There is a story about Archimedes that says a king asked him to work out if a crown was pure gold, or a mix of gold and silver. Archimedes found out by measuring the mass and volume of the crown. Gold weighs $20$ grams for every millilitre, and silver weighs $10$ grams for every millilitre. If the crown weighed $250$ grams, and its volume was $15$ millilitres, was it pure gold? Show answer

If the crown were pure gold, it would weigh $20\times15 = 300$ grams. It is lighter than that, at $250$ grams, so it can’t be pure gold.

Guess how many millilitres of gold and how many of silver went into the crown, and see if you are right. (This question is answered with algebra in Question 7.5.36.) Show hint

How heavy would it be if it were pure gold? What if it were pure silver? Does it seem to be mostly gold, or mostly silver?We already worked out that if the crown were pure gold, it would weigh $300$ grams. If it were pure silver, it would weigh $10\times15 = 150$ grams. $250$ is closer to $300$ than it is to $150$. In fact, $250$ is only $50$ away from $300$, but is $100$ away from $150$, and $100$ is twice $50$, so my guess would be that there is twice as much gold as there is silver, by volume. I am guessing that there were $10$ millilitres of gold, and $5$ millilitres of silver.

I can check if this could be correct by working out how much the crown would weigh if my guess were true. The $10$ millilitres of gold would weigh $20\times10 = 200$ grams and the $5$ millilitres of silver would weigh $10\times5 = 50$ grams, making $200 + 50 = 250$ grams together. So my guess may be right.