Solve the system of equations to find the values of x, y and z: xy/(x+y)=1/2, yz/(y+z)=1/3, zx/(z+x)=1/7
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?
Two solutions were received to this problem, both using the same
method, and both from Yorkshire. Thank you to Vassil Vassilev of
Lawnswood High School in Leeds and Catherine Harrison of The Mount
School in York.
First, we need to see what type of jewels each jeweller has:
Then, we convert this to equations:
( r = ruby, s = sapphire, p = pearl, d = diamond )
5r + s + p + d = r + 7s + p + d => 4r = 6s
r + 7s + p + d = r + s + 97p + d => 6s = 96p
r + s + 97p + d = r + s + p + 2d => 96p = d
From here, we can see that:
1 diamond = 96 pearls
1 sapphire = 16 pearls
1 ruby = 24 pearls
If we say that a pearl has a price of 1, then:
After the sharing of jewels, each jeweller has a stock worth