Schools Mathematics Grand Challenge
Week three's Puzzles
Problem 7:
The solution was:
We'll use B for the amount of cola in a bottle and G as the amount in
a glass. When John fills 6 glasses using 6 bottles, he has one bottle
left over,
6B = 6G + 1B,
so 5B = 6G, or B = 6/5 G. Now, we want to know what 36 bottles will
fill. 36B = 36*6/5 G = 43.2 G. So we can fill 43 whole bottles and have
a little left at the end.
Problem 8:
The solution was:
The area of a circle is pi*r*r, where pi is 3.14159... and r is the
radius of the circle. The area of a semicircle is just half this,
or pi*r*r/2, and the area of a quarter circle is one quarter this,
pi*r*r/4.
Let the r be the distance from a to b. Then the area of the semicircle
centered at a is pi*r*r/2, and the area of the semicircle centered
at c is the same. The area of the big quarter circle is pi*2r*2r/4 = pi*r*r,
because its radius is twice as big as the small circles.
We've marked in the area of each white bit as A. The area of the
semicircles is:
pi*r*r/2 = I + A
So, the area of I = pi*r*r/2  A. The area of the big quarter
circle is:
pi*r*r = I + II + A + A
if we subtract off the area of a semicircle circle, then we get,
pi*r*r  pi*r*r/2 = I + II + A + A  I  A
or
pi*r*r/2 = II + A
so II = pi*r*r/2  A. This means that the area of I and II are the
same, so the ratio of the areas is 1.
Problem 9:
The solution was:
When the driver drives from the start to the end, suppose he drives
a distance U uphill, D downhill and F on the flat. It takes him 2 hours,
so:
U/112 + F/126 + D/144 = 2
on the way back again, he is going the opposite direction, so the uphill
bits become downhill and the downhill bits become uphill. It takes him
2 hours 20 minutes, which is two and a third hours:
U/144 + F/126 + D/112 = 7/3
We want to know how long the track is, which is U+F+D. This looks
a bit like simultaneous equations, but we have three things we don't
know, and only two equations, so we can't actually find U, F and
D, but we can find their sum. If we add the two equations, we get:
U/112 + U/144 + F/126 + F/126 + D/144 + D/112 = 2 + 7/3
Then we add up all the fractions: 1/112 + 1/144 = 1/63, 1/126 +
1/126 = 1/63 and 2+7/3 = 13/3. So we get:
(U + F + D)/63 = 13/3,
or U + F + D = 63*13/3 = 273, so the answer is 273 kilometres.


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