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?

  1. 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?
  2. 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?

  1. Arsenal’s gate receipts per year are £3.2 x 107 and there are 6.2 x 107 peo­ple liv­ing in the UK. How much does the aver­age UK per­son con­tribute to Arsenal’s receipts?
  2. 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.
  3. 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?
  4. 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?
  5. 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 1.055. 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.

  1. 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.

  1. 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.

  1. 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:

101 =10 and 10–1 = 0.1 =1/10

102 =100 and 10–2 = 0.01 = 1/100

103 =1,000 and 10–3 = 0.001 = 1/1,000

104 =10,000 and 10–4 = 0.0001 = 1/10,000

105 =100,000 and 10–5 = 0.00001 = 1/100,000

106 =1,000,000 and 10–6 = 0.000001 = 1/1,000,000

107 =10,000,000 and 10–7 = 0.0000001 = 1/10,000,000

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

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

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

When you are mul­ti­ply­ing by 10x 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 103 = 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.

  1. Where you could answer with dif­fer­ent units of mea­sure (say cen­time­tres, metres or kilo­me­tres) use stan­dard form in your answer.
  2. 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.
  3. Nom­i­nal value is the sup­posed value with no errors

Absolute error = Actual value — nom­i­nal value

Per­cent­age error = Absolute error/Nominal Value x 100%

  1. 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 1.055 = £995,500

Con­tract B will pay £850,000 x 1.035 = £985,383

There­fore after 5 years Con­tract A will pay the most.

  1. £700,000 x  Y5 = £1,005,000

Y5 = £1,005,000/£700,000

Y5 = 1.4357

Y = 5√1.4357 = 1.075

There­fore com­pound inter­est rate = 7.5%

  1. 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

  1. £3.2 x 107 /6.2 x 107   = £3.2/6.2 = £0.52 per person
  2. 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 = 8.330

  1. 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.

  1. 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%

  1. 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.