How to multiply fractions...& find out what your scrap gold is worth at the same time!

Part of a series of hubs aimed at demystifying tricky topics in everyday mathematics

"Why should I learn fractions? They're never going to be any use!" is something I heard several times from students when I was teaching adult numeracy classes. I must confess that this question had me stumped back then, and I was never able to give an answer other than the rather wimpy "Well, it helps keep your brain active…" I have to admit that as someone who had fractions dinned into me at school (in the days before calculators became universal), I often find it easier to work things out using fractions rather than via other methods. Want to divide a large number with lots of zeros at the end by another, smaller number, also with lots of zeros at the end? Turn it into a fraction, and cancel the zeros. For example, one million divided by one thousand looks like this as a fraction:

1000000
1000

Cancel by crossing three zeros off and you end up with

1000
1

i.e. an answer of 1,000.

Finally, I've found a use for fractions!

Now, two years after having left teaching, I now think I have the answer to the "What's the use of knowing about fractions?" question my students were asking. If you learn about fractions (specifically how to multiply and cancel fractions), you can use your new-found knowledge to work out the price of your scrap gold should you wish to sell it. With the price of gold at an all-time high, the world and his dog has turned into a dealer, clamouring to buy your scrap gold. But a lot of them are offering way under the odds. If you can calculate the price of gold yourself before trying to sell it, you're well on the way to protecting yourself against being ripped off.

If you've got some gold you want to sell, there are three things you need to know before working out how much it's worth:

• How much it weighs - buy or borrow a pair of digital scales that can work things out to the nearest tenth of a gram (0.1 g)
• How pure it is (how many carats, in other words?)
• The current price of gold.

First, let's talk about how to find the current price (the "spot price") of pure gold.  At the time of writing, it's approximately £680 pounds per ounce. The "ounce" being talked about isn't the ounce used when weighing out sugar or flour (known as the avoirdupois ounce), but the so-called troy ounce, which is a bit heavier. A troy ounce is 31.1 grams (or 31 grams if you don't mind being a bit less precise. This is the figure I'll be using in my calculation below, to make things a bit easier).

As you'll be aware, any gold jewellery or coins you have in your collection are almost certainly not made from pure gold, but from an alloy whose gold purity is quoted in carats. Pure gold is 24 carats. So for example nine carat gold actually has 9 parts gold out of 24 total (in other words, it contains 9 parts gold and 15 parts base metal).

Let's say you have some scrap 9 carat gold that you want to sell. It weighs 10 grams. How would you use fractions to work out its value?

The first hurdle to get over is that the spot price of pure gold given above (£680) is the price per troy ounce, not the price per gram. If you want to work out the price per gram, you need to divide this by 31. Expressed as a fraction, it looks like this:

680
31

Don't forget also that 9 carat gold is only 9 parts in 24 pure, which as a fraction is written like this:

9
24


To find the price of one gram of 9 carat gold, your fraction calculation would look like this:

And if you're working out the price of ten grams of gold, the calculation becomes:

You can simplify this by cancelling some of the fractions. Thus 9/24 can be cancelled down to 3/8 (by dividing both the numerator and denominator by 3), as follows:

When cancelling, you can also do it diagonally across two different fractions as well as up and down within the same fraction.  So in this calculation, the 680 and the 8 can be cancelled:

Once you get down to this stage, you can either whip the calculator out, do a spot of manual long division, or - if you don't mind losing a bit of accuracy - you can round the numbers off a bit.  So if you round the 31 to a 30, you get 2550/30, which cancels down to 255/3, or 85.  Thus your 10 g of 9 carat gold is worth a bit less than £85 at current spot prices. 

Obviously anyone you sell to is going to want to make a bit of profit, so they're not going to give you that.  But at least you know that if they offer you - say - £40 as opposed to £70 or more, you're being ripped off.

Some other maths stuff to do with gold

Sometimes you won't see a carat value stamped on the gold, but something like "0.375".  This is 375 parts gold per thousand which, written as a fraction, looks like this:

 375
1000

Cancel this down.  You can do this by jumping straight in and dividing the numerator and denominator by 25 or if that's too much to take on in one go, keep dividing by 5, then by 5 again and you end up with 3/8.  In other words (going back to my example above), this is 9 carat gold.

Similarly, the stamp "0.75" or "0.750" means 750 parts per thousand, which cancels down to 3/4.  If you multiply both numerator and denominator by 6 so that the denominator is 24, you get 18/24, i.e. 18 carat gold.