In this question, assume that phosphorus has only two isotopes, one with a mass number of 30 and one with a mass number of 32.
The average mass of a naturally occurring sample of P is found to be 30.97376. What are the relative
abundances (percentages) of the two phosphorous isotopes in the naturally occurring sample?

³²P = 48.688%

³⁰P = 51.312%

Explanation:

The average mass is defined as the sum of each mass atom times its abundance. For the phosphorus with 2 isotopes with mass 30 and 32 uma, the average mass is defined as:

Average mass: Mass ³²P * Abundance + Mass ³⁰P * Abundance

30.97376 = 32 * X + 30 * Y (1)

Where X is abundance of ³²P and Y abundance of ³⁰P

Also, the sum of both abundance is 100% = 1:

1 = X + Y (2)

Replacing (2) in (1):

30.97376 = 32X + 30Y

30.97376 = 32X + 30(1-X)

30.97376 = 32X + 30-30X

30.97376 = 2X +  30

0.97376 = 2X

0.48688 = X

The percentage of ³²P is 0.48688*100 = 48.688%

And ³⁰P is 100 - 48.688 = 51.312%

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