1. This weeks

2. Year 7 & 8 Middle and Upper Band
Year 7 & 8 Fast Track
Year 7 & 8 High Achievers
Year 9 & 10 Middle and Upper Band
Year 9 & 10 Fast Track
Year 9 & 10 High Achievers

3. Year 7 & 8 Middle and Upper Band
Loose Change
In how many ways can you give change for a ten pence piece?
ANSWER
If there are two 5 p pieces, this gives one way. If there is one 5 p piece, then there are two 2 p coins, one 2 p coin or no 2 p coins, with all the other coins as 1 p coins. If there are no 5 p coins, there are between 0 and 5 coins that are 2 p, and all the rest are 1 p. Therefore there are 1+3+6=10 ways to give change.

4. Year 7 & 8 Fast track
Find from Factors
A certain number has exactly eight factors including 1 and itself.
Two of its factors are 21 and 35. What is the number?
ANSWER
105 If 21 and 35 are factors of the number, then 3, 5 and 7 must all be included amongst its prime factors. This means that the required number must be a multiple of 105 and, as the complete list of factors of 105 is 1, 3, 5, 7, 15, 21, 35 and 105, the answer is 105.

5. Centred What is the size of angle BAC? Year 7 & 8 High Achievers
In the triangle ABC, AD=BD=CD.
What is the size of angle BAC?
ANSWER
∠BAC=90∘ A , B and C are all equidistant from D and therefore lie on a circle whose centre is D . BC is a diameter of the circle and ∠BAC is therefore the angle subtended by a diameter at a point on the circumference (the angle in the semicircle). Alternatively, suppose ∠ACD=x . Then, ∠BAC=x also, as DAC is isosceles. This means ∠CDA=180−2x by angles in a triangle, so ∠ABC=2x . Then, as ADB is also isosceles, ∠BAD=∠DBA=12(180−2x)=90−x . Therefore, ∠BAC=∠BAD+∠DAC=90−x+x=90∘

6. Year 9 & 10 Middle and Upper Band
Find from Factors
A certain number has exactly eight factors including 1 and itself.
Two of its factors are 21 and 35. What is the number?
ANSWER
105 If 21 and 35 are factors of the number, then 3, 5 and 7 must all be included amongst its prime factors. This means that the required number must be a multiple of 105 and, as the complete list of factors of 105 is 1, 3, 5, 7, 15, 21, 35 and 105, the answer is 105.

7. Centred What is the size of angle BAC? Year 9 & 10 Fast Track ANSWER
In the triangle ABC, AD=BD=CD.
What is the size of angle BAC?
ANSWER
∠BAC=90∘ A , B and C are all equidistant from D and therefore lie on a circle whose centre is D . BC is a diameter of the circle and ∠BAC is therefore the angle subtended by a diameter at a point on the circumference (the angle in the semicircle). Alternatively, suppose ∠ACD=x . Then, ∠BAC=x also, as DAC is isosceles. This means ∠CDA=180−2x by angles in a triangle, so ∠ABC=2x . Then, as ADB is also isosceles, ∠BAD=∠DBA=12(180−2x)=90−x . Therefore, ∠BAC=∠BAD+∠DAC=90−x+x=90∘

8. Weighing the Baby Year 9 & 10 High Achievers
Weighing the baby at the clinic was a problem. The baby would not keep still and caused the scales to wobble. So I held the baby and stood on the scales while the nurse read off 78kg. Then the nurse held the baby while I read off 69kg. Finally I held the nurse while the baby read off 137kg. What was the combined weight of all three?
ANSWER:
Let m be my weight, b be the baby's weight and n be the nurse's weight. We have m+b = 78
b+n = 69
m+n = 137.
We can add all three equations together like this: (m+b)+(b+n)+(m+n)=
It follows that 2(m+b+n)=284 and hence m+b+n=142 . So the combined weight is 142kg . For an extra challenge can you find the weights of each person?