Compound Interest, Reverse Percentage and Compound Proportional Change (Senior 3 Mathematics): free lessons | Ganzaa

Compound Interest, Reverse Percentage and Compound Proportional Change

Lessons

Senior 3 · Mathematics

Compound Interest, Reverse Percentage and Compound Proportional Change

4 lessons

Lesson 1 of 4

Multipliers, simple interest and reverse percentages

25 min

What was the price before VAT?

A receipt from a shop in Kigali shows a total of 11,800 Frw for a school bag, and says the price includes VAT at 18%. What was the price before VAT was added?

Many people take 18% of 11,800 Frw away and get 9,676 Frw. That is wrong: the 18% was worked out on the price before VAT, not on 11,800 Frw. This lesson shows how to work backwards from a new amount to the original one, using multipliers. Multipliers are also the key to compound interest in the next two lessons.

Definition

Multiplier: the number you multiply an amount by to apply a percentage change in one step.

\text{increase by } r\% : \times\left(1 + \frac{r}{100}\right), \qquad \text{decrease by } r\% : \times\left(1 - \frac{r}{100}\right).

Example: an increase of 18% has multiplier 1.18; a decrease of 15% has multiplier 0.85, because $100\% - 15\% = 85\%$ .

Change	New amount as a percentage of the original	Multiplier
increase by 5%	105%	1.05
increase by 18% (VAT)	118%	1.18
increase by 100% (doubling)	200%	2
decrease by 15%	85%	0.85
decrease by 2.5%	97.5%	0.975

A multiplier greater than 1 means an increase; a multiplier between 0 and 1 means a decrease.

Definition

Simple interest (review from S1): interest paid only on the original sum, the principal $P$ . At a rate of $R %$ per year for $T$ years,

I = \frac{P \times R \times T}{100}, \qquad \text{amount } A = P + I.

Example: 200,000 Frw saved at 6% simple interest for 3 years earns $I = \frac{200\,000 \times 6 \times 3}{100} = 36\,000$ Frw, so $A = 236\,000$ Frw.

Worked example 1: a loan with simple interest

A trader borrows 450,000 Frw from a SACCO at 12% simple interest per year for 18 months (illustrative terms). How much does she repay?

Solution 18 months is $T = 1.5$ years.

I = \frac{450\,000 \times 12 \times 1.5}{100} = 81\,000\ \text{Frw}, \qquad A = 450\,000 + 81\,000 = 531\,000\ \text{Frw}.

Definition

Reverse percentage: a problem in which you know the amount after a percentage change and must find the original amount.

\text{new} = \text{original} \times \text{multiplier} \quad\Longrightarrow\quad \text{original} = \frac{\text{new}}{\text{multiplier}}.

Example: after a 25% increase a price is 5,000 Frw, so the original price was $\frac{5000}{1.25} = 4000$ Frw.

Important

Divide by the multiplier; never subtract the percentage of the new amount. The percentage was calculated on the original, and the new amount is not the original. Dividing undoes the multiplication exactly.

Worked example 2: the price before VAT

The school bag costs 11,800 Frw including VAT at 18%. Find the price before VAT and the amount of VAT.

Solution The price paid is 118% of the price before VAT.

\text{price before VAT} = \frac{11\,800}{1.18} = 10\,000\ \text{Frw}, \qquad \text{VAT} = 11\,800 - 10\,000 = 1800\ \text{Frw}.

A bar model: 118% of the unknown price is 11,800 Frw.

Check: 18% of 10,000 Frw is 1,800 Frw ✓. (The wrong method, $11\,800 - 0.18 \times 11\,800 = 9676$ , gives too little.)

Worked example 3: reverse percentages after a decrease

(a) In a sale, a pair of shoes costs 25,500 Frw after a 15% discount. What was the price before the sale? (b) After a 6% pay rise, a teacher's monthly salary is 212,000 Frw (illustrative). What was it before?

Solution (a) The sale price is 85% of the original: $\frac{25\,500}{0.85} = 30\,000$ Frw.

(b) The new salary is 106% of the old: $\frac{212\,000}{1.06} = 200\,000$ Frw.

Check (a): 15% of 30,000 is 4,500, and $30\,000 - 4500 = 25\,500$ ✓.

Worked example 4: working back from a simple-interest amount

A farmer repays 330,000 Frw to clear a loan taken 2 years earlier at 5% simple interest per year. How much did he borrow?

Solution Over 2 years the interest is $2 \times 5\% = 10\%$ of the principal, so the repayment is 110% of it:

P = \frac{330\,000}{1.10} = 300\,000\ \text{Frw}.

Check: $I = \frac{300\,000 \times 5 \times 2}{100} = 30\,000$ and $300\,000 + 30\,000 = 330\,000$ ✓.

Warning

The base of a percentage is the original amount. "Reduced by 15% to 25,500 Frw" does not mean the original was $25\,500 + 15\% \text{ of } 25\,500 = 29\,325$ Frw. Check any answer by applying the percentage change to it: you must get back the amount you were given.

Tip

Write the equation first: original $\times$ multiplier $=$ new. Then the only step left is to divide. For VAT, remember the pair: $\times 1.18$ adds VAT, $\div 1.18$ removes it.

Corbettmaths: Reverse Percentages. Watch how the new amount is matched to its percentage of the original, such as 85% or 120%, and divided back to 100%.

✦

Practice

The last three are harder.

Write the multiplier for (a) an increase of 12% (b) a decrease of 30% (c) an increase of 7.5%.
Find the simple interest on 150,000 Frw at 8% per year for 4 years, and the total amount.
A phone costs 59,000 Frw including 18% VAT. Find the price before VAT.
After a 20% discount a radio costs 36,000 Frw. What was the original price?
A town's population grew by 4% to 20,800 in one year. What was it before?
A car's value fell by 25% in a year to 9,000,000 Frw. A pupil says it was worth 11,250,000 Frw before. Is the pupil right? Find the correct value.
A saver has 460,000 Frw after 3 years at 5% simple interest per year. How much did she deposit?
A price includes VAT at 18%. What percentage of the price paid is VAT? Give your answer to 1 decimal place.

Show answersHide answers

(a) 1.12 (b) 0.70 (c) 1.075.
$I = \frac{150\,000 \times 8 \times 4}{100} = 48\,000$ Frw; $A = 198\,000$ Frw.
$\frac{59\,000}{1.18} = 50\,000$ Frw.
$\frac{36\,000}{0.8} = 45\,000$ Frw.
$\frac{20\,800}{1.04} = 20\,000$ .
No: the pupil added 25% of 9,000,000. The correct value is $\frac{9\,000\,000}{0.75} = 12\,000\,000$ Frw.
The amount is 115% of the deposit: $\frac{460\,000}{1.15} = 400\,000$ Frw.
VAT is $\frac{18}{118}$ of the price paid: $\frac{18}{118} \times 100 \approx 15.3\%$ .

Sources and credits

Written by Ganzaa.
VAT rate: the standard rate of 18% is set by Law N° 049 of 5 September 2023 establishing value added tax (Article 4), which is linked from the Rwanda Revenue Authority's About VAT page; the rate is also summarised in PwC Worldwide Tax Summaries: Rwanda, other taxes.
Video: Reverse Percentages, Corbettmaths, embedded from YouTube.

The bar model, worked examples, practice items and quiz questions were written by Ganzaa. Prices, salaries and loan terms are illustrative.

1What was the price before VAT?

What was the price before VAT?