Consumer arithmetic
Clement Radcliffe, Contributor

Let us begin by checking the answers to last week's practice exercise.

1. Calculate the value of 4 ¹/2 x ²/3 - ¹/4.

Solution

4 ¹/2 x ²/3 - ¹/4.

The first step, of course, is to evaluate the product.

... 4 ¹/2 x ²/3 = ⁹/2 x ²/3= 3.

Completing, 3 - ¹/4 = 2 ²/3or ¹¹/4

2. Simplify (2 ¹/3 - 1 ⁵/8) ÷ ^{1 1}/3

Solution

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

2 ¹/3 -1⁵/8 = ⁷/3 - ¹3/8 (LCM of 3 and 8 is 24)

= ( (8 x 7) - (3 x 13) )/ 24 = (56 - 39)/24 = 17/24

Dividing next, ¹⁷/24 ÷ 1¹/3 = ¹⁷/24 ÷ ⁴/3 = ¹⁷/24 x ³/4 = ¹⁷/32

3. Find the value of: 18.75 - (2.11)²

Solution

18.75 - (2.11)² = 18.75 - (2.11 x 2.11)

= 18.75 - 4.4521 = 14.298

The use of a calculator is recommended.

4. Find the value of : (2¹/4 x ⁴/5)/ ³/5 - ¹/2

Solution

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

2 ⁹/5/4 x ⁴/5 = ⁹/4 x ⁴/5 = ⁹/5

Evaluating the denominator: ³/5 - ¹/2 (The LCM of 5 and 2 is 10)

( (2 x 3) - (5 x 1) )/10 = (6 - 5)/10 = 1/10

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 consumer arithmetic. Some popular topics are cost price, selling price, discount, sales tax, hire purchase, simple and compound interest.

Points to note

• Selling price = cost price + profit
  It follows that cost price = selling price - profit
  and profit = selling price - cost price
• Percentage profit = profit/cost price x 100
  It follows that percentage loss = loss/cost price x 100
• Discount is a reduction in selling price.
  It may be expressed as a dollar value or as a percentage.

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

Definition

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

It should be noted that a percentage may be expressed as a decimal fraction or as a vulgar fraction, for example, 40% = 40/100 = 2/5 = .4

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 60% of $800.

Solution

60/100 x $800 = $480.

This is the basis of finding values such as profit and loss, sales tax, general consumption tax, discount, etc.

Example 2

Schools were offered a 15% discount on the purchase of football gear. If a set of gear is valued at $180,000, how much less was paid?

Solution

15% of $180,000 = 15/100 x $180,000 = $27,000.

... The school paid $27,000 less.

NB: The amount the school paid is found as follows: 85/100 x $180,000 = $153,000.

The next situation is:

(B) Finding percentages, given the values.

Example 1

Express 5 m as a percentage of 8 m.

(a) 30%

(b) 45%

(c) 62.5%

(d) 160%

Solution

5/8 x 100 = 62.5%

... 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 costs $2,500, was sold for $2,200. Find the percentage loss.

Solution

profit = selling price - cost price

= $2,200 - $2,500 = - $300 ... the loss = $300

The percentage loss = loss/cost price x 100

300/2,500 x 100 = 12%

... The percentage loss is 12%.

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%.

... 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 x $6,300 = $945.

... The amount paid is $6,300 + $945 = $7,245

NB: As the amount represents 115%, you could also have found it as follows:

115/100 x $6,300 = $7,245.

In summarising, 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 )%.

Now, for your homework.

1. Mr Williams bought a plot of land for $40,000. The value of the land appreciated by 7% each year. Calculate the value of the land after one year.

2. In a certain country, electricity charges are made up of a fixed fuel charge of 45 cents per unit and an energy charge computed under three schemes as follows:

The meter reading of a certain business place is as follows:

Calculate

(i) The number of units used

(ii) The energy charge

(iii) The fuel charge

(iv) Calculate the total amount paid by the business place.

Clement Radcliffe is the principal of Glenmuir High School in May Pen.