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Whole Numbers Only

Can you work out how many of each kind of pencil this student bought?

Squaring the Circle

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make an estimate.


What number could replace * so that */5 is between 3 and 4?

Does This Sound about Right?

Age 11 to 14
Challenge Level

1. Yes - As $ 1 \textrm{m} = 100 \textrm{cm} $ a crate with dimensions 100cm by 50cm by 50cm will have volume $ V = 250,000 \textrm{cm}^3 $. Hence, the number of water bottles required to fill this up is

$$ N = \frac{250,000}{500} = 500 $$


2. Yes - If the diameter of the tree is almost half a meter, its radius is almost 25 cm. Now, if the tree's radius grows by 0.6 cm in each year, in 42 years the radius should be about $ 42 \cdot 0.6 = 25.2 \textrm{cm} $, and this is about right. However, to make a safer estimate (as it is unlikely that the tree will be growing by exactly 0.6 cm each year) a range of the form 38 - 45 years would be better.


3. No - While $365 \cdot 0.3 = 10.95 \textrm{kg}$, which is close to the estimate, perhaps it would be sensible to consider that the scientist only has crisps on the working days of the year (since she is having a pack in her morning break). Now, a person is on average working 44 weeks each year, so they have $44 \cdot 5 = 220 $ working days (and perhaps slightly less, if we take into account bank holidays).

Therefore, the scientist is more likely to be consuming $220 \cdot 0.3 = 6.6 \textrm{kg} $ of crisps each year.


4. No - A teaspoon of sugar weighs about 5 grams. So, the scientist and her friend consume together 8 cups of coffee each day, with 16 teaspoons of sugar. So, each day they consume $16 \cdot 5 = 80 \textrm{g}$ of sugar. Thus, in a two - month period, they would need $ 60 \cdot 80 = 4800 \textrm{g} $ of sugar, which is much more than a packet!


5. No - Once again, the calculation $365 \cdot 22 = 8030 $ is correct, but 365 is not the right number to use, as she is not working every day of the year. Using our previous estimate for the number of working days in a year, we see that the scientist is actually going to claim around $220 \cdot 22 = 4840 $ miles from work.


6. Yes - In this case, it is highly likely that the lab will be working all year round, so the estimate $560 \cdot 12 = 6720 $ is about right.


7. Yes - We can model the rubble by a large cone, whose height is 2m and base radius is 1.5m. The volume of such a cone is $$ V = \frac{1}{3}\cdot \pi \cdot r^2 \cdot h = \frac{1}{3} \cdot \pi \cdot 1.5^2 \cdot 2 \approx 4.7 m^3 $$ So the estimation is correct, and a 5 cubic meter skip should be large enough.


8. No - $124 \ cdot 170 = 21080 m^3 $ mean that each minute there are 21080 cubic meters of water flowing. Hence, over the course of an hour, there will be $21080 \cdot 60 = 1,264,800 m^3$ flowing, which is equivalent to 1,264.8 metric tonnes.


9. Yes - The area of the vegetable plot is $ A = 9.5 \cdot 11 = 104.5 m^2 $. Now, with the test data as described, we expect that on average there are $\frac {42 + 53}{2} = 47.5 $ worms per square meter.

Hence, a good estimate for the number of worms in the whole plot is $ N = 47.5 \cdot 104.5 = 4963.8 $, so the scientist is correct in this case.


Well done to the students in Class 5C of the Brooklands Primary School, for sending us some very well-reasoned and clear answers to most of these short problems!