Henry Ernest Dudeney/Puzzles and Curious Problems/300 - Fresh Fruits/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $300$

Fresh Fruits
Some fresh fruit was being weighed for some domestic purpose.
It was found that the apples, pears and plums exactly balanced each other as follows:
One pear and three apples weigh the same as $10$ plums;
and one apple and six plums weigh the same as one pear.
How many plums alone would weigh the same as one pear?


Solution

One pear weighs the same as $7$ plums.


Proof

Let $a$, $p$ and $m$ denote the weights of one apple, one pear and one plum respectively.

We have:

\(\text {(1)}: \quad\) \(\ds p + 3 a\) \(=\) \(\ds 10 m\) One pear and three apples weigh the same as $10$ plums;
\(\text {(2)}: \quad\) \(\ds a + 6 m\) \(=\) \(\ds p\) and one apple and six plums weigh the same as one pear.
\(\ds \leadsto \ \ \) \(\ds a + 6 m + 3 a\) \(=\) \(\ds 10 m\) substituting for $p$ from $(2)$ into $(1)$
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds m\) simplifying
\(\ds \leadsto \ \ \) \(\ds p\) \(=\) \(\ds 7 m\) substituting for $a$ in $(2)$ and simplifying

$\blacksquare$


Sources

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