Mathematics

Key Unit Competence: By the end of this unit, learners should be able to solve
problems involving compound interest, reverse percentage and proportional
change using multipliers.

Unit Outline

• Reverse percentage.

• Compound interest.

• Compound proportional change.

Introduction

Unit Focus Activity

Lucie, a farmer from the village has saved 1 000 000 FRW from her tea and
livestock farming. She wants to invest the money in financial institution for
three years to safeguard it and also get some interest. She visited two different
financial institutions to get advice. The first institution told her that she could
invest her money with them at 10% p.a simple interest for the 3 years. The
second institution told her that she could invest her money with them at
10% p.a compound interest for the 3 years.
Unfortunately, Lucie did not fully understand the difference and so she
does not know the best option to take. She has come to you for proper advice
in regard to:
(a) What is (i) simple interest (ii)
compound interest?

(b) Which of the two types of investment better?

(c) What would be the difference in the total interest generated through the
    two types of investments after the 3 years?

By performing the necessary calculations, kindly write down on a piece of paper
your full advice to Lucie regarding her three queries.

7.1 Reverse Percentage

Activity 7.1

Consider an iron box that costs 450 FRW after a 25% increase in its original price.

1. Explain how you can determine the original price of the iron box. Then
    determine that price.

2. Compare the current price and the original price of the iron box.
    Which is higher? Why?

Reverse percentage involves working out the original quantity of an item
backwards after the increase or decrease in its quantity. This method is applied
when given a quantity after a percentage increase or decrease and one is required
to find the original quantity.

Example 7.1
A radio is sold at 620 FRW after a 40% increase in the price. Find the original price.

Example 7.2

For purposes of sales promotion, the price of a book has been reduced by 20% to 3 600 FRW.
What was the price before the reduction?

Exercise 7.1

1. A man’s daily wage was increased by 25% to 500 FRW. Find how much it
    was before the increase.

2. The price of an article is decreased by 5% to 1 900 FRW. What was the
    price before the decrease?

3. A company produced 23 000 shirts in September. This was 8% less than the
    August production. How many shirts did the company produce in August?

4. A new car falls in value by 30% a year. After a year, it is worth 84 000 FRW.
    Find the price of the car when it was new.

7.2 Compound interest

7.2.1 Definition of compound interest

Activity 7.2

Find out from reference books or the
internet the:

1. definition of interest and some of the area of its application.

2. ways of calculating compound interest.

3. difference between compound interest and simple interest.

When money is borrowed from or deposited in a financial institution, it
earns an interest at the end of each interest period as specified in the terms
of investment, for example at the end of each year, half year etc. Instead of paying
the interest to the owner, it is added to the principal at the end of each period
i.e. compounded with the principal. The resulting amount is then taken to be the

principal for the next interest period. The interest earned in this period is higher
than in the previous one. The interest so earned is called compound interest.

Therefore, compound interest defines the interest calculated on the initial principal
and also on the accumulated interest of previous periods of a deposit or loan.
There are two ways of calculating compound interest:

(a) the step by step method

(b) the compound interest formula.

7.2.2 Step by Step Method

Activity 7.3

Using a step by step method, determine
to amount accumulated by a principal of 500 000 FRW invested at 8%
compound interest for 4 years. What is the total interest realised?
Compound interest can be calculated step by step through compounding the
interest generated with the principal.

Example 7.5

Daka borrows 3 800 FRW from Jane at 10% p.a. compound interest. At the end of
each year, he pays back 910 FRW. How much does he owe Jane at the beginning of
the third year?

Example 7.6

Find the accumulated amount of money after 112
years, for 10 500 FRW, invested at the rate of 8% p.a. compounded semiannually.

Exercise 7.2
1. Find the amount and the compound  interest for each of the following
   correct to the nearest FRW:

(a) 8 000 FRW invested for 3 years at 4% p.a. compound interest.

(b) 48 000 FRW invested for 2 years at 6% p.a. compound interest.

(c) 36 000 FRW invested for 2 years at 5% p.a. compound interest.

2. Determine the difference between the simple interest and compound
    interest earned on 15 000 FRW for 2 years at 3% p.a.

3. Kampire invested P FRW, which amounted to 8 420 FRW in 3 years
   at a rate of 4.5% p.a. compound interest. Find the value of P.

4. Determine the compound interest earned on 45 000 FRW after 3 years
    at the rate of 6% p.a.

5. Mugisha borrows a sum of 8 000 FRW at 10% p.a. simple interest and
    lends that to Neza at the same rate compound interest. How much will
     Mugisha gain from this transaction after 3 years?

7.2.3 The compound interest formula

Activity 7.4

1. Consider the case in which 10 000 FRW is invested in a bank
   for 3 years at the rate of 5% p.a.compound interest.

2. Using the step by step method calculate the amount of money
  accumulated after every year.

Assuming 5 000 FRW is the principal amount compounded at 6% p.a.The
amount of money accumulated after every year is calculated as follows:

Amount after 1st year

Considering the case in which P is invested in a bank for n-interest periods
at the rate of r% p.a. The accumulated amount (A) after the given time is given
by:
A = P (1 + r/100)n
where n is the number of interest periods.
This is called the compound interest formula. It is conveniently used in
solving problems of compound interest especially those involving long periods
of investments or payment.

In this method, the accrued compound interest is obtained by subtracting the
original principal from the final amount. Thus, Compound interest =
Accumulated amount(A) – Principal(P) Note that the principal and the interest
earned increased after each interest period. We can also deduce that;

Example 7.8

Find the accumulated amount that 30 000 FRW will yield if deposited for 2 years at
6% p.a. compounded semi-annually.

Solution

For every year there will be two interest
periods;
Thus, in 2 years we will have (2 × 2) interest periods i.e. n = 4
The rate per interest period = 6 ÷ 2 = 3%
Thus, A = 30 000 FRW (1 + 3/100)⁴

Exercise 7.3

1. Determine the accumulated amount for each of the following:
(a) 19 000 FRW invested for 2 years at 3% p.a. compound interest.

(b) 8 300 FRW invested for 3 years at 4% p.a. compound interest.

2. A businesswoman invested 4 500 FRW for 2 years in a savings account. She
   was paid 4% per annual compound interest. How much did she have in
   her savings account after 2 years?

3. Adams invests 4 500 FRW at a
   compound interest rate of 5% per annum. At the end of n complete
  years the investment has grown to 5 469.78 FRW. Find the value of n.

4. A company bought a car that had
   a value of 12 000 FRW. Each year the value of the car depreciates with
   25%. Work out the value of the car at the end of three years.

5. Erick invested 60 000 FRW for 3 years
   in a savings account. He gets a 3% per annum compound interest. How
   much money will Erick have in the savings account at the end of 3-years?

7.3 Compound proportional  change

 Activity 7.5

Consider that 3 people working at the
same rate can plough 2 acres of land in 3 days.

What do you think will happen if the working days are increased to five
working at the same rate? Discuss. Sometimes, a quantity may be
proportional to two or more other

quantities. In such a case, the quantities are said to be in compound proportion.
Problems involving rates of work, and other similar problems, often contain quantities
that are in compound proportion. Such problems are solved using either the ratio
or unitary method.

Example 7.11
Eighteen labourers dig a ditch 80 metres long in 5 days. How long will it take 24
labourers to dig 64 metres long? What assumptions have you made?

Solution

Notice that the number of days depends on
the number of labourers as well as on the length of the ditch. Thus, this is a problem
in compound proportion.

1. Ratio method

Length of ditch decrease from 80 m to 64
m, i.e. in the ratio 64 : 80.

A shorter ditch takes a shorter time.

∴ multiply days by 64/80 .
Number of labourers increases from 18 to
24, i.e. in the ratio 24 : 18.
More labourers take a shorter time.

∴ multiply days by 18/24 .
80 metres of ditch are dug by 18 labourers in 5 days.

64 metres of ditch are dug by 24 labourers in
5 × 64/80 × 18/24 days = 3 days.

Example 7.12

To plant a certain number of tree seedlings Mutoni takes 5 hours. Gahigi takes 7
hours to plant the same number of seedlings. If Mutoni and Gahigi worked together,
how long would they take to plant the same number of seedlings?

Note: We first found out the fraction of the number of seedlings that each
person plants in 1 hour, then the fraction of the number of seedlings that
they plant together in 1 hour. Invert this fraction, and the result is
the number of hours they take working together.

Exercise 7.4

1. 16 men dig a trench 92 m long in 9 days. What length of trench can 12
     men dig in 15 days?

2. To unclog a silted drain 85 m long, 15 workers take 10 days. Find how
    many workers are required to unclog a similarly silted drain 51 m long
    drain in 5 days.

3. Six people pay 40 320 FRW for a 7 day stay at a hotel. How much would
    eight people pay for a 3 day stay?

4. A car hire company with 24 cars uses 2 940 litres of petrol in 5 days. How
    long would 4 116 litres of petrol last if the company had 28 cars and the
     consumption rate does not change?

5. A transport company charges 54 800 FRW to move a load of 2.8 tonnes
    for 350 km. For what load will the corporation charge 47 040 FRW for 400 km?

6. A man, standing next to a railway line, finds that it takes 6 seconds fora train, 105 m long,

travelling at 63km/h, to pass him. If another train,100 m long, takes 5 seconds to pass
him, at what speed is it moving?

7. It takes 15 days for 24 lorries, each of which carries 8 tonnes, to move
   1 384 tonnes of gravel to a construction site. How long will it take 18 lorries,
   each of which carries 10 tonnes, to move 1 903 tonnes of the gravel?

8. A car moving at 65 km/h takes 2 h 24 min to travel 156 km. What distance
   does the car travel in 48 min moving at 55 km/h?

9. Twelve men, working 8 hours a day, can do a piece of work in 15 days.
    How many hours a day must 20 men work in order to do it in 8 days?

Unit Summary

• Reverse percentage is the working out of original price of a product
   backwards after the increase.

• Compound interest: is the interesst calculated on both the amount
   borrowed and any accumulated previous interests.

• The compound interest formula states as shown:
   A = P(1 + r/100 )n where n is the number of interest periods, r is the
  rate, A is the initial amount and P is the principal

Unit 7 Test

1. After the prices of fuel increased by 15%, a family’s annual heating bill
    was 1 654 FRW. What would the bill have been without the increase in
    price?

2. An insurance company offers a no claim discount of 55% for drivers who have
    not had an accident for 4 years. If the discounted premium for such a driver
    is 3 340 FRW, how much did the driver save?

3. After a long-haul flight, the total  weight of a passenger jet had
   decreased by 27% to 305 000 kg. What weight of fuel was the aircraft
   carrying at take off?

4. Lange borrows 16 000 FRW to buy a coloured TV set at 10% p.a.
    compound interest. He repays 980  FRW at the end of each year. How
    much does he still owe at the end of 3 years?

5. If 12 000 FRW is invested at 12%  p.a. compounded quarterly, find the
   accumulated amount after one year to the nearest Francs.

6. Mwiza borrowed 2 000 FRW at 5% p.a. compound interest from a
    microfinance company. She paid back 350 FRW at the end of each
    year. How much does Mwila still owe the company at the end of the second
    year?

7. A sum of money is invested at compound interest and it amounts
    to 420 FRW at the end of the first year and 441 FRW at the end of the
     second year. Determine;

    (a) the rate in percent

   (b) the sum of money invested.

8. Calculate the amount after 3 years if 7 800 FRW is invested at 121/2 % p.a.
    compound interest (give your answer to the nearest francs).

9. If you deposit 4 000 FRW into an account paying 6% annual interest

    compounded quarterly, how muchmoney will be in the account after 5 years?

10. If you deposit 6 500 FRW compounded monthly into an account
      paying 8% annual interest compounded monthly, how much
      money will be in the account after 7 years?

11. If you deposit 8 000 FRW into an account paying 7% annual interest
      compounded quarterly, how long will it take to have 12 400 FRW in the
      account?

12. One pipe can fill a bath in 5 minutes and another can empty the same bath
      in 10 minutes. Both pipes are opened  at the same time and after 5 minutes,
      the second pipe is turned off. What fraction of the bath is then full? How
       long will it take for the first pipe to fill the bath completely from then?

13. A man can do a job in 41/2 days. Another man can do the same job in
     9 days. How long will the two men take on the job if they work together?

14. Workmen A and B, working together, do a certain job in 1 hour. Workman

   A alone does the job in 3 hours. How long does it take workman B alone to
  do the job?

15. Working alone, Alex can do some job in 6 days. John, also working alone,
     can do the same job in 9 days. Alex starts alone, but is joined by John
   after 1 day. How long do they take to finish the work together?

16. Three persons, Peter, Mike and James, build a certain length of wall
      in 2 days. Peter and Mike together could build the same length of wall in
      4 days, and Mike and James together would take 3 days.

(a) Find the fraction of the length  of the wall that Mike and James
      build in 2 days and hence find how long Peter would take by
      himself.

(b) Find the fraction of the length of the wall that Peter and Mike
     build in 2 days and hence find how long James would take by  himself.

(c) Similarly, find how long Mike would take by himself.