Understanding directed numbers (cont'd)
Clement Radcliffe, Contributor

As we continue our review of directed numbers, I will share with you the answers to the problems given for homework last week.

Evaluate the following:

(i) -4 x -3 = 12

(ii) -21 ÷ 7 = -3

(iii) ¹¹/12 + ⁵/6 ^{- 2}/3 (Please note that the LCM of 12, 6 and 3 is 12)

= ( (11 x 1) + (5 x 2) - (2 x 4) )/12

= (11 + 10 - 8)/12 = 13/12

(iv) -8 -7 + 6 = -9

(v) 3a x -6b = - 18ab

(vi) ¹²/25 x ⁵/9 ÷ ⁵/18

12/25 x 5/9 ÷ 5/18 = 12/25 x 5/9 x 18/5 = 24/25

If the above posed no difficulty, then you are ready to consider exam-type questions.

Application of the Four Arithmetic Operations to Vulgar Fractions

In applying the four basic operations to vulgar fractions, students are required to observe the correct law with respect to the order of operation as follows:

B - Brackets

O - Of (Multiply)

M - Multiply

D - Divide

A - Add

S - Subtract

BOMDAS identifies the order in which the operations should be carried out and must always be obeyed. If an expression has multiple operations, then brackets are evaluated before division. Multiplication is done before subtraction and so on.

Let's practise the use of BOMDAS.

(a) Practice 1

Calculate the value of:

1¹/2 + 5 x 2 ÷ 1²/3

Convert to common fraction

= ³/2 + 5 x 2 ÷ ⁵/3

We do the multiplication first:

³/2 + 10 ÷ ⁵/3

We then do the division:

³/2 + (10 x ³/5)

= ³/2 + 6

= 7¹/2

(b Practice 2

4 x (2 ¹/3+ ¹/2) We first do the brackets (despite the fact we are required to add):

... (2 ¹/3+ ¹/2) = ⁷/3 + ¹/2 .

Using the LCM of 2 and 3, that is 6, we get:

= ( ( 2 x 7) + (3 x 1) )/6

= (14 + 3)/6 = ¹⁷/6

To complete the problem, we now multiply:

4 x ¹⁷/6 = ⁶⁸/6 = ³⁴/3 = 11¹/3

(c) Practice 3

Calculate the value of: (6¹/3 - 1⁵/6)/1¹/2 x 2²/3

The line represents brackets and so the numerator may be evaluated first.

6¹/3 - 1⁵/6 = ¹⁹/3 - ¹¹/6

= ( ( 2 x 19) - (1 x 11) )/6

= (38 - 11)/6 = 27/6

Evaluating the denominator:

³/2 x ⁸/3 = ²⁴/6 = 4

Dividing: = ²⁷/6 ÷ 4 = ²⁷/6 x ¹/4 = ⁹/8

Points to note

• In solving a problem such as Practice 3, you may first evaluate either the numerator or the denominator.
• Finding the LCM correctly is a very important step in the solution.
• As Practice 3 requires the exact value, you are not allowed to express the fraction in decimal form. If this is done, then your answer would be different from 9/8 and you may be penalised.
• Your working must be always clearly shown in logical sequence.

Let us now work the following together:

Using a calculator, or otherwise, determine the exact value of:

(3.7)² - (6.24 - 1.3).

Solution

(3.7)² - (6.24 - 1.3)

Using the recommended approach, we first evaluate the brackets:

(3.7)² = 13.69 and (6.24 - 1.3) = 4.80

= 13.69 - 4.80 = 8.89

Ans = 8.89

I close this week with the following:

1. Calculate the value of: 4¹/2 x ³/4 - ¹/4.

2. Evaluate: ⁷/10 ÷ (²/5 + ⁴/15 x ³/5)

3. Simplify: 2¹/3 - ¹⁵/8 ÷ 1¹/3?

4. Find the value of: 18.75 - (2.11)2

(No. 1 (a) (ii), CXC January 2006)

5. Find the value of: 2¹/4 x ⁴/5

³/5 - ¹/2

(No. 1 (a) (i), CXC January 2006)

Finally, let me urge you to keep all of these lessons together in a scrapbook so that you can always refer to them. If you require previous copies, you should be able to access these from the Gleaner Company.

Students listen attentively in a math class during the The Gleaner's Youthlink CXC seminar in Westmoreland, last year. - FILE

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