Consumer arithmetic

HAVING REVIEWED the application of arithmetic operations to fractions, let us complete it by checking the answers to last week’s practice exercise. If you are having difficulty, you may check the information presented previously. Let us now review a few problems presented in last week’s assignment. 1. Calculate the value of 4 1/2 x 7/3 +1/4. SOLUTION 4 1/2 x 7/3+1/4.

The first step, of course, is to evaluate within the brackets according to BOMDAS.

7/3 +1/4

As the LCM of the denominators 3 and 4 is 12, the two fractions are converted to:

7 x4 + 1 x 3 /12 = 28 + 3 /12 = 31/12

Completing, 4 1/2 x 31/12 = 9/2 x 31/12 =93/8

Answer : 93/8 2. Find the value of: (1.5)3 - (0.7) 2 SOLUTION Using the calculator: (1.5)3 - (0.7) 2 = 3.375 - 0.49 = 2.885 = 2.885 3. Calculate the value of: 4 1/3 - 1 5/6 / 2 1/2 x 2 2/3 SOLUTION The line represents brackets and so the numerator may be evaluated first. 4 1/3 - 1 5/6 = 13/3 - 11/6 The LCM of 3 and 6 is 6 = ( 2 x 13) - (1 x 11) /6 = 26 - 11/6 =15/6

Evaluating the denominator, we first convert to fractions:

5/2 x 8/3 = 40/6 = 20/3 Dividing: = 15/6 ÷ 20/3 = 15/6 x 3/20 = 3/8

NB: The line in the question represents brackets and so the denominator could have been first evaluated. 5. Find the value of: (4.27)2 - 7.6 ʚ3.6 ÷ 0.08 SOLUTION Using the calculator to evaluate the numerator, (4.27)2 - 7.6 =18.233 - 7.6 = 10.633 Evaluating the denominator: ʚ 3.6 ÷ 0.08 = 1.897 ÷ 0.08 = 23.713 Dividing: 10.633 ÷ 23.713 = 0.448 Practice Calculate the values of: i) 3.122 - 1.053 ii) 3 2/3 + 5 1/2 x1/4 / 4 1/2 - 2 1/5

The lesson, this week 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.

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

The following extract was taken from the syllabus:

SPECIFIC OBJECTIVES

Calculate: discount, sales tax, profit and loss. Also, percentage profit and loss.

Express a profit, loss, discount markup and purchase tax as a percentage of some value.

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,

60% = 60/100 = 3/5 = .60

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 75% of $ 9,600.

SOLUTION

75/100 x $ 9,600 = $7,200.

This is the basis of finding values such as profit and loss, sales tax, General Consumption Tax, discount, etc.

EXAMPLE 2

Jamaica charges 15% tax on the purchase of football gear. If a complete set is valued at $250,000. How much was paid?

SOLUTION

15% of $250,000 = 15/100 x $ 250,000 = $37,500.

The school paid $37,500 more, therefore, the amount paid is $ 250,000 + $ 37,500 = $ 287, 500

Answer : $ 287,500

NB: The amount the school paid may also be found as follows:

Since there is a 15% tax payable, then the amount paid represents 115%. 115/100 x $250,000 = $287,500.00 The next situation is: (B) Finding percentages, given the values.

EXAMPLE 1

Express 30cm as a percentage of 80cm. (a) 30% (b) 37.5% (c) 62.5% (d)130%

SOLUTION

30/80 x100 = 37.5% The answer is (b). This is the basis of finding values such as percentage loss or gain, percentage tax, discount, etc.

EXAMPLE 2

A television set, which costs $1,064 is sold for $1,399. If the television set is purchased in cash, there is a 5% discount. Find a) the selling price on a cash sale, b) the profit or loss as a percentage of cost price.

SOLUTION

a) The marked price is $1,399. Since there is a 5% discount on cash sale.

Then 5% of $1399 = 5/100 x $ 1,399 = $ 69.95 The amount paid = $1,399 - $69.95 = $ 1,329.05

ALTERNATIVELY

The marked price is $1,399. A 5% discount represents 95%.

The amount paid = $1,399 x 95/100 = $ 1,329.05 b) The Profit = Selling Price - Cost Price = $ 1,329.05 - $ 1,064 = $ 265.05 The percentage profit = Profit /Cost Price x 100 = $265 .05/1,064 x 100 = 24.9% The third situation is: (C) Problems involving percentages. EXAMPLE 1 Milk is sold in three containers as follows: Container size Selling price a) 350ml $4.20 b) 450ml $5.35 c) 500ml $5.80 Which size of milk is the most cost effective?

SOLUTION

The cost per ml for each is as follows: a) $4.20 ÷ 350 = 1.2 cents b) $5.35 ÷ 450 = 1.19 cents

c) $5.80 ÷ 500 = 1.16 cents The 500ml container is the most cost effective as it is the lowest cost per ml.

EXAMPLE 2:

The cash price of a radio is $1,299. It can be bought on hire purchase by making a deposit of $350 and 10 monthly payments of $120 each.

1.What is the total hire purchase price of the radio?

2.How much is saved by buying the radio for cash?

SOLUTION

1. Total hire purchase is : deposit + total monthly payments = $350 + 10 x $120 = $350 + $1,200 = $1,550

2. Savings by paying cost price = $1,550 $1,299 = $251

Answer : $251 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 )%.

HOMEWORK

1. Mr James bought and sold a cellular phone. The cost price is $1,064 and the marked price is $1,399. If a discount of 5% is applied, calculate the: The selling price. The profit or loss as a percentage of the cost price.

2. In a certain country, electricity charges are calculated based on the following table: Fixed charge Charge per kwH used $3,500 $2.30

(i) Calculate the electricity charges for a customer who used 1,200kwH.

There is a government tax of 17.5% on the electricity charges.

(ii) Calculate the tax on the customer’s electricity charges, giving your answer to the nearest cent.

(iii) Calculate the total amount paid by the customer.

3. How much simple interest is due on a loan of $1,200 for two years if the annual rate of interest is 5 1/2 per cent.?

a) $120 b) $132 c) $264 d) $330.

4. A tourist exchanged US $ 300 for Jamaican currency at the rate of US $1 = Ja $115.

Government tax of 15% of the amount exchanged is payable. Calculate in Jamaican currency: i) The tax paid. ii) The amount the tourist received.

5. A sum of $1,498 is invested at 6% simple interest per annum. Calculate: i) The interest earned after six months. ii) The total amount of money in his account after three years.

iii) How long it will be before his investment earns $449.40. Have a productive week.