Depreciation, costing and conversions
Cian's van cost €20,000 new. A year later he checks what it is worth, and it is a lot less. A tradesperson quoting for a job must also add up materials, waste and labour. And a trip to Belfast means changing euro into sterling. In this page you will learn the three tools behind all of these: depreciation, costing and conversions. By the end you will be able to say exactly how much Cian's van is worth after four years.
Depreciation (reducing balance)
Most vehicles and machines lose value as they get older. This loss of value is called depreciation.
With the reducing balance method, the item loses a fixed percentage of its current value each year. So the amount lost gets smaller every year.
Why does this work? Losing 15% leaves 85% of the value, so you multiply by $0.85$. Do that every year and you multiply by $0.85$ repeatedly. This is the compound interest formula $F = P(1+i)^t$ from the Formulae and Tables booklet, with a negative rate, $i \to -i$.
A van bought in Dundalk for €20,000 depreciates by $15\%$ in its first year (reducing balance). Find its value after one year.
A van bought in Dundalk for €20,000 depreciates by $15\%$ in its first year (reducing balance). Find its value after one year.
A machine costing €45,000 depreciates by $20\%$ per year on the reducing balance method. a) Calculate its value after 3 years. b) Calculate the total depreciation over…
A machine costing €45,000 depreciates by $20\%$ per year on the reducing balance method.
- a) Calculate its value after 3 years.
- b) Calculate the total depreciation over the 3 years.
Reducing balance or straight line?
With straight-line depreciation the item loses the same amount every year. For an item costing €80,000 losing €14,400 per year, the value after $t$ years is $80{,}000 - 14{,}400t$.
Reducing balance drops faster at the start, which matches how real vehicles lose value.
To find when a value first falls below a target, try $t = 1, 2, 3, \dots$ with your calculator until the value passes the target. The answer is a whole number of years.
A farm tractor bought for €80,000 depreciates at $18\%$ per year on the reducing balance. a) Find its value after 4 years, to the nearest euro. b) After how many…
A farm tractor bought for €80,000 depreciates at $18\%$ per year on the reducing balance.
- a) Find its value after 4 years, to the nearest euro.
- b) After how many complete years will its value first fall below €30,000?
- c) Compare this with straight-line depreciation at €14,400 per year over the same period and comment.
Costing a job
A costing adds up everything a job needs: materials, waste and labour.
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Materials = area (or quantity) × price per unit.
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Waste allowance: increase the quantity by a percentage, for example 10% extra means multiply by $1.10$.
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Labour = hours × rate per hour.
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Total cost = materials + labour.
For painting you need the area of the walls. Four walls of a room form a rectangle when unrolled: area $=$ perimeter of the floor $\times$ height. Then subtract doors and windows.
A carpenter in Kilkenny is making a rectangular wooden table top $2.4$ m long and $1.2$ m wide. The wood costs €45 per square metre and $10\%$ extra is needed to allow…
A carpenter in Kilkenny is making a rectangular wooden table top $2.4$ m long and $1.2$ m wide. The wood costs €45 per square metre and $10\%$ extra is needed to allow for waste. Labour is 6 hours at €28 per hour.
- a) Calculate the area of wood needed including waste.
- b) Calculate the total cost of the table top.
A painter in Athlone is to paint the four walls of a room $5$ m by $4$ m with height $2.5$ m. There is one door of $2\text{ m}^2$ and one window of $3\text{ m}^2$ which…
A painter in Athlone is to paint the four walls of a room $5$ m by $4$ m with height $2.5$ m. There is one door of $2\text{ m}^2$ and one window of $3\text{ m}^2$ which are not painted. One litre of paint covers $8\text{ m}^2$ and two coats are needed. Paint is sold in 5-litre tins at €32 each. Labour costs €25 per hour and the job takes 7 hours. A $5\%$ allowance is added to the paint quantity for waste.
- a) Calculate the area to be painted.
- b) Calculate the number of tins of paint required.
- c) Find the total cost of the job.
Conversions and speed
A conversion factor tells you how many of one unit equal one of another. To convert, multiply or divide by the factor. In the exam, the factor is given in the question.
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Miles to km: multiply by $1.609$. Km to miles: divide.
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Inches to cm: multiply by $2.54$.
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Currency: with €1 = £0.85, euro to pounds: multiply by $0.85$. Pounds to euro: divide by $0.85$.
A quick check: if the new unit is smaller (inches to cm), the number gets bigger. If you change into a bigger unit, the number gets smaller.
Average speed
$$\text{average speed} = \frac{\text{total distance}}{\text{total time}}$$So $\text{distance} = \text{speed} \times \text{time}$ and $\text{time} = \dfrac{\text{distance}}{\text{speed}}$. Time must be in hours when speed is in km/h. Convert minutes by dividing by 60: 30 minutes is $0.5$ h, and 50 minutes is $\tfrac{50}{60}$ h.
A car travels 150 km in 2 hours and 30 minutes. Calculate its average speed in km/h.
A car travels 150 km in 2 hours and 30 minutes. Calculate its average speed in km/h.
On a trip to Belfast, Niamh changes €300 into pounds sterling at a rate of €1 = £0.85. How many pounds does she receive?
On a trip to Belfast, Niamh changes €300 into pounds sterling at a rate of €1 = £0.85. How many pounds does she receive?
A road sign in Co. Donegal shows a distance of 40 miles. Use $1\text{ mile} = 1.609\text{ km}$ and $1\text{ inch} = 2.54\text{ cm}$. a) Convert 40 miles to kilometres,…
A road sign in Co. Donegal shows a distance of 40 miles. Use $1\text{ mile} = 1.609\text{ km}$ and $1\text{ inch} = 2.54\text{ cm}$.
- a) Convert 40 miles to kilometres, correct to one decimal place.
- b) A car travels this distance in 50 minutes. Find its average speed in km/h.
- c) A tourist's tablet has a screen of width 9.7 inches. Convert this to centimetres.
Cian drives from Dublin to Rosslare, a distance of 165 km, at an average speed of 90 km/h. He then takes a ferry to Wales. a) Calculate the time of the first journey in…
Cian drives from Dublin to Rosslare, a distance of 165 km, at an average speed of 90 km/h. He then takes a ferry to Wales.
- a) Calculate the time of the first journey in hours and minutes.
- b) On the return, Cian drives the first 80 km at 100 km/h and the remaining 85 km at 70 km/h. Calculate his average speed for the whole return journey.
- c) In Wales he buys fuel at £1.45 per litre, using €1 = £0.86. Express the price in euro per litre and find the cost in euro of 40 litres.
Summary
Back to the start: Cian's van, bought for €20,000 and losing 15% per year, is worth $20{,}000 \times 0.85^4 = €10{,}440.13$ after four years, to the nearest cent. Not the €8,000 you would get by taking 15% of €20,000 four times.