Think of a Number/Examples/Rhind Papyrus 30

Example of Think of a Number puzzles

Problem $30$ of the Rhind Papyrus is as follows:

If the scribe says to thee:

$10$ has become $\dfrac 2 3 + \dfrac 1 {10}$ of what?

This can more clearly be expressed as:

I think of a number.

$\dfrac 2 3$ of that number plus $\dfrac 1 {10}$ of it make $10$.

What number did I think of?

The number was $13 + \dfrac 1 {23}$.

Let $x$ be the number first thought of.

We have:

$\ds \dfrac 2 3 x + \dfrac 1 {10} x$

$=$

$\ds 10$

$\ds \leadsto \ \ $

$\ds x \paren {\dfrac 2 3 + \dfrac 1 {10} }$

$=$

$\ds 10$

$\ds \leadsto \ \ $

$\ds x \paren {\dfrac {20 + 3} {30} }$

$=$

$\ds 10$

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds \dfrac {300} {23}$

$\ds $

$=$

$\ds 13 + \dfrac 1 {23}$

$\blacksquare$