Let's take it that Frances joined the business n months after Abiola started it.

Abiola invested $1,250 for 12 months for the same profit as Frances who invested $1875 for n months.

Since the profit are shared equally, then Abiola's contribution is the same as Frances' contribution.

Contribution = Amount invested x Number of months in the business

Abiola's contribution = $1,250 x 12

Number of months that Frances was in the business = 12 - n

Frances' contribution = $1875 x (12 - n)

1250 x 12 = 1875 x (12 - n)

Divide both sides by 1875

12 - n = $\frac{\mathrm{1250\; x\; 12}}{1875}$

12 - n = 8

12 - 8 = n

n = 4 months

∴ Frances joined the business 4 months after Abiola started.