At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.
At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening?
According to Plutarch, the Greeks found all the rectangles with
integer sides, whose areas are equal to their perimeters. Can you
find them? What rectangular boxes, with integer sides, have their
surface areas equal to their volumes?
Lots of great solutions were submitted to this
problem, using a wide variety of approaches. The problem prompted
you to use fractions, ratios, percentages, and graphs. In addition,
you could investigate and consider which methods worked most
effectively in different situations.
The first part of the task is to
determine the mixture with the stronger tasting lemonade
Mahir, from Saltus Grammar School
converted the values given so that there is the same amount of
water in each glass:
For the first glass, we now have:
$400$ml lemon juice
and $600$ml water
For the second glass, we now have:
$300$ml lemon juice
and $600$ml water
From this, we can now tell that the mixture in
the first glass must taste stronger: for the same amount of water,
there is more lemon juice.
Jonathan, from Wilson's School, used a similar
method. However, he instead made the amounts of lemon juice equal,
and then saw which glass had more water. The glass with less water
for the same amount of juice will be stronger, as the juice is less
Another related method, used by many people
was to use ratios, fractions and/or percentages. Will K. from
Wilson's School gave a lovely explanation:
Will expressed the strength of the
lemonade as a percentage, as these can be easily compared.
Sharumilan, also from Wilson's School converted the fractions so
that they had a common denominator. In this way, they can be more
easily compared. In fact, this is the same as converting to
percentages; percentages are fractions with a denominator of $100$!
Here is Sharumilan's answer:
What about a more visual approach? Iona,
from Whitby Maths Club compared the two solutions by drawing
Dominic, from Wilson's School, also suggested a graphical
Another suggestion made by several people
would be to convert the amounts so that there are equivalent
amounts of water or lemon juice, as described above. Then, you
could draw a glass and visually see which is the stronger
Dulan, from Wilson's School, suggested a very
nice method, which can display graphically multiple different
He thought that a graph could be constructed with $x$ and $y$
axes. On the $x$ axis could be "amount of lemon juice", and on the
$y$ axis, "amount of water". Try constructing this for
What does it mean if two mixtures have the same $x$ coordinate, but
different $y$ coordinates?
What if they have the same $y$ coordinates, but different $x$
You should be able to construct straight lines from the origin to
the various points representing different mixtures and compare
Along each line the strength of all of the mixtures is the same as
the proportions do not change.
We have now seen different examples of
approaches to this problem. Do the different methods always work?
Which method is most efficient?
Nathan, from Wilson's School noted that
different methods are more efficient in different situations:
Several people had their own preference of
method, depending on what they felt most comfortable using.
Sharumilan explained this, and also examined
the combination of the different mixtures:
Try this out for yourself, with some squash,
Thank you very much to everyone who submitted
solutions. There were many correct solutions, and so we could not
mention them all. Well done!
If you enjoyed this problem, try the
extension problem Ratios and Dilutions.