Think of a Number/Examples/Rhind Papyrus 33

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Example of Think of a Number puzzles

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

A number, plus its two-thirds, and plus its half, plus its seventh, makes $37$. What is the number?

It is to be expressed in Egyptian form.


Solution

This can more clearly be expressed as:

I think of a number.
If you add $\dfrac 2 3$ of that number plus $\dfrac 1 2$ of that number and $\dfrac 1 7$ of that number, you get $37$.
What number did I think of?


The number was $16 + \dfrac 1 {56} + \dfrac 1 {679} + \dfrac 1 {776}$.


Proof

Let $x$ be the number first thought of.

We have:

\(\ds x + \dfrac 2 3 x + \dfrac 1 2 x + \dfrac 1 7 x\) \(=\) \(\ds 37\)
\(\ds \leadsto \ \ \) \(\ds x \paren {1 + \dfrac 2 3 x + \dfrac 1 2 x + \dfrac 1 7}\) \(=\) \(\ds 37\)
\(\ds \leadsto \ \ \) \(\ds x \paren {\dfrac {42 + 28 + 21 + 6} {42} }\) \(=\) \(\ds 37\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds 37 \times \dfrac {42} {97}\)
\(\ds \) \(=\) \(\ds \dfrac {1554} {97}\)
\(\ds \) \(=\) \(\ds 16 + \dfrac 2 {97}\)
\(\ds \) \(=\) \(\ds 16 + \dfrac 1 {56} + \dfrac 1 {679} + \dfrac 1 {776}\) from a table of Egyptian fractions

$\blacksquare$


Sources

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