Directed numbers
Clement Radcliffe, Contributor

Let us begin by checking the answers to last week's Practice Exercise.

In summarising, the following points should be noted:

1. Calculate the value of: 2 1/3 - 1 1/2, expressing your answer as a fraction.

Solution

2. Evaluate: 16/25 x 5/7 ÷ 8/25

Solution

16/25 x 5/7 ÷ 8/25 = 16/25x57x28/8 = 10/7

3. Calculate the value of (2 1/3 -1 5/8) ÷ 1 1/3

Using the order indicated by BOMDAS, we evaluate within the brackets:

Dividing next, 17/24 ÷ 1 1/3 17/24 x 3/4 = 17/32

Solution

Using the order indicated by BOMDAS, we first evaluate the numerator.

2 1/4 x 4/5 = 9/4 x 4/5 = 9/5

Dividing, 9/5 ÷ 1/10 = 9/5 x 10/1 = 9 x 2 = 18

NB. The line in the question represents brackets and so the denominator could have been first evaluated.

The lesson today will continue with a review of selected areas of sonsumer arithmetic. Some popular topics are: cost price, selling price, discounts, sales tax, hire purchase, simple and compound interest.

The concept of Percentage is fundamental to these topics as our review will illustrate.

Definition

Percentage is a fraction with its denominator being 100, therefore a% = a/100

It should be noted that a percentage may be expressed as a decimal fraction or as a vulgar graction, for example, 25% = 25/100 = 1/4 = .25

I will illustrate by looking at three situations in which the problems may be presented:

(a) Finding the value representing a certain percentage.

Example 1: Find 30% of $800

Solution: 30/100 x $800 = $240.00

This is the basis of finding values such as Profit and Loss, Sales tax, General Consumption tax, Discount etc.

Example 2: Mrs. King bought a component set for $48,600. If a tax of 15% is payable, find the value of the tax.

Solution: 15% of $48,600 = 15/100 x $48,600 = $7,290.00

Therefore the tax is $7,290.00

Example 3: Schools were offered a 15% discount on the purchase of football gear. If a set of gear is valued at $80,000, how much less was paid?

Solution: 15% of $80,000 = 15/100 x $80,000 = $12,000

Therefore the school paid $12,000.00 less

(b) Finding the percentage given the value.

Example 1: Express 5 m as a percentage of 8 m.

(a) 200%
(b) 40.0%
(c) 62.5%
(d) 130%

Solution: 5/8 x 100 = 62.5%

Therefore the answer is (c).

This is the basis of finding values such as Percentage loss or gain, Percentage tax, Discount etc.

Example 2: A radio cassette, which cost $2,500, was sold for $2,000. Find the percentage loss.

Loss = Cost Price - Selling Price

= $2,500 - $2000 = $500

Therefore the loss = $500

Therefore percentage loss is 20%.

Please note that percentage gain and loss are calculated as a fraction of cost price. A common error is to use the selling price.

The third situation is:

(c) Problems involving percentages.

Example 1: If 30% of a number is 69, then the number is

(a) 90
(b) 230
(c) 189
(d) 139

Solution: If 30% of a number is 69, then the number is equivalent to 100%

Therefore 100% represents 100/30 x 69 = 230
The answer is (b)

This is the basis of finding values such as Cost Price and Selling Price, Hire Purchase etc.

Example 2: A set of tools is priced at $6,300 plus GCT (General Consumption Tax) of 15%. How much is actually paid for the tools?

Solution: Cost Price is $6,300. Since the tax is 15%,
then 15/100 x $6300 = $945.00

Therefore the amount paid is $6,300 + $945.00 = $7,245.00

In summarizing, the following points should be noted:

• Percentage is a fraction of 100.
• The whole is represented by 100%
• If the whole is increased by x%, then the value becomes (100 + x)%
• If the whole is reduced by x%, then the value becomes (100 - x )%

For your Homework

1. The cash price of a dining-room suite consisting of a table and six identical chairs is $880. If the price of the table is $250 what is the price for each chair?

2. The dining-room suite may be bought on hire purchase for a deposit of $216 plus monthly payments of $35 for a period of two years. Calculate

(i) The total hire purchase price of the suite
(ii) The extra cost of buying on hire purchase as a percentage of the cash price.

(CXC January 2004, 2000, 1b, c)

Clement Radcliffe teaches at Glenmuir High School.