Mohan takes $16$ days less than Manoj to do a piece of work. If both working together can do it in $15$ days, in how many days will Mohan alone complete the work?

Let Manoj take x days, so Mohan takes (x - 16) days. Using the combined work formula, 15 times x plus 15 times (x - 16) equals x times (x - 16), which simplifies to x squared minus 46x plus 240 equals zero. Factoring gives x equal to 40 or 6, but x equal to 6 is invalid because x minus 16 would be negative. Manoj takes 40 days and Mohan takes 40 minus 16 days, resulting in 24 days.

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