingroup$ Ignore that last sentence: the £12920 figure is wrong. $\endgroup$
    – Brian M. Scott
    Feb 18, 2012 at 23:48
  • $\begingroup$ Table 1 is missing. . . How can we answer this if we don't know "number of occasions when canoe/pedalo is hired for the period shown"? That affects the answer entirely. $\endgroup$
    – 000
    Feb 19, 2012 at 0:26

2 Answers 2

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If a single canoe lasts 5 years and costs 200, you need to add 200/5=40 to the depreciation fund for each boat, or 400/year for the fleet of 10. The others are similar: 230*12/5=552 for double canoes, 150*12/3=600 for pedaloes, 2*400/5=160 for large canoes (assuming they last 5 years), giving me 1712. If the large canoes last only 3 years, they need 266.7 per year, giving 1818.7

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You can work out the annual replacement costs, as Ross did in his answer, or you can first work out the total replacement cost over a convenient period, as I did in my comment. Since $15$ is the least common multiple of $3$ and $5$, $15$ years is a convenient period: each pedalo must be replaced $5$ times, and each canoe $3$ times. Thus, the total replacement cost in pounds is

$$5\cdot 150\cdot 12+3\Big(200\cdot 10+230\cdot 12+400\cdot 2\Big)=25680$$

over $15$ years, or $\dfrac{25680}{15}=1712$ pounds a year.

This answer assumes that the depreciation fund does not earn interest, since nothing is said about interest in the problem. And contrary to what I said in my comment, the fund is always sufficient to cover the replacement costs. Assuming that the replacement fund starts at $0$, here’s a table of the total replacement costs and the amount in the fund at the points at which craft are replaced:

$$\begin{array}{r|r|r} \text{Year}&\text{Total Costs}&\text{Paid to Fund}\\ \hline 0&0&0\\ 3&1800&5136\\ 5&7360&8560\\ 6&9160&10272\\ 9&10960&15408\\ 10&16520&17120\\ 12&18320&20540\\ 15&25680&25680 \end{array}$$

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