Directed Numbers
Clement Radcliffe,Contributor

Having reviewed important aspects of the structure of the CSEC examinations, I will now consider some fundamental concepts of mathematics. These should have been done in the lower forms (grades seven to nine), but they are worth reviewing.

Prior to doing so, please let us together determine the solutions to last week's homework.

1. 4² - 2² =

(a) 2

(b) 4

(c) 12

(d) 14

SOLUTION

It is best to evaluate the answer as follows:

4² - 2² = 16 - 4 = 12. The answer is (c)

2. The least number of sweets which can be shared equally among 5, 10 or 15 children is

(a) 15

(b) 30

(c) 45

(d) 60

SOLUTION

The least number to be divided equally among the three numbers is the highest common factor (HCF). The HCF of 5, 10 and 15 is 30.
Therefore, the answer is (b).

You could have tested each answer also. For example, 10 sweets cannot be shared equally among 15 children. This is also the case for 45. While 60 is an answer, it is not the least number.

3. ²⁄₅ expressed as a percentage is

(a) 5%

(b) 20%

(c) 25%

(d) 40%

SOLUTION

²⁄₅ expressed as a percentage is ²⁄₅ x 100 = 40%. The answer is, therefore, (d)

4. 7853 to the nearest hundred is

a) 7800

(b) 7900

(c) 7950

(d) 8000

SOLUTION

7853 is between 7800 and 7900, therefore, the nearest hundred to 7853 is 7900. The answer is, therefore, (b).

Now let us continue this week's lesson by reviewing the topic Directed Numbers.

I do believe that it is worth emphasising the importance of this topic, as weakness in this area will affect your ability to solve problems involving the application of the four arithmetic operations (+, - , x , ÷) to real numbers.

Your performance in a wide variety of topics, including many in algebra, could also be significantly affected. The number line is quite useful in helping students to understand this topic. The following method is also recommended:

EXAMPLE

Evaluate 8 - 11

SOLUTION

I have 8 items but owe 11

I, therefore, owe three items which may be expressed as 8 - 11 = -3

Using either approach, if necessary, you should be able to evaluate the following examples.

(1) 3 + 9 = 12

(2) -3 + 14 = 11

(3) -29 + 2 = -27

(4) 8 - (-6) = 14

(5) -6- 9 = -15

(6) 5 - 8 -3 = -6

Let us now proceed to look at the multiplication and division of integers. Review the following examples with a view to identifying obvious patterns.

(1) -4 x -3 = 12

(2) -22 ÷ -2 = 11

(3) 12 ÷ -3 = -4

(4) -2 x 8 = -16

(5) 5a x -5b = -25ab

(6) 8 x p x q = 8pq

From the examples given above, the following should be noted:

Positive x Positive = Positive
Negative x Positive = Negative
Positive x Negative = Negative
Negative x Negative = Positive

This above pattern is also true when dividing. I strongly suggest that this be committed to memory. More important, you should ensure that all future calculations satisfy these rules.

Let us now review the addition and subtraction of fractions. This is usually the first question on the paper. It is in your best interest to begin on a successful note. Practice is, therefore, key.

ADDITION AND SUBTRACTION OF FRACTIONS

The method requires that you are comfortable with finding LCM.
Please review if necessary. It is also important that you are comfortable with converting a fraction to one with a different denominator.

E.g. 3/2 is equivalent to 3 x 2/2 x 2 = 6/4

The method is illustrated as follows:

Find 5/6 + 1/4 As the LCM of 6 and 4 is 12

⁵⁄₆ + ¹⁄₄ =

( (2 x 5) + (3 x 1) )/12 = ( 10 + 3 )/12. Answer is ¹³⁄₁₂

Now let us attempt the following together:

2²⁄₃ - ⁷⁄₅

In this case it is recommended that mixed numbers 2²⁄₃ be inverted to a fraction.

2²⁄₃ - ⁷⁄₅ = ⁸⁄₃ - ⁷⁄₅

The LCM of 3 and 5 is 15.
therefore ⁸⁄₃ - ⁷⁄₅ = ( (5 x 8) - (3 x 7) )/15 = ( 40 - 21 )/15. Answer = ¹⁹⁄₁₅

The multiplication and the division of fractions are also important fundamental concepts. Please review the following, noting that the rules relating to positive and negative numbers are also applicable:

1. 1/3 x - 5/3 = - 5/9

2. - 3/4 ÷ - 1/2 = - 3/4 x - 2/1 = 3/2

3. 1/6 x 7/3 ÷ 5/12 = 1/6 x 7/3 x 12/5 = 14/15

Constant practice is crucial to your success in mathematics so I will end this lesson with your homework.

Evaluate the following:

(i) -4 x -3

(ii) -21 ÷ 7

(iii) 11/12 + 5/6 - 2/3

(iv) -8 -7 + 6

(v) 3a x -6b

(vi) 12/25 x 5/9 ÷ 5/18

Now that you have completed the homework, I do recommend that you continue to do similar examples from your textbook.

Clement Radcliffe is an independent contributor. Send questions and comments to kerry-ann.hepburn@gleanerjm.com