Simple interest, loans, savings and taxes – Percentage, Discount, Profit and Loss (Senior 1 Mathematics) | Ganzaa

Percentage, Discount, Profit and Loss

Lessons

Senior 1 · Mathematics

Percentage, Discount, Profit and Loss

5 lessons

Lesson 5 of 5

Simple interest, loans, savings and taxes

25 min

Let's begin

Habimana puts 200,000 Frw in a savings account at his Umurenge SACCO. At the end of each year the SACCO pays him a percentage of that money for letting it use his savings. His neighbour borrows money from the same SACCO to buy a sewing machine, and she pays a percentage back on top of what she borrowed. Both are paying or earning interest. Governments also take percentages, as taxes. This lesson shows how all three are worked out.

1. Simple interest

Definition

Principal $P$ : the amount of money saved or borrowed.

Interest $I$ : the extra money paid for the use of the principal: earned by a saver, paid by a borrower.

Rate $R$ : the interest for one year as a percentage of the principal (" $6 %$ per annum" means $6 %$ per year).

Time $T$ : the number of years.

Amount $A$ : the principal plus the interest, $A = P + I$ .

Definition

Simple interest: interest worked out on the original principal only, the same amount every year: $I = \frac{P \times R \times T}{100}.$

Example: 200,000 Frw at $6 %$ per year for 3 years earns $\frac{200{,}000 \times 6 \times 3}{100} = 36{,}000$ Frw.

Why the formula works. Each year the interest is $R %$ of $P$ , which is $\frac{R}{100} \times P$ . With simple interest this is the same every year, so $T$ years give $T$ times as much: $I = \frac{P R}{100} \times T$ .

Year	Interest that year (Frw)	Amount at the end of the year (Frw)
0	–	200,000
1	12,000	212,000
2	12,000	224,000
3	12,000	236,000
4	12,000	248,000
5	12,000	260,000

The amount grows by the same 12,000 Frw each year, so it is a linear function of time, $A = 200,000 + 12,000 T$ , just as in Unit 3. The gradient, 12,000 Frw per year, is the yearly interest, and the intercept is the principal.

Notice that the vertical axis starts at 180,000 Frw, not at 0. That makes the yearly growth easy to see, but it also makes the growth look bigger than it is. Always read the labels on the axes before you judge a graph.

Worked example

Savings. Habimana saves 200,000 Frw at $6 %$ per year simple interest. How much is in the account after 3 years?

I = \frac{200{,}000 \times 6 \times 3}{100} = 36{,}000 \text{ Frw}

A = 200{,}000 + 36{,}000 = 236{,}000 \text{ Frw}.

Worked example

A loan. A tailor borrows 480,000 Frw at $15 %$ per year simple interest, to be repaid in equal monthly instalments over 2 years. Find the interest, the total to repay and the monthly instalment

I = \frac{480{,}000 \times 15 \times 2}{100} = 144{,}000 \text{ Frw}

A = 480{,}000 + 144{,}000 = 624{,}000 \text{ Frw}.

Over $2 \times 12 = 24$ months: $624{,}000 \div 24 = 26{,}000$ Frw per month.

Worked example

Finding $R$ or $T$ . (a) 150,000 Frw earns 27,000 Frw interest in 2 years. Find the rate. (b) How long will 300,000 Frw take to earn 60,000 Frw at $8 %$ per year?

Rearrange $I = \frac{PRT}{100}$ to make the unknown the subject.

(a) $R = \dfrac{100I}{PT} = \dfrac{100 \times 27{,}000}{150{,}000 \times 2} = 9\%$ per year.

(b) $T = \dfrac{100I}{PR} = \dfrac{100 \times 60{,}000}{300{,}000 \times 8} = 2.5$ years, which is 2 years and 6 months.

Worked example

Time in months. Find the simple interest on 400,000 Frw at $12 %$ per year for 9 months

The rate is per year, so the time must be in years: $T = \frac{9}{12} = 0.75$ .

I = \frac{400{,}000 \times 12 \times 0.75}{100} = 36{,}000 \text{ Frw}.

Warning

Match the units of $R$ and $T$ . With a yearly rate, 9 months is $T = 0.75$ , not $T = 9$ . Using 9 would make the interest twelve times too big: 432,000 Frw instead of 36,000 Frw.

2. Taxes: value added tax (VAT)

Definition

Tax: money that people and businesses must pay to the government, often worked out as a percentage. In Rwanda taxes are collected by the Rwanda Revenue Authority (RRA).

Value added tax (VAT): a tax added to the price of most goods and services. Rwanda's standard VAT rate is $18 %$ (some items are zero-rated or exempt).

Example: a shirt priced 10,000 Frw before VAT carries $0.18 \times 10{,}000 = 1{,}800$ Frw of VAT, so the customer pays 11,800 Frw.

Adding VAT is a percentage increase with multiplier $1.18$ :

\text{price with VAT} = 1.18 \times \text{price before VAT}

\text{price before VAT} = \frac{\text{price with VAT}}{1.18}.

Worked example

A carpenter charges 25,000 Frw for a table before VAT. How much VAT is added, and what does the customer pay?

VAT $= 0.18 \times 25{,}000 = 4{,}500$ Frw. The customer pays $25,000 + 4,500 = 29,500$ Frw, or in one step $1.18 \times 25{,}000 = 29{,}500$ Frw.

Worked example

A phone costs 177,000 Frw, including VAT at $18 %$ . Find the price before VAT and the VAT paid

\text{price before VAT} = \frac{177{,}000}{1.18} = 150{,}000 \text{ Frw}

\text{VAT} = 177{,}000 - 150{,}000 = 27{,}000 \text{ Frw}.

Check: $18 %$ of 150,000 is 27,000. ✓

Warning

VAT is a percentage of the price before VAT. For the 177,000 Frw phone, $18 %$ of 177,000 is 31,860 Frw, which is not the VAT: that would tax the tax. Divide by 1.18 to remove VAT; never subtract $18 %$ of the VAT-inclusive price.

Tip

Every problem in this unit is one of three questions: find a percentage of a whole (multiply), find what percentage a part is (divide by the whole), or find the whole (divide by the multiplier). Decide which one you are answering before you calculate.

Cognito: "How to Calculate Simple Interest". Watch how the interest for one year is found first and then multiplied by the number of years. (The prices in the video are in pounds.)

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Practice

Give money answers to the nearest franc.

Find the simple interest on 50,000 Frw at $4 %$ per year for 3 years.
Uwase saves 120,000 Frw at $7 %$ per year simple interest. How much will she have after 4 years?
Find the interest on 600,000 Frw at $10 %$ per year for 18 months.
At what rate does 250,000 Frw earn 45,000 Frw simple interest in 3 years?
A loan of 360,000 Frw at $12 %$ per year is repaid in 12 equal monthly instalments over 1 year. Find each instalment.
A service costs 40,000 Frw before VAT at $18 %$ . Find the VAT and the total price.
A receipt shows 23,600 Frw including VAT at $18 %$ . How much of it is VAT?
Which gives more interest on 100,000 Frw: $8 %$ per year for 3 years, or $12 %$ per year for 2 years?

Show answersHide answers

$\frac{50{,}000 \times 4 \times 3}{100} = 6{,}000$ Frw
$I = 33,600$ , so $A = 153,600$ Frw.
$T = 1.5$ : $I = 90,000$ Frw.
$R = \frac{100 \times 45{,}000}{250{,}000 \times 3} = 6\%$
$I = 43,200$ ; total 403,200; $403{,}200 \div 12 = 33{,}600$ Frw.
VAT 7,200 Frw; total 47,200 Frw.
Before VAT $= 23{,}600 \div 1.18 = 20{,}000$ ; VAT $= 3,600$ Frw.
$8 %$ for 3 years gives 24,000 Frw, and $12 %$ for 2 years gives 24,000 Frw too. They are equal, because $R \times T = 24$ in both.

11. Simple interest

22. Taxes: value added tax (VAT)

1. Simple interest

2. Taxes: value added tax (VAT)