Helping Ada Lovelace with her Homework: Classroom Exercises from a Victorian Calculus Course – Solution to Problem 1

De Morgan enclosed his solution to Problem 1 in a letter to Lovelace of November 14, 1840 [LB 170, 14 Nov. 1840, f. 20v]. Beginning with the initial equation
$$\frac{1+b}{1-b}=\frac{1+x}{x}$$
he multiplied both sides by $1-b$ and $x$:
$$(1+b)x=(1-b)(1+x).$$ Then, distributing
$$x+bx=1+x-b-bx$$ cancelling an $x$ from both sides
$$bx=1-b-bx$$ collecting like terms
$$2bx+b=1$$ factoring out the $b$
$$(2x+1)b=1$$ and finally dividing by $2x+1$ gave
$$b=\frac{1}{2x+1}.$$

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