This problem is from the Algebra of Abu Kamil Divide 10 into

This problem is from the Algebra of Abu Kamil.

Divide 10 into two parts in such a way that when a certain one of these parts is divided by the other and the fraction is multiplied by its numerator, the result gives 9.

There should be two answers. Thank you!

Solution

Let one part be x. Then the other part is 10-x. Then there are two cases:

Case 1:

[x/(10-x)]*x = 9 or, x² = 9(10-x) or, x² = 90-9x or, x² +9x -90 = 0 or, x² +15x-6x -90 = 0 or, x(x+15)-6(x+15) = 0 or, (x+15)(x-6)= 0. Therefore, x = 6 or x = -15. If we want two numbers smaller than 10, then x = 6

Case 2:

[(10-x)/x] *(10-x) = 9 or, (100-20x+x²)/x = 9 or, x² – 20x +100 = 9x or, x² -29x +100 = 0 or, x²-25x -4x +100 = 0 or, x(x-25) -4(x-25) = 0 or, (x-4)(x-25) = 0. If we want two numbers smaller than 10, then x =4.

Thus the answer is that the two parts of 10 satisfying the given condition are 6 and 4.

Note: We may derive the other part of 10 i.e. 4 from Case 1 also by subtracting 6 from 10. The 2^nd Case is only to demonstrate that the two possibilities give the same result.