Henry Ernest Dudeney/Puzzles and Curious Problems/143 - Lilivati, A.D. 1150/Solution

From ProofWiki
Jump to navigation Jump to search

Puzzles and Curious Problems by Henry Ernest Dudeney: $143$

"Lilivati", A.D. $1150$
Beautiful maiden, with beaming eyes, tell me which is the number that, multiplied by $3$,
then increased by three-fourths of the product,
divided by $7$,
diminished by one-third of the quotient,
multiplied by itself,
diminished by $52$,
the square root found,
addition of $8$,
division by $10$,
gives the number $2$?


Solution

$28$


Proof

We simply reverse the order of operations, calculating as we go:

\(\ds 2 \times 10\) \(=\) \(\ds 20\) division by $10$
\(\ds 20 - 8\) \(=\) \(\ds 12\) addition of $8$
\(\ds 12^2\) \(=\) \(\ds 144\) the square root found
\(\ds 144 + 52\) \(=\) \(\ds 196\) diminished by $52$
\(\ds \sqrt {196}\) \(=\) \(\ds 14\) multiplied by itself
\(\ds 14 \div \paren {1 - \dfrac 1 3}\) \(=\) \(\ds 21\) diminished by one-third of the quotient
\(\ds 21 \times 7\) \(=\) \(\ds 147\) divided by $7$
\(\ds 147 \div \paren {1 + \dfrac 3 4}\) \(=\) \(\ds 84\) then increased by three-fourths of the product,
\(\ds 84 \div 3\) \(=\) \(\ds 28\) multiplied by $3$

Hence the result.

$\blacksquare$


Historical Note

Henry Ernest Dudeney quotes this in its entirety as an example of one of the problems in Lilavati, by Bhaskara $\text {II}$ Acharya, dating from approximately $\text {1150}$ $\text {C.E.}$

In Dudeney's words:

Here is another little morning problem from Lilivati (A.D. $1150$)...
... This, like so many of those old things, is absurdly easy when properly attacked.

It will be noted that he refers to it as Lilivati, which is most probably incorrect.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Henry_Ernest_Dudeney/Puzzles_and_Curious_Problems/143_-_Lilivati,_A.D._1150/Solution&oldid=561753"