# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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1. Reflection in two parallel mirrors. Suppose M is the transformation reflection in the mirror m and N is the reflection in the mirror n, where m and n are parallel mirrors placed 5 units apart. The object is a flag F that is placed 3 units in front of the first mirror m. The first transformation of the flag F is reflection in the mirror m to give the image F'. This is written as M(F) = F', where F' is the image of F. Then

100 COMPOUND INTEREST

m n

F F ' F"

|\ /1 w

:

(a)

F' becomes the object for the second transformation N, which is a reflection in the mirror n, and is written as N(F') = F".

The composite transformation M followed by N on the flag F to give the final image F" is written as NM(F) = F". Note that in the composite transformation NM(F), the reflection M is done first, followed by N. It can be seen that the composite transformation NM(F) is equivalent to a translation of ) (seethe entry Combinations for this notation), where 10 units is twice the distance between the mirrors.

If F is reflected in the mirror n first and then in m, the composite transformation MN(F) is equivalent to a translation of (_q°).

2. Reflection in two perpendicular mirrors. As before, M is a reflection in the mirror m and A is a reflection in the mirror n, but on this occasion the mirrors are at right angles instead of parallel (see figure b). The composite transformation NM(F) is equivalent to a rotation of 180° about the point of intersection of the two mirrors.

(b)

If the angle between the mirrors was 6° instead of 90°, the composite transformation would be a rotation of 20° about the point of intersection of the mirrors.

References: Enlargement, Parallel, Perpendicular Lines, Reflection, Rotation, Transformation Geometry, Translation.

COMPOUND INTEREST

Compound interest is an application of percentages to the investment of money in the business world. The terminology used is defined as follows:

♦ The money that a customer invests in the bank is called the principal, denoted by P.

COMPOUND INTEREST 101

♦ The principal earns interest, and this interest is money the bank gives to the customer as a reward for using their money, and is denoted by I.

♦ The rate of interest is the percentage of their principal given to the customer, denoted by R%.

♦ Per annum means “for each year.”

♦ The amount is the principal plus the interest, and is denoted by A.

A straightforward example of calculating interest is worked through here before compound interest is explained.

Example 1. Jo invests $200 in the bank at 8% per annum. What interest will this investment earn after 1 year?

Solution. Write

The interest after 1 year is $16.

For a simple interest investment the interest on $200 at 8% per annum is $16 every year. If the principal of $200 is invested at a compound interest rate of 8% per annum, the interest grows bigger each year. The idea behind compound interest is that the interest is added to the principal and thereafter also earns interest. If the interest of $16 at the end of the first year is added to the principal, then Jo has a new principal of $216 to invest for the second year. The interest grows each successive year, and we say the interest is compounded. A useful formula for calculating compound interest is

In this formula A is the amount in the bank (principal + interest) after an investment period of n years when P is invested at compound interest of R% per annum.

I = 8% x 200

/ = R x P for 1 year

I = 0.08 x 200

8% = 0.08

I = 16

Example 2. Jo invests $200 in her bank at an interest rate of 8% per annum for 3 years, and the interest is compounded annually. What will her investment amount to after 3 years?

102 CONCAVE

Solution. The following formula can be applied to Jo’s investment:

A = 200 x ( 1 H——^ Substituting P = 200, R = 8

1 +-------

100

Substituting P = 200, R = 8, and n = 3

A = 200 x (1 + 0.08)3 8 -r- 100 = 0.08

A = 200 x 1.083

A = $251.94 (to 2 dp) Using a calculator

In the examples we have studied so far the unit of time has been 1 year. But the interest rate can be compounded for a different unit of time, such as every 6 months as in the next example.

Example 3. Mr. and Mrs. Millar deposited $6000 with a savings firm. The money is to accrue interest at the rate of 6% per annum, compounded 6-monthly. Find the maturity value at the end of 7 years if no further deposits or withdrawals are made.

Solution. Write

A = $9075.54 (to 2 dp) Using a calculator

The maturity value after 7 years = $9075.54.

If the same principal of $6000 is invested at the same interest rate of 6%, but is compounded annually instead of 6-monthly, a smaller answer is obtained for the amount:

Formula for compound interest

Substitute P = 6000, R = 3 (divide annual rate by two for 6-monthly rate), n = 14 (the time period is 6 months and there are 14 time periods in 7 years)

A = $9021.78 (to 2 dp)

References: Percentage, Simple Interest.

CONCAVE

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