discovering mathematics week 14 – unit 13 exponential mu123 dr. hassan sharafuddin

Discovering Mathematics

Week 14 – Unit 13

Exponential

MU123

Dr. Hassan Sharafuddin

Learning Outcomes

This unit includes the following:

- Exponential growth & decay - Working with exponential - Calculate Depreciation

Week 12

1 Exponential growth and decay

People sometimes talk about things growing exponentially – What does this mean?

The idea of exponential growth can be understood by thinking about chain letters (or, more commonly nowadays, chain emails or chain text messages).

This type of growth, where the numbers increase by the same factor at each stage, is called exponential growth.

1 Exponential growth and decay:Exponential growth is growth that arises from repeated multiplicationby the same number. The number that you start with is called thestarting number, and the number that you multiply by is called the scale factor (or multiplication factor)

Week 12


1.2 Multiplying by a scale factor:In real-life examples of exponential growth and decay, scale factors are often given in the form of percentage increases or decreases.

Suppose, for example, that you want to increase the price of an item costing £18 by 15%. Since 100% + 15% = 115%, this means that the new price is 115% of the old price. So you can calculate the new price by multiplying the old price by the fraction 115/100, which is equal to 1.15. The new price is £18 × 1.15 = £20.70.

By the same principle:to increase a number by 27%, multiply it by the scale factor 1.27;to increase a number by 80%, multiply it by the scale factor 1.80, orsimply 1.8; and so on.

Week 121.2 Multiplying by a scale factor


The principle above also applies to percentage decreases. Suppose that you want to apply a 15% discount to the price of an item costing £18. Since 100% − 15% = 85%, the new price is 85% of the old price. So you can calculate the new price by multiplying the old price by the fraction 85/100 , which is equal to 0.85. The new price is £18 × 0.85 = £15.30.

By the same principle:to decrease a number by 27%, multiply it by the scale factor 0.73(since 100% − 27% = 73%);to decrease a number by 80%, multiply by the scale factor 0.2(since 100% − 80% = 20%);and so on.



Solution: (a) (i) The scale factor for a 10% increase is 1.1.

(ii) The scale factor for a 3% increase is 1.03.(iii) The scale factor for a 0.5% increase is 1.005.(iv) The scale factor for a 15% decrease is 0.85.(v) The scale factor for a 2% decrease is 0.98.(vi) The scale factor for a 1.5% decrease is 0.985.

(b) (i) A scale factor of 1.08 gives a 8% increase.(ii) A scale factor of 0.91 gives a 9% decrease.(iii) A scale factor of 1.072 gives a 7.2% increase.



There are some practical situations where a number is repeatedly multiplied by a scale factor, leading to exponential growth. For example:

A sum of £240 is invested in a deposit account that gives a 4% per yearrate of return. There are no further transactions.(a) How much will the investment be worth after the following times?(i) 1 year (ii) 5 years

(b) Suppose that £V is the value of the investment after n years. Writedown a formula for V in terms of n.

Solution:(i)The value of the investment after 1 year is £240 × 1.04 = £249.60.(iii) The value after 5 years is £240 × 1.04 × 1.04 × 1.04 × 1.04 × 1.04

= £240 × 1.045 = £291.97(b) A formula for the value £V of the investment after n years is: V = 240 × 1.04n

Solve activity 5 and 6, page 123


Calculate Depreciation

In this subsection you will learn how to calculate depreciation:

Compute depreciation using the straight-line method.Compute depreciation using the unit of production.

What is Depreciation?Depreciation is the decrease in the value of assets owned by a business, such as automobiles, buildings, and computers.

Depreciation is caused by wear or by obsolescence (becoming out-of-date).

An automobile will wear out after a number of years or miles of use.

Week 12


Buildings lose value as wood, electrical wiring, and fixtures deteriorate and as design characteristics and owners’ needs change.

A business computer frequently becomes obsolete in 3 to 5 years.

In business, depreciation is figured on almost all physical assets owned and in use. Depreciation is deducted from gross profits as an expense.

These assets are called tangible assets and the calculation is called Depreciation. BUT, If the asset is intangible; for example, a patent or goodwill; it's called amortization.

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There are five common methods of calculating depreciation: 1. the straight-line, 2. units-of-production,3. declining-balance, 4. sum-of-the-years-digits, 5. Modified Accelerated Cost Recovery System methods.

In this course, we present only the first two methods.

Week 12


1. the straight-line:It distributes depreciation evenly over the useful life of an asset, assigning equal amounts. In this method, the company estimates the salvage (scrap) value of the asset at the end of the period during which it will be used to generate revenues.

Week 121. The straight-line method

A Volkswagon Beetle whose value has certainly depreciated significantly! Its scrap value is almost $zero

What is salvage value?The salvage value (or scrap value) is an estimate of the value of the asset at the time it will be sold or disposed of; it may be zero or even negative. The company will then charge the same amount to depreciation each year over that period, until the value shown for the asset has reduced from the original cost to the salvage value.


Let us take an example: suppose in 2010 a business buys $100,000 worth of machinery that is expected to have a useful life of 4 years, after which the machine will become totally worthless (salvage value of zero).

Note:In business, it is not allowed to count the entire $100,000 amount as an expense. Instead, only the extent to which the asset loses its value (depreciates) is counted as an expense.

The simplest way to depreciate an asset is to reduce its value equally over its life. So in our example, this means the business will be able to deduct $25,000 each year.



From the previous example, the three factors used to compute depreciation by the straight-line method are:

1.The original cost, which includes the price paid for an item and any freight charges and expenses for installation.

2.The estimated service life, which is the length of time the buyer expects to be able to use an asset. The estimated service life may be stated in terms of years or months that normally may be expected during the life of the asset.

3.The estimated scrap value (SV), which is the amount the owner of an asset expects to receive upon disposing of it at the end of its estimated service life.



The basic formula for computing the amount of depreciation under the straight-line method is:

Depreciation for one time period = (Original cost - Scrap value) ÷ Estimated service life in periods of time

Example:An office computer costing $12,500 has an estimated life of 5 years and an estimated scrap value of $900.What is the annual depreciation amount?

Depreciation = (original cost – scrap value) ÷ service life



Example:An office computer costing $12,500 has an estimated life of 5 years and an estimated scrap value of $900.What is the annual depreciation amount?

Depreciation = (original cost – scrap value) ÷ service lifeDepreciation = (12,500 – 900) ÷ 5Depreciation = 11,600 ÷ 5 Depreciation = $2,320 annual depreciation

(Solve activity 14, Book D, page 137)



Example:A machine costing $10,000 has an estimated life of 60,000 hours of operation and an estimated scrap value of $400. If it was operated for 2,800 hours during the first year, how much depreciation expense will be shown for the first year?

Solution:Depreciation for one unit = (Original cost - Scrap value) ÷ Estimated life in service unitsDepreciation = ($10,000 - $400) ÷ 60,000 hours Hourly depreciation = $0.16 2,800 hours operated × $0.16 = $448 first year’s depreciation.

Week 122. Units of production

Thank you

discovering mathematics week 14 – unit 13 exponential mu123 dr. hassan sharafuddin

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