Alan

Average speed = total distance/total time.

Let t be the total time and d be the total distance. Then

d = 60*t/2 + 100*t/2

so d = t(30 + 50)

d = t*80

so average speed = d/t = 80 km/hr

If the question were 60km/hr for half the distance and 100 km/hr for the other half distance, then

t = d/(2*60) + d/(2*100) 

t = d*(1/120 + 1/200) 

t = d/75

so average speed = d/t = 75 km/hr

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I conjecture that the answer is 256 (= 2^9-1) based on extrapolating results for sets {1,2}, {1,2,3} and {1,2,3,4}.  However, I have yet to find an elegant proof!

Partial reasoning:

If we have a number of arrangements for the set A = {1,...,n-1} then we can immediately get that many again for the set B = {1, ...,n} by simply putting n after all the arrangements for A.  Any extra arrangements for B must end in 1, because if they end in anything other than 1 or n, the end value will be bigger than 1 and smaller than n, violating the criteria.  It seems that there are just as many arrangements ending in 1 as there are ending in n, hence, since there are just 2 arrangements, for the set {1.2} each extra digit doubles the number of arrangements.  There must be a simple proof of the fact that there are just as many arrangements ending in 1 as there are ending in n (if it's true!), but I'm having a mental block trying to see it!

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Nice, elegant reasoning Bertie!

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It means you don't include a numerical approximation for pi in your answer, you just leave pi as pi.  

For example, if you were asked to turn 45 degrees into radians, leaving your answer in terms of pi, you would express the answer as pi/4 radians, you wouldn't go on to say this is approximately 3.14/4 =  0.785 radians.

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First consider the number of ways they can be seated so that no boys are next to each other.  The only way this can happen is if the boys are on seats 1, 3, 5, 7 and 9.  There are 5 boys who could go on seat 1.  For each of these there are 4 boys who could go on seat 3 ...etc., so there are 5! ways of arranging the 5 boys on these seats.  For each of these there are 4! ways of arranging the girls on seats 2, 4, 6 and 8.  So in total there are 5!*4! ways of arranging everyone on 9 chairs such that no boys sit together.

Now there are 9! ways of arranging everyone on the 9 chairs, regardless of who sits next to whom.

This means there must be 9! - 5!*4! ways of arranging everyone on the 9 chairs such that at least two boys are next to each other.

$${\mathtt{9}}{!}{\mathtt{\,-\,}}{\mathtt{5}}{!}{\mathtt{\,\times\,}}{\mathtt{4}}{!} = {\mathtt{360\,000}}$$

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Let p be cost of a pillow and c be cost of a case.  Then

10p + 15c = 175

p = c + 6.5

Replace the p in the first equation using the second equation:

10(c + 6.5) + 15c = 175

25c + 65 = 175

25c = 110

c = 4.4

So a pillow case costs £4.40

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$$S=\frac{U\times V}{U+V}$$

Multiply both sides by U + V:

$$S\times (U+V)=U\times V$$

Expand the left-hand side:

$$S\times U + S\times V = U\times V$$

Subtract U*V and S*V from both sides:

$$S\times U-U\times V=-S\times V$$

Factor out U on the left-hand side:

$$U\times (S-V)=-S\times V$$

Divide both sides by S - V:

$$U=-\frac{S\times V}{S-V}$$

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If N(t) is the population after t years, we write:

N(t) = 895*k^t where k is found from the fact that when t = 1 year N(1) = 1.07*895, so:

1.07*895 = 895*k¹  so k = 1.07

Hence N(t) = 895*1.07^t

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Use asin (press the 2nd button first)

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