Can you work out the individual prices of two menu items, knowing only about the value and ordered items from two orders?

**The Problem**

Jack orders two coffees and six macaroons and pays £6.20. Kate orders three coffees and four macaroons and pays £5.55. How much does a coffee cost and how much does a macaroon cost?

Show Solution

**The Solution**

This is simply a 'wordy' simultaneous equations problem. To solve, we should start by writing down the simultaneous equations. Let ??c?? equal the price of a coffee and ??m?? equal the price of a macaroon. Then we can say that:
$$ 2c + 6m = 6.20 \:\:\: \mathrm{(Equation 1.)}$$
$$ 3c + 4m = 5.55 \:\:\: \mathrm{(Equation 2.)}$$

Now lets rearrange Equation 1. to make ??c?? the subject:
$$ 2c + 6m = 6.20$$
$$\implies 2c = 6.20 - 6m$$
$$\implies c = 3.10 - 3m$$

Next we must substitute in for this expression for ??c?? into Equation 2.:
$$3c + 4m = 5.55$$
$$\therefore 3(3.10 - 3m) + 4m = 5.55$$
$$\implies 9.30 - 9m + 4m = 5.55$$
$$\implies 9m - 4m = 9.30 - 5.55$$
$$\implies 5m = 3.75$$
$$\implies m = 0.75$$

Now that we know how much a macaroon costs, we can substitute that into either of the equations, lets substitute it into Equation 1.:
$$2c + 6m = 6.20$$
$$\therefore 2c + 6(0.75) = 6.20$$
$$\implies 2c + 4.50 = 6.20$$
$$\implies 2c = 6.20 - 4.50$$
$$\implies 2c = 1.70$$
$$\implies c = 0.85$$

That's it! A macaroon costs £0.75 and a coffee £0.85 - the question can be solved quite easily by writing the simultaneous equations and solving them by
substitution.