A business woman borrowed money in a village SACCO at 21/2% rate per month. After 5 months, she paid back a total amount of sh. 900,000 to the SACCO. Calculate the amount of money the woman borrowed.

Let $P$ be the principal amount borrowed.
Let $R$ be the interest rate per month.
Let $T$ be the time in months.
Let $I$ be the simple interest.
Let $A$ be the total amount paid back.

Given:
Rate $R = 2\frac{1}{2}\% = 2.5\%$ per month.
Time $T = 5$ months.
Total amount paid back $A = \text{sh. } 900,000$.

The formulas for simple interest are:

1. Simple Interest: $I = \frac{P \times R \times T}{100}$
2. Total Amount: $A = P + I$

Substitute the given values of $R$ and $T$ into the simple interest formula:

$$I = \frac{P \times 2.5 \times 5}{100}$$ $$I = \frac{12.5P}{100}$$ $$I = 0.125P$$

Substitute the given total amount $A$ and the expression for $I$ from Step 2 into the total amount formula:

$$A = P + I$$ $$900,000 = P + 0.125P$$

Combine the terms involving $P$:

$$900,000 = (1 + 0.125)P$$ $$900,000 = 1.125P$$

Divide both sides by 1.125:

$$P = \frac{900,000}{1.125}$$ $$P = 800,000$$

The amount of money the woman borrowed was sh. 800,000.

The final answer is $\boxed{\text{sh. } 800,000}$.

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