Ratio and approximation

AS WE continue to review consumer arithmetic, we will begin this week with the solution to last week’s practice exercises. They are as follows:

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

SOLUTION

Simple Interest = Principal x Time x Rate/100 = $1200 x 2 x 5 1/2 /100 = $ 1200 x 2 x 11/2 /100 = $ 132.00 Answer is b)

2. A tourist exchanged US$300.00 for Jamaican currency at the rate of US$1.00 = J$115.00. Government tax of 15% of the amount exchanged is payable. Calculate in Jamaican currency: i) The tax paid. ii) The amount the tourist received.

SOLUTION

i) US$300 = J$300 x 115 = $34,500 Tax at 15% = 15/100 x J$34,500 = J$5,175.00

Answer: J$5,175

ii) Amount is $300 x 115 = J$34, 500. Since tax = J$5, 175

The amount received is J$34,500 - J$5,175 = J$29,325

Answer: J$29,325.00

3. In Jamaica, electricity charges are calculated based on the following: Fixed charge; $3,500

Charge per kwH used: $2.30

1. Calculate the electricity charges for a customer who used 1,200 KwH. There is a government tax of 17.5% on the electricity charges.

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

3. Calculate the total amount paid by the customer.

SOLUTION

1. The electricity charge is computed as: Fixed Charge + Charge for KwH used.

As 1200 KwH was used, then the charge is: = $3,500 + 1,200 x $2.30

= $3,500 + $1,200 x 2.30 = $3,500. + $ 2,760.

Answer: $ 6,260.

2, Since there is a government tax on the customer’s electricity charges:

The charge is = $ 6,260.

17.5% tax = $6,260 x 17.5 /100 = $1,095.50 Tax = $1,095.50

The total amount paid by the customer is: Electricity Charge + Tax

$ 6,260. + $ 1,095.50

Total amount is $7,355.50

We will continue this lesson with a review of Ratio.

The following is an extract from the syllabus:

SPECIFIC OBJECTIVES

1. Compare two quantities using ratios.

2. Divide a quantity in a given ratio.

CONTENT

1. Ratio and proportion.

2. Ratio and proportion of no more than three parts.

RATIO

If two values are in the ratio 5:4, then each represents, respectively, the fraction of: 5/9 and 4/9 of the whole.

PROOF

In this case, the whole is taken as 5 + 4 = 9. The fractions are 5/9 and 4/9 . NB: 5/9 + 4/9 = 9/9 = 1

It is vital for you to be able to convert ratios to fractions in all cases.

EXAMPLE

A number is divided in the ratio 7:3. What fraction does the larger ratio represents?

As the number is divided into the ratio 7:3, then 7+3 = 10.

The fractions are 7/10 and 3/10

The answer is 7/10.

EXAMPLE

A sum of money is to be divided among A, B and C in the ratio 3:4:5. The largest portion amounts to $1,800.

CALCULATE

a) The total sum of money to be shared. b) A’s share.

Since the money is shared in the ratio 3:4:5 and the whole is represented by 3 + 4 + 5 = 12, the respective portions are as follows:

A = 3 /12 or 1 /4

B = 4/12 or 1/3

C = 5/12

If the largest share = $1, 800, then this represents C’s share - the total sum is 1,800 x 12 /5.

The total sum is $4,320.

A’s share represents 1/4 of the total. This is equal to 1/4 x $4,320 = $1,080

A’s share is $1,080.

PRACTICE

Mr Brown left $350,000 and $150,000 for his two children. Calculate the ratio representing the share.

Total amount is $ 350,000 + $ 50,000 = $500,000. The factions are 350,000/500,00 & 150,000/500,000. That is 7/10 & 3/10 or ratio 7:3

Answer: 7:3 Finally, we will now review briefly aspects of approximation.

APPROXIMATION

This topic highlights the various degrees of accuracy to which a value may be expressed. While counting always gives an accurate value, it is measurement which lends itself to approximation, depending on the nature of the instrument used. Please note the following:

1. An electronic balance can measure the weight of a very small sample, for example, 42.00347934gm

2. This degree of accuracy is not always required.

3. You, therefore, have the option of giving a value to the degree of accuracy you require.

The three methods which are usually used at this level are:

1. Decimal places

2. Significant figures

3. Scientific notation

DECIMAL PLACES

Numbers may be expressed correct to a specified number of decimal places as in the case of the following:

Express 73.367032

(i) Correct to one decimal place (ii) Correct to five decimal places.

SOLUTION

(i) 73.4 (Start by looking first at the number which is holding the second place after the decimal point. It is 6. Since 6 is greater than 5, the number 3, which comes one place after the decimal point is increased by 1 to 4.)

NB: If it were 2 instead of 6 holding the second place after the decimal, then 3 would remain unchanged.

(ii) 73.36703

EXAMPLE

Express 0.0375 correct to three decimal places.

SOLUTION

(iii) 0.038 (The number holding the 4th place is 5. One is added to 7.)

SIGNIFICANT FIGURES

The degree of accuracy to which a value is required may be determined by the number of figures in the value. For example, a value expressed correct to two significant figures may be in the form of 24, 1200km or 0.036 litres. Please note that in the latter two cases, the zeros are not counted.

All three represent the respective value correct to two significant figures. The value 0.3009 is four significant figures. In this case, the zeros between 3 and 9 are counted. Please note the pattern.

EXAMPLE

Express 842.6590 correct to:

(i) Three significant figures.

(ii) One significant figure.

(iii) Five significant figures.

(iv) Express 3.47 x 0.15 correct to 2 significant figures.

SOLUTION

(i) 843 (The number holding the fourth place is 6 so 1 is added to 2.)

(ii) 800 (NB: Since the second number is 4, then the 8 remains unchanged.) Some students may give the answer as 8, but note that 842.6539 is approximately equal to 800.

(iii) 842.66 (NB: The sixth number is 9, so 1 is added to 5.)

(iv) The product of 3.47 x 0.15 = 0.5205. Correct to 2 significant figures is 0.52

SCIENTIFIC NOTATION

This is a very effective method, especially for very large or very small values. The scientific notation is A x 10n, where A is a number between 1 and 10 and n, the power of 10, is an integer (positive or negative whole number, or zero).

EXAMPLE

Express 3,715,382 in scientific notation. a) 372 x 106 b) 3.72 x 10-6 c) 3.72 x 106 d) 3.72 x 107

Since the scientific notation is A x 10n, using the above definition of A and n, then A is 3.72.

It should be noted that CXC accepts values expressed correct to three significant figures. The other figures are simply ignored.

In the number 3,715,382, since the decimal place is after the 2, then it is moved 6 places to the left, to between the numbers 3 and 7, consistent with the definition of A, therefore, n = 6.

The scientific notation is, therefore, 3.72 x 106. The answer is (c).

PRACTICE

Given the values: i) 0.07653 ii) 9,653.1274 iii) 53.9471

Express each in scientific notation, two decimal places and three significant figures.

HOMEWORK

1. $750,000 is divided among three sisters in the ratio 5:8:2, respectively. Calculate the amount each received.

2. Find the following numbers correct to two decimal places. a) 4.028 b) 0.055 c) 6.999

3. Divide 56 by 13. Give your answer to four decimal places.

4. Express the number 105.7064 correct to the number of significant figures stated below. a) 6 b) 4 c) 2 d) 5 e) 3 f) 1

5. 36,549 written correct to three significant figures: a) 37,000 b) 36,000 c) 36,500 d) 36,600

Please remember to schedule the review of these routine questions as when they are included on the paper, you need to maximise the marks earned.