Tools - Math 'Plausible Estimation' Estimating for Amazing Facts Tasks, Set #3 (solutions)

Estimates for a Million, Set #1 (solutions)
Estimating for Amazing Facts: Set #2 (solutions) || Set #3 (solutions)
Estimates for the USA, Set #4 (solutions)

Malcolm Swan
Mathematics Education
University of Nottingham
Malcolm.Swan@nottingham.ac.uk

Jim Ridgway
School of Education
University of Durham
Jim.Ridgway@durham.ac.uk

 The aim of this assessment is to provide the opportunity for you to: develop a chain of reasoning that will enable you to estimate quantities to an appropriate degree of accuracy choose suitable units for your estimate communicate the assumptions upon which your estimate is based.

1. High stack
 Suppose you have a very large sheet of paper. You tear it in half and put one half on top of the other. You now have a stack of two sheets. You now tear the whole stack in half and place one half on top of the other to make a new stack. You repeat this process, tearing 50 times. (Yes I know its impossible - just imagine you could). How high would the stack be? 50 feet? 100 feet? A mile? or more...? Make a sensible estimate, based on careful reasoning.

Solution:

 Assumptions A ream of paper (500 sheets) is about 2 inches thick Reasoning Each sheet is therefore approx 1/250th of an inch thick. After 50 tears, the stack will contain 250 sheets of paper. This is equal to approximately 1.1259 x 1015 sheets or 4.5 x 1012 inches = 71million miles. Answer: 71 million miles!

1. The swimming pool and the glass.
 How long would it take you to empty an olympic size swimming pool with a glass?

Solution:

 Assumptions Dimensions of swimming pool = 50 meters x 20 meters Depth of water = 6 feet (2 meters) The glass holds half a pint. You empty the pool at one glass per second. Reasoning This gives volume of pool as 2000 cubic meters or 2 x 106 liters. One half pint glass has a volume of about 25 x 10-5 cubic meters. Thus approximately 8 million glasses will empty the pool. This would take 2,215 hours or about 90 days. Answer: 92 days or about 3 months, working day and night!

1. The briefcase of cents
 Suppose you filled a briefcase with one cent coins. How much would the money be worth?

Solution:

 Assumptions A cent coin has a diameter of 20 mm (0.75 inches) and a thickness of 1.5 mm The briefcase is approx 100mm x 350mm x 500mm Reasoning The volume of the coin is therefore given by r2h = 3.14 x (20/2)2 x 1.5 ≈ 470 mm3 The volume of the case is approx 500 x 100 x 350 = 17,500,000 mm3 Thus the number of coins that will fit in (assuming no gaps) will be about 17,500,000 ÷ 470 = 37,000 (approx). There would be some gaps between coins, so Answer: The money will be worth around \$350. (Note that the total weight of the case would be almost 100 kilograms - far too heavy!)

Estimates for a Million, Set #1 (solutions)
Estimating for Amazing Facts: Set #2 (solutions) || Set #3 (solutions)
Estimates for the USA, Set #4 (solutions)

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