last question to do Watch

hi, I'm hoping someone would be able to help. Myself and the rest of my class are all stuck on this last question.

A new element has three isotopes with mass numbers of 288, 290 and 291. The weight averaged atomic mass of this element is 289.2 and the percent abundance of the heaviest isotope is 35%, what are the percent abundances of the other two isotopes?

The element in question is one the tutor has made up. Several of us have all come to different answers, one not even using algebra, we need help to make it as clear as possible and as simple as possible to understand please.

Hi there!
Firstly, it is important to understand that the Average Atomic Mass = The Sum of (Each Atomic Mass x Percentage Abundance) = (M1 x P1) + (M2 x P2) + (M3 x P3)

289.2 = (288 x P1) + (290 x P2) + (291 x 0.35)
289.2 = 288P1 + 290P2 + 101.85
187.35 = 288P1 + 290P2
P1 + P2 + P3 = 1 (as all percentages add up to 100%)
1-0.35 = 0.65
Hence, P1 + P2 = 0.65
Hence, there are a system of simultaneous equations to solve:

187.35 = 288P1 + 290P2
P1 + P2 = 0.65—> (Rearrange) P1 = 0.65 - P2
187.35 = 288(0.65-P2) + 290P2 —>
187.35 = 187.2 - 288P2 + 290P2 —> 2P2 = 0.15
:. P2 = 0.075
As P1 = 0.65 - P2
P1 = 0.65 - 0.075 = 0.575
Hence, P2 = 7.5% (of 290 isotope)
P1 = 57.5% (of 288 isotope)

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