Answer

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**Hint**: In this problem the lady spends money sequentially at different spaces. She spends a fraction of what is left on next purchase. This is the problem that will take us into a fractional equation giving the last solution that is money left one rupee.

Try to solve the problem along with the journey of the lady.

**:**

__Complete step-by-step answer__Consider the amount lady had initially $ = x\,\,Rupees$

First, she spent half she had for hankie and gave one rupee to the beggar

Money spent:

= $ \dfrac{x}{2} + 1$

Money left:

$

= x - \left( {\dfrac{x}{2} + 1} \right) \\

\Rightarrow \dfrac{x}{2} - 1 \;

$

Second, she spent half left on lunch with two rupee tip

Money spent:

\[

\Rightarrow \dfrac{{\dfrac{x}{2} - 1}}{2} + 2 \\

\Rightarrow \dfrac{x}{4} + \dfrac{3}{2} \;

\]

Money left:

$ $ \[

\Rightarrow \dfrac{x}{2} - 1 - \left( {\dfrac{x}{4} + \dfrac{3}{2}} \right) \\

\Rightarrow \dfrac{x}{4} - \dfrac{5}{2} \;

\]

Third, purchase she spent half of what was left on a book and 3 rupees on bus fare

Money Spent:

$

\Rightarrow \left( {\dfrac{{\dfrac{x}{4} - \dfrac{5}{2}}}{2} + 3} \right) \\

\Rightarrow \dfrac{x}{8} + \dfrac{7}{4} \;

$

Money left:

\[

\Rightarrow \dfrac{x}{4} - \dfrac{5}{2} - \left( {\dfrac{x}{8} + \dfrac{7}{4}} \right) \\

\Rightarrow \dfrac{x}{8} - \dfrac{{17}}{4} \;

\]

Money she left with is one rupee

\[

\Rightarrow \dfrac{x}{8} - \dfrac{{17}}{4} = 1 \\

\Rightarrow \dfrac{x}{8} = \dfrac{{21}}{4} \\

\Rightarrow x = 42\,Rupees \;

\]

Hence, she had 42 rupees at the start of the journey.

**So, the correct answer is “42 rupees”.**

**Note**: The concept of fraction is widely used in many areas like calculating age, estimating profit loss, getting the share of profit of one among the three, etc. The simplification involves all mathematical operations on one variable. The use of multiple operations makes the problem tricky.

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