### Overview

Complex Calculations and Accuracy covers using knowledge and your calculator to manipulate numbers. Topics include compound interest, percentage change, standard form, upper and lower bounds and absolute and percentage error.

I think half the battle for these questions is knowing how to use your calculator for roots and powers.

### Complex Calculations and Accuracy Questions

1. Gervinho is negotiating his new contract with Mr. Wenger. He is offered 2 contracts:

- Contract A. Salary now £780,000 with guaranteed increases of 5% per year for the next five years.
- Contract B. Salary now £850,000 with guaranteed increases of 3% per year for the next five years.

Which contract will pay the most after 5 years?

2. Gervinho is also offered another contract were he is offered £700,000 rising to £1,005,000 by the end of year 5. What is the rate of compound interest?

3. Arsenal’s gate receipt are under threat. In the 2011/2012 season they are expected to be £30 million. However in the 2012/2013 season they are expected to decline by 5% and in the 2013/2104 season they are expected to decline by a further 7.5%. What are the expected gate receipts in 2012/13 and 2013/14?

Manchester United receipts for 2011/2012 were £35million which was a 7.5% decrease from the previous season 2010/2011. What were Manchester United’s receipts in 2010/2011?

4. Arsenal’s gate receipts per year are £3.2 x 10^{7} and there are 6.2 x 10^{7} people living in the UK. How much does the average UK person contribute to Arsenal’s receipts?

5. The Manchester United groundsman uses 5 kg of fertiliser on the Old Trafford pitch. The pitch measures 100 metres by 60 metres. How much fertiliser did he spread per square kilometre, per square metre and per square centimetre? Use standard form in your answers.

6. The groundsman now has to give the pitch an all purpose feed. The pitch dimensions are 100 metres x 60 metres and each dimension is accurate to the nearest metre. The recommended amount of feed to apply is 5 kilograms (to the nearest kg.) per 100m². The groundsman says that he needs to buy 270 kilograms of feed to ensure that the pitch receives at least the recommended amount. Is he correct?

7. The net weight of a kilogram box of cornflakes may may have up to a 2.5% error. What are minimum and maximum weights of cornflakes that a box may contain? What is the nominal weight of cornflakes? If a box contains 982 grams what are the absolute and percentage errors?

8. The manufacturers of hula-hoops are concerned about the amount of hula-hoop material (the mix of potatoes and their secret recipe of seasoning and flavourings) that is used to produce each hula-hoop. The production director explains that not all hula-hoops are uniform. The nominal values for the exterior diameter, interior diameter and length of each hula-hoop are 1.2 cms, 1.0 cms and 1.5 cms respectively. The dimensions for the diameters are only accurate to one decimal place. However the length dimension is very accurate and consistent so there is no need to allow for error in length. What is the maximum percentage error in the volume of material required to make each hula-hoop?

### Complex Calculations and Accuracy Approach

1. As with many questions in this area, the key is to be confident in the use of your calculator. In this example you enter the starting salary: 780,000 and then multiplying by 1.05^{5}. Just make sure you are so familiar with your calculator that you can do this without having to search for the keys.

This involves compound interest where interest is added to the initial amount, this revised amount is used to calculate to the following period’s interest.

2. Here you have to work backwards to calculate the compound interest rate. Break it down step by step– see answer, below, for detailed method.

Calculator skills are again vital. You cannot quickly calculate ^{5}√1.4357 without having good calculator skills.

3. To decline by a percentage, say 5%, multiply by (1−0.05) = 0.95.

To work out reverse percentages (the Manchester United question) to calculate an original or starting value understand the logic. In this case lets call 2010/2011 receipts ‘R” then:

R x (1–0.75%) = £35,000,000

Then you can use this statement to work out “R”- see answer below.

4. The standard form is a number between 1 and 10 multiplied by 10 to a power. This power can be positive or negative. So 10² =100 and 10^{–2} = 0.01.

See the pattern:

10^{1} =10 and 10^{–1} = 0.1 =1/10

10^{2} =100 and 10^{–2} = 0.01 = 1/100

10^{3} =1,000 and 10^{–3 }= 0.001 = 1/1,000

10^{4} =10,000 and 10^{–4 }= 0.0001 = 1/10,000

10^{5} =100,000 and 10^{–5 }= 0.00001 = 1/100,000

10^{6} =1,000,000 and 10^{–6 }= 0.000001 = 1/1,000,000

10^{7} =10,000,000 and 10^{–7 }= 0.0000001 = 1/10,000,000

So 10^{x} =1 followed by x number of zeros and 10^{–x }= 1/10^{x}

So any number, no matter how small or large can be shown in the standard form:-

Y x 10^{x} or Y x 10^{–x} where Y = a number between 1 and 10.

When you are multiplying by 10^{x} you count x places to the right from the decimal point

When you are multiplying by 10^{–x} you count x places to the left from the decimal point.

S0 3.2 x 10^{3} = 3,200.0 &

S0 3.2 x 10^{–3} = 0.0032

The reason for using standard form is to be able to write very small or very large numbers in a quick and easy to understand way.

5. Where you could answer with different units of measure (say centimetres, metres or kilometres) use standard form in your answer.

6. This is about upper and lower bounds, so the length of the pitch has a lower bound of 99.5 metres and an upper bound of 100.5 metres. Work out the bounds for the width and the amount of feed. Then to ensure at least the recommended amount you need the maximum pitch size applied with the minimum amount of feed.

7. Nominal value is the supposed value with no errors

Absolute error = Actual value — nominal value

Percentage error = Absolute error/Nominal Value x 100%

8. Here you need to calculate the maximum amount of material. This would be where the external diameter is maximum, the internal diameter is minimum (this would give the thickest hoop).

This is an A* question and requires a methodical approach and you also need to apply your knowledge of radius, diameter and how to calculate the area of a circle and a cylinder.

### Complex Calculations and Accuracy Answers

1. Contract A will pay £780,000 x 1.05 x 1.05 x 1.05 x 1.05 x 1.05 = £780,000 x 1.05^{5} = £995,500

Contract B will pay £850,000 x 1.03^{5} = £985,383

Therefore after 5 years Contract A will pay the most.

2. £700,000 x Y^{5} = £1,005,000

Y^{5} = £1,005,000/£700,000

Y^{5} = 1.4357

Y = ^{5}√1.4357 = 1.075

Therefore compound interest rate = 7.5%

3. Gate reecipts ® have declined by 7.5%. Therefore

R x 0.925 = £35,000,000

R= £35,000,000/0.925 = £37,837,838

Gate receipts in 2010/2011 were £37,837,838

4. £3.2 x 10^{7} /6.2 x 10^{7 }= £3.2/6.2 = £0.52 per person

5. The amount of fertiliser per square metre = 5kg/(100 x 60) = 8.33 x 10^{–4 }

Square metres per square kilometer = 1,000,000 (1,000 x 1000). Therefore the amount per square kilometre = 8.33 x 10^{–4 }/ 1,000,000 = 8.33 x 10^{–10 }

Square centimetres per square meter = 10,000 (100 x 100). Therefore the amount per square metre = 8.33 x 10^{–4 }x 10,000 = 8.33^{0}

6. The maximum size of the pitch is 100.5 x 60.5 metres = 6,080.25m². The minimum amount of feed required per 100m² = 4.5kgs. Therefore the minimum amount of feed required to ensure that the pitch receives at least the recommended amount =

(6,080.25m² /100) x 4.5 = 273.6 kgs or 274 kgs to the nearest kg.

The groundsman is incorrect in saying that 270kgs will ensure that the pitch receives at least the recommended amount.

7. The minimum weight that a box may contain is 1kg x (100–2.5%) = 1kg x .975 = 0.975 kilogrammes or 975 grammes.

The maximum weight that a box may contain is 1kg x (100+2.5%) = 1kg x 1.025 = 1.025 kilogrammes or 1,025 grammes.

The nominal weight of cornflakes is 1kg.

If a box contains 982 grammes, the absolute error is 1,000 grammes — 982 grammes = 18 grammes.

The percentage error is 18/1000 = 1.8%

8. The nominal value for the volume of material required=

Volume based on External diameter — Volume based on Internal diameter.

Volume based on external value = ∏ x (1÷2 diameter = radius)² x Length = ∏ x 0.6² x 1.5 = 1.696cm³

Volume based on internal value = ∏ x (1÷2 diameter = radius)² x Length = ∏ x 0.5² x 1.5 = 1.178cm³

Nominal value for the volume = 1.696cm³ — 1.178cm³ = 0.518cm³ = 0.52cm³ (to 2 d.p.)

The maximum value for the volume of material required =

Maximum volume external diameter = ∏ x (1.25/2)² x 1.5 = 1.841cm³

Minimum volume internal diameter = ∏ x (0.95/2)² x 1.5 = 1.063cm³

Maximum value for the volume = 1.8411cm³ — 1.063cm³ = 0.78cm³ (to 2 d.p.)

The maximum percentage error = (0.78−0.52)÷0.52 = 50%

After solving this long hand I realised that you could do it far more simply. The nominal width of hula-hoop wall (external radius– internal radius) = 0.6cms — 0.5 cms = 0.1 cms (that’s a mighty thin hula-hoop!). The largest width of hula-hoop wall = 0.625cms –0.475cms = 0.15cms. As the length is constant, then the maximum volume of hula-hoop will be (0.15−0.1)÷0.1 = 50% greater than the nominal volume.

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