# Using Units to Deal With Density

Glancing over sample papers for the new GCSE, I stumbled on this:

Zahra mixes 150g of metal A and 150g of metal B to make 300g of an alloy. Metal A has a density of $19.3 \unit{g/cm^3}$. Metal B has a density of $8.9 \unit{g/cm^3}$. Work out the density of the alloy.

I don’t think I’m being mean to say that this would stump the majority of students. It’s probably designed to.

### Ahoy there, Mathematical Pirate!

“Aharr! I spies a compound unit!”

“Are you going to talk in that ridiculous manner for the whole blog post? I happen know you’re from Windsor.”

“*The rough end* of Windsor.”

“Fine. But let’s talk normally so as not to distract the readers.”

“Fine. You see that unit there, the $\unit{g/cm^3}$?”

“I do.”

“That’s a dead giveaway that you can use a formula triangle.”

“Teachers don’t like those, do they?”

“Many don’t. I’m in the ‘anything that works is OK at this stage’ camp. Say, you notice how when I talk normally, it’s really hard to keep track of who’s saying what?”

“That’s only because I don’t pepper the conversation with ‘said the Pirate.’”

“Arr. Now, what do you measure in grams? It’s mass. That’s on the top of the unit, so it goes on the top of the triangle. What’s in centimetres cubed? That’s volume, so it goes on the bottom. You’ve got one space left, so you may as well put the thing you’re measuring, density, in there.”

“So $M$ on top, and $V$ and $D$ on the bottom row, in either order.”

“In either order. So to find out the *volume* of 150g of each of the metals, we work out $150 \div 19.3$ and $150 \div 8.9$,” peppered the Pirate.

“If the Ninja’s not around, I’ll work those out on the calculator: 7.772 and 16.854 centimetres cubed, respectively.”

“So the volume of the alloy is those added together…”

“… $24.626\unit{cm^3}$ …”

“… and the density (referring back to the triangle) is 300 divided by that number you just said…”

“… $12.18\unit{g/cm^3}$.”

“Easy as pie-racy, arr. Ut-oh!”

### Enter the Ninja

“Er… hello, sensei, have you done something different with your bandanna?”

“Did someone mention a formula triangle?”

“Did they? I think you must have misheard.”

“Shenanigans, I say, shenanigans!”

“Strong language, sensei!”

“Deservedly so. In any case, it is simple algebra: when the masses are the same, you just need the harmonic mean: $\br{\frac{d_1^{-1}+d_2^{-1}}{2}}^{-1}$.”

“… obviously.”

“Or better, $\frac{2d_1 d_2}{d_1+d_2}$.”

“I’ll concede that that is quite pretty.”

“Not with these numbers. But $19.3 \times 8.9$ is 180 less about 5%, and their sum is 28.2. Let’s call it $360 \div 28$…”

“… some say that 360 is a nice number because it has lots of factors, sensei…”

“Some are fools. $360 \div 28$ is $90 \div 7$…”

“A shade less than 13…”

“12.857142 recurring, if you want to get snotty about it, but we don’t have that kind of accuracy here. We need to lose about 5% from the top and and extra 1% from the bottom, so I’d reckon taking off 0.6 or 0.7 from that would be ok.”

“Making 12.2, give or take.”

The Mathematical Ninja nodded.