Direct numbers
AS WE continue our review of selected topics from the syllabus with Directed Numbers, I will share with you the answers to the problems given for homework last week. Evaluate the following: (i) -3 x -4 = 12 ii) - 48 / 8 = - 6 iii) - 2 - 6 + 7 = - 1 iv) 3a x -5b = - 15ab (v) 7/12 + 5/6 - 2/3
The solution to # (v) is based on the conversion of the three fractions to the same denominator. This denominator is 12, the LCM of the existing denominators 12, 6 and 3.
(Please note that the LCM of 12, 6 and 3 is 12.)
7/12 + 5/6 - 2/3
= (7 x 1) + (5 x 2) - (2 x 4)/12 = 7/12 + 10/12 - 8/12 = 9 /12 = 3/4
If the above posed no difficulty, then you are ready to consider exam-type questions.
Prior to doing so, I do recommend that you pursue a comprehensive review of the number system. You should no doubt be familiar with odd and even numbers, but what about natural numbers (1, 2, 3...), whole numbers (0, 1, 2 .... ), integers (..- 2, - 1, 0, 1, 2...), rational or irrational numbers, fractions and decimals? In reviewing these, please pay attention to:
I Proper fractions (5/6 )
I Improper fractions ( 17/12 )
I Mixed numbers ( 3 7/12 )
APPLICATION OF THE FOUR ARITHMETIC OPERATIONS TO VULGAR FRACTIONS
Having reviewed the application of the four arithmetic operations to integers, select fractions and decimals, we may now consider exam-type questions.
In applying multiple operations to vulgar fractions, you are required to observe the correct law with respect to applying the order of the operations as follows:
1) B – Brackets
2) O – Of (Multiply)
3) M – Multiply
4) D – Divide
5) A – Add
6) S – Subtract
BOMDAS identifies the order in which the operations should be carried out and must always be obeyed. Where an expression has multiple operations, then the operations within the brackets are evaluated first, if they exist. Multiplication or Of is done before division, while division is done before addition, and so on.
Let’s now practise the use of BOMDAS.
(A) PRACTICE 1
Calculate the value of: 2 1/3 + 9 x 2 , 1 1/2
Convert the mixed numbers to fractions = 7/3 + 9 x 2 , 3/2
In this case, three operations are involved. Using BOMDAS we do the multiplication first:
That is 9 x 2 = 18 = 7/3 + 18 , 3/2
We then do the division: = 7/3 + 18 , 3/2
= 7/3 + 18 x 2/2
= 7/3 + 12
And finally the addition: = 7/3 + 12 = 2 1/3 + 12 = 14 1/3
(B) PRACTICE 2
6 x ( 2 1/2 + 1/3). BOMDAS directs that we evaluate the brackets first (despite the fact that we are required to add):
(2 1 /2+ 1/3) = 5/2 + 1/3 .
Using the LCM of 2 and 3, that is 6, we get = ( 3 x 5) + (2 x 1)/6
= 15 + 2/6 = 17/6
To complete the problem, we now multiply: 6 x 17/6 = 17 6
(C) PRACTICE 3
Calculate the value of: 6 1/3 - 1 5/6 / 1 1/2 x 2 2/3 This is a typical exam-type question, so please note it well.
Using BOMDAS, we first note that the line represents brackets and so the numerator may be evaluated first. (It is also appropriate to evaluate the denominator first.)
6 1/3 - 1 5/6 = 19/3 - 11/6 (Initially, convert mixed numbers to a fraction.)
The LCM of 3 and 6 is 6
= ( 2 x 19) - (1 x 11) /6
= 38 - 11 /6 = 27/6
Evaluating the denominator: 1 1/2 x 2 2/3
Convert to common fractions and cancel 3/2 x 8/3 = 24/6 = 4
Dividing: = 27/6 , 4 = 27/6 x 1/4 = 9/8
The above assumes that you are able to manipulate fractions. If you are not able to, including cancelling, you need to get help in this specific area.
POINTS TO NOTE
In solving a problem such as Practice 3, you may first evaluate either the numerator or the denominator. You may verify this by finding the solution beginning with the denominator.
Finding the LCM CORRECTLY is a very important step in the solution. If you have difficulty with this step, you should resolve these at this time.
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 always be clearly shown in logical sequence as presented above.
Let us now work the following together:
EXAMPLE
Using a calculator, or otherwise, determine the exact value of : (4.3)2 - (7.24 - 5.31).
Solution
(4.3)2 - (7.24 - 5.31).
Using the recommended approach, we first evaluate the brackets using the calculator.
(4.3)2 = 18.49 and (7.24 - 5.31) = 1.93
As there are brackets, you may do the second bracket first as long as the substitution is done appropriately.
= 18.49 - 1.93 = 16.56
Ans = 16.56
EXAMPLE
Determine the values of:
1. 6.35 , 1.024
2. √8.6091
(0.3)3 + (0.5)2 5.62 x 3.7462
SOLUTION
Using the calculator:
1. 6.35 , 1.024 = 6.201
2. √8.6091 = 2.934
3. (0.3)3 + (0.4)2 = 0. 187
4. 5.62 x 3.7462 = 5.62 x 14.033 = 78.865
Please be reminded that it is important to get this the first question on the exam paper correct. It naturally builds your confidence. Always remember to apply BOMBAS. Even if the individual operations are done correctly, the appropriate order is required to get the correct answer.
I close this week with the following:
1. 4 x 6 + 12 / 3 - 5 =
2. Calculate the value of 41/2 x (7/3 + 1/4)
3. Simplify 3 1/3 - 1 5/8 / 1 1/3
4. Simplify 4 + 1 5/8 x 1 1/3
5. ( 1.8)3 - (0.5)2 =
6. Using a calculator, determine the value of (3.29)2 - 5.5 / √(1.5 x 0.06)
7. Find the value of : 4 1/3 - 1 5/6 / 2 1/2 x 2 2/3
8. Find the value of: 4.27 - (7.6)2 / √3.6 / 0.08
Finally, let me urge you to keep all of these lessons together in your notebook so that you can always refer to them. Your notebook should also include solutions to other similar questions. If you require previous copies of solutions to similar questions you should be able to access these from The Gleaner Company.
Clement Radcliffe is an independent contributor. Send feedback to kerry-ann.hepburn@gleanerjm.com.