R9 - Reverse percentages and simple interest

Finding an original value after a decrease

A bicycle costs $\pounds 420$ after a $30\%$ discount. Is its original price $\pounds 546$, found by adding $30\%$ to $420$? If that seems reasonable, try reducing $546$ by $30\%$. We will find a price that really works.

The sale price is $70\%$ of the unknown original price. It is not a new $100\%$ for working backwards. If $70\%$ is $\pounds 420$, divide by $70$ to find $1\%$, then multiply by $100$ to recover the whole.

[DIAGRAM: asset_name: Rebuilding the original from the sale price; asset_slug: aqa8300_r05_reverse_bar; caption: Seven equal parts cost 420 pounds, so all ten parts cost 600 pounds; description: A horizontal bar is split into ten equal sections, each worth 10 per cent of the original price. The first seven are blue and labelled sale price, 70 per cent, 420 pounds. The final three are pale clay and labelled discount, 30 per cent. A bracket under one section says 10 per cent is 60 pounds. A bracket below all ten sections says original price, 100 per cent, 600 pounds.]

Dividing by the decimal multiplier does the same job in one step. Here $420 \div 0.70 = 600$, since $0.70 \times 600 = 420$.

A fraction can make the non-calculator division easier. After a $20\%$ reduction, the sale price is four fifths of the original.

The next example changes only the discount rate. The sale price is still $\pounds 48$, but now it represents three quarters of the original.

The opening bicycle follows the same rule: $420 \div 70 \times 100 = 600$. Its original price is $\pounds 600$. Reducing that by $30\%$ removes $\pounds 180$, leaving $\pounds 420$.

Now change the sale price as well as choosing the percentage that remains. A calculator is useful for this less tidy multiplier.

Finding an original value after an increase

A price after an increase is more than the original whole, so reverse it by dividing by a multiplier greater than $1$.

Suppose a repair costs $\pounds 144$ including VAT at a stated rate of $20\%$. VAT, value added tax, is a tax added to the price before VAT. The total is $120\%$ of that price, so the price before VAT is $144 \div 1.20 = 120$. The VAT is $\pounds 24$. Use the rate given in the question; different situations can use different rates.

Profit questions use the same idea when the profit percentage is based on the price paid for the item.

For example, buying for $\pounds 50$ and selling for $\pounds 60$ gives a profit of $\pounds 10$. Relative to the cost price this is $20\%$. Selling for $\pounds 40$ instead gives a $\pounds 10$ loss, also $20\%$ of the cost price. Check the question's stated base whenever it names a percentage.

The next example changes only the profit rate. The selling price stays the same, so a higher percentage profit means a smaller cost price.

Reversing an increase makes a positive new value smaller. Reversing a decrease makes it larger. This is a quick way to spot the wrong multiplier.

A VAT problem changes the context, but the original-value decision is the same as in the profit examples.

Simple interest on the original amount

Simple interest adds the same cash amount each year when the starting amount and rate stay fixed.

The principal is the original money saved or borrowed. Interest is money earned on savings or charged for borrowing. An interest rate gives the interest as a percentage for a stated period. “Per annum” means per year. At $5\%$ simple interest per annum, $\pounds 200$ earns $\pounds 10$ each year: $\pounds 210$ after one year, $\pounds 220$ after two.