Financial Mathematics: Money, Interest and Budgeting

Ordinary arithmetic wearing clothes

Money questions are not a special kind of mathematics. Every one of them comes down to arithmetic you can already do, and the difficulty is almost never the sum itself. It is working out WHICH sum, because a real situation hands you several numbers that are not comparable until you make them comparable. A large pack and a small pack carry two prices and two weights, and until both are expressed for the same amount there is nothing to compare. A wage and the money that reaches a bank account are two different numbers, and mixing them up is the single commonest slip in this topic. A rate of interest means nothing until you know what it is applied to and how often. Three habits do most of the work here, and they carry straight into the exam. Bring things to the same unit before comparing anything. Subtract what has already gone before deciding what is left. And when you meet a rate, read what it is a rate OF. Every figure in this module is given to you inside its question, exactly as an exam supplies one. Nothing here is a real rate, a real price or a real product, and nothing should be treated as one.

Five money words

Three of these are about being paid and two are about comparing. Learn them before you use them.

Which calculation does each need?

Which is the better value?

A shop sells the same product in two sizes. The small box holds 200 grams and costs 90 pence. The large box holds 500 grams and costs 2 pounds 10 pence. Which is better value, and how would you know?

• The large box, because 500 grams for 210 pence works out at 42 pence per 100 grams and the small box works out at 45
• The small box, because it costs less
• The large box, because the larger size is always the better value
• There is not enough information to say

Two ways interest can be worked out

The word interest covers two different things, and the difference is not small once more than one year has passed. Both are worked out from a rate, and the rate is always given to you.

What reaches the bank account

Someone has a gross monthly pay of 1600 pounds. The deductions for that month come to one fifth of the gross amount. Work out the net pay, in pounds.

Compare two sizes in this order

Five steps for deciding which of two different-sized packs is better value. Put them in the order that works every time.

• Choose one fixed amount that you will price both packs for, such as 100 grams
• Put both prices into the same units, so that pounds and pence are not mixed
• Work out how many of the fixed amount each pack contains
• Divide each price by that number, to get the price of the fixed amount
• Compare the two results, and say which is lower and by how much

The trap in every money question

Almost every mark lost in this topic comes from comparing two numbers that were never comparable. It happens in the same few places. A price is in pounds and the other price is in pence, so one of them is a hundred times out and the answer looks reasonable anyway. Two packs are compared on price alone when they hold different amounts. A wage is compared with what somebody else takes home, which is comparing a gross figure with a net one. A rate of interest is applied to the original amount when the question said the amount held at the time, or the other way round. And a monthly figure is compared with a weekly one, which is the version that catches people out in a budget. The habit that prevents all of it is the same habit. Before comparing anything, say out loud what each number is a number OF, including its unit and its period. If the two descriptions do not match word for word, something has to be converted before any comparison is worth making. That takes a few seconds and it is the difference between an answer that is out by a factor of a hundred and one that is right.

Complete the money sentences

The amount earned before anything is taken off is called _____. The amounts taken off it, such as tax or a pension, are the _____. What is left, and what actually reaches a bank account, is _____. Interest worked out each time on the original amount only is _____.

Price it per 100 grams

A pack holds 750 grams and costs 3 pounds. Work out the cost per 100 grams, in PENCE.

One month, written down

Here is a budget for a single month, done the only way that works: everything brought to the same period before anything is compared. Money coming in is 1600 pounds gross, and the deductions that month come to one fifth of that, so 1280 pounds is what actually arrives. That 1280 is the number the rest of the month is built on, and using the 1600 instead is the mistake that makes a budget look comfortable when it is not. Now what is already committed. Rent is 620 pounds a month. Council tax and utilities together are 210 pounds a month. A phone contract is 25 pounds a month. Travel is quoted weekly at 22 pounds, which is the figure that has to be converted before it can sit alongside the others: over a four-week month that is 88 pounds, and over a five-week month it is 110, so the careful version uses the larger figure. Committed spending is therefore 620 plus 210 plus 25 plus 110, which is 965 pounds. Take that from 1280 and 315 pounds is left for everything not yet listed: food, anything that breaks, and saving. Notice what the arithmetic did. It did not decide anything. It turned a vague feeling about whether the month works into a single number, and the decisions are made against that number rather than against a guess.

Find the comparison that does not work

Five statements from a conversation about money. Tap the ONE that compares two numbers which are not comparable.

• The large bag works out at 42 pence per 100 grams and the small one at 45, so the large is better value.
• My rent is 620 a month and my travel is 22 a week, which is about 95 a month, so together that is roughly 715.
• She earns 1800 a month gross and I take home 1400, so she is paid more than me.
• The account pays simple interest, so three years at the same rate adds three times the one-year amount.
• Both jars cost the same, but one holds twice as much, so the bigger jar is better value.

Justify the better buy

An answer that just names the cheaper pack earns little. Choose the option in each gap to turn it into reasoning.

The money run

Five quick questions on money mathematics. Three lives.

Three decisions in one month

You have a gross monthly pay of 1600 pounds, deductions of one fifth, rent of 620 a month, and a travel cost quoted at 22 pounds a week. You are working out whether you can commit to something new. Work through three decisions.

• Which figure do you build the whole calculation on?
• The travel cost is weekly and everything else is monthly. What do you do with it?
• You find that 315 pounds is left once everything committed has been taken off. What does that number let you say?

Explain the comparison

A friend says that the 500 gram box at 2 pounds 10 pence must be worse value than the 200 gram box at 90 pence, because it costs more. Explain why that reasoning does not work and how to settle the question properly. Show the working in your explanation.

• Say what is wrong with comparing the two prices on their own
• Explain what a unit price is and why it makes the comparison fair
• Work both boxes out for the same amount, showing the figures
• State which is better value and by how much for that amount
• Finish with the general rule this case illustrates