Let 'x' and 'y' represent the two numbers, and let y>=x.

One number is 5 times another means:

y = 5x

The sum of the reciprocals is 24/5 means:

1/x + 1/y = 24/5

Plug y = 5x into 1/x + 1/y = 24/5:

1/x + 1/(5x) = 24/5

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click here to see the equation solved for x

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x = 1/4

5*1/4 = 5/4

Indeed:

1/(1/4) + 1/(5/4) = 4 + 4/5 = 24/5

The two numbers are 1/4 and 5/4.