When a crystal is forming, it's mass is increasing evenly.
Scientists noticed that when two crystals were forming, first's mass increase in 3 months was equal to second's mass increase in 7 months.
After one year first crystal's mass was 4% bigger and second's - 5%. Find ratio of crystals' initial masses.

Question

When a crystal is forming, it's mass is increasing evenly. 
Scientists noticed that when two crystals were forming, first's mass increase in 3 months was equal to second's mass increase in 7 months. 
After one year first crystal's mass was 4% bigger and second's - 5%. Find ratio of crystals' initial masses.

zyberg · Answer

The question is quite clear to me, however, I can't think of a mathematical formula between percentage increase, mass and time (I would think of a formula before solving initial problem with distance or work problems). 
All that I can think of is something like this:
$$m_{new}= m_{old}(1+i)$$ Where i is increase in percentages. The problem with this is that it doesn't take account of time...

zyberg · Answer

Noticed that it is possible to play around with powers, but then, there are another unknown member - i:

$$m_{After12months} = m_{old}(1+i)^{12}$$

zyberg · Answer

With my formula, I would get answer as:
$$\frac{ m_{starting1} }{ m_{starting2} } = \frac{ (1+i)^{12}-1.05 }{ (1+i)^{12}-1.04 }$$
But it is not a good answer as I got a new variable i involved... Anyone have any ideas?

mww · Answer

Interested in what mass increasing 'evenly' means - is this a constant rate of increase or does the rate of increase depend on the mass of the crystal?
This is a little ambiguous as to what model the increase takes.

zyberg · Answer

@mww, the problem states that it's increasing evenly / gradually / uniformly. Not sure what problem actually means. Could you help me with both ways of solving it depending on interpretation? (I think that it's better to solve it in both ways than to leave it unsolved)

acespeedfighter · Answer

2

mww · Answer

This one's a protracted question. I'm a bit tired right now so I'll simply tell you how the linear structure would work:

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Let a(t) and b(t) represent the amount of crystal at time t
The gradient for a and b is given by 
$$\frac{ a(3)-a(0) }{ 3 }$$
$$\frac{ b(7)-b(0) }{ 7 }$$
where a(0) and b(0) are initial amounts of crystal mass.

Thus the equations for the lines would be 
$$a(t) =  \frac{ a(3)-a(0) }{ 3 }t + a(0)$$
$$b(t) = \frac{ b(7)-b(0) }{ 7 }t + b(0)$$
following the form y = mx + b

Also the increase in a(t) for 3 months is mirrored by that of b(t) in 7
Thus we can write
$$a(3) -a(0)=b(7)-b(0)$$

We also have after one year, a(12) = 1.04 x a(0) and b(12) = 1.05 x b(0)
I think plugging in some of these around will get to the answer (say sub in t = 12 into line (At) and b(t) and see if you can reduce the variables.