This year Rana will save certain amount of his income, and he will spend the rest. Next year Rana will have no income, but for each taka that he saves this year, he will have \((1+r)\) taka available to spend. In terms of r, what fraction of his income should Rana save this year so that next year the amount available for his spending will be equal to half the amount that he spends this year?

Explanation: Let \(X\) be the amount spent and \(Y\) be the amount saved this year. Income is \(X+Y\). Next year's spending available: \(Y(1+r)\). Next year's spending available = \(1/2\) of this year's spending: \(Y(1+r) = X/2\). We need to find the fraction of income saved this year: \(Y/(X+Y)\). From the condition: \(X = 2Y(1+r)\). Substitute \(X\) into the fraction: \(Y/(X+Y) = Y / (2Y(1+r) + Y) = Y / (Y(2(1+r) + 1)) = 1 / (2(1+r) + 1) = 1 / (2+2r+1) = 1 / (2r+3)\).