Revision
WE WILL continue to review some basic topics which should not provide challenges for grade-11 students. It is important to develop the skill of showing working. This skill includes presenting the important steps in logical order. The following is the solution to the homework given last week.
1. $750,000 is divided among three daughters in the ratio 5:8:2, respectively. Calculate the amount each received.
SOLUTION
As $750,000 is divided in the ratio 5:8:2, then the total is represented by 5 + 8 + 2 = 15, therefore, the respective fractions are 5/15 = 1/3, 8/15 and 2/15.
THE ANSWERS ARE:
(a 1/3 x $750,000 = $250,000 (b) 8/15 x $750,000 = $400,000 (c) 2/15 x $750,000 = $100,000
It is always a good practice that in cases, as above, where the total is known, we should check the answer. In this case, $250,000 + $400,000 + $100,000 = $750,000
2. Find the following numbers correct to two decimal places.
a) 4.0287 b) 0.055 c) 6.99933
SOLUTION
(a) 4.0287 = 4.03 (b) 0.055 = 0.06 (c) 6.99933 = 7.00
POINTS TO NOTE:
In a) 1 is added to 2 as 8, the value holding the third place, is greater than or equal to 5.
In b) the method directs that the zero after the decimal point is counted in the number of decimal places. Your answer always has two digits after the decimal point. 3. Divide 56 by 13. Give your answer correct to three decimal places.
SOLUTION
56 ÷ 13 = 4.30769. The answer to three decimal places is therefore 4.308
4. Express the number 15.7064 correct to the number of significant figures stated below.
a) 3 b) 1 c) 2
SOLUTION
a) 15.7 (b) 20 (c) 16
NB: Some students are inclined to give the answer to (b) as 2. The reason is that while 2 is correct to one significant figure, you should always note that 2 is not an approximation of 15.7064. It is clear that 20 is. You should always check that the number and the answer are approximately equal. Note also that in significant figures, the zero(s) before the decimal point is not counted. 5. Express 493.3785 in scientific notation.
SOLUTION
4.93 x 102 You should, when possible, expand to verify that your answer is correct. In this case, 4.93 x 102
= 4.93 x 100 = 493, which is approximately equal to 493.3785. We will complete this lesson by reviewing a very interesting area, ALGEBRA.
The important areas which will be considered for the syllabus content are:
Apply the distributive law to factorising or expanding algebraic expressions. Simplify algebraic fractions. Solve linear equations in one unknown. Change the subject of formulae. Solve a simple linear inequality in one unknown.
Solve simultaneous linear equations in two unknowns algebraically.
Factorise algebraic expressions, for example, a2 - b2, a2 ± 2ab + b2, ax + bx +ay + by and ax2 + bx + c, where a, b, and c are integers and a ≠ 0.
Students, you will recall that many of these topics were done in the lower forms and are not usually effectively revised. I must, again, remind you of the need to include these in your revision syllabus.
PRODUCT
The product of a x (p + q) is found by multiplying a by each term in the bracket and adding both products.
a x ( p + q) = ap + aq
EXPANDING TWO BRACKETS
The product of (a + b) (x + y) is found by multiplying each term in the first bracket by the terms in the second, and then adding the four products. This is the way to do it. (a + b) (x + y) = ax + ay + bx + by As usual, we will look at some examples. EXAMPLE 1 Expand 3x x (4x + 3y)
SOLUTION
3x x ( 4x + y) = 3x x 4x + 3x x y = 12x2 + 3xy NB : The 3x must multiply both terms in the bracket.
Also, I am sure you agree that 2a x ( 3n - 2m) = 6an - 4am.
And do you agree that 2x(x + 5) - 3(x - 4) = 2x2 + 10x - 3x + 12
= 2x2 + 7x + 12? EXAMPLE 2 Evaluate (4x -1) (x + 3)
SOLUTION
(4x - 1) (x +3) = 4x2 - x + 12x - 3 = 4x2 + 11x - 3 Answer = 4x2 + 11x - 3
Here are a few common errors that some students make:
1. Some students ignore the negative sign, if there is one.
2. Some students do an incorrect addition of the products.
Please avoid the common error of saying either 3 x -1 = 3 or -1 x x = x. Avoid this.
EXAMPLE 3
(3m - 2)2 = (a) 3m2 - 2 (b) 9m2 + 4 (c) 9m2 - 12m + 4 (d) 9m2 - 12m - 4
SOLUTION
(3m - 2)2 = (3m - 2)(3m - 2) = 9m2 - 6m - 6m + 4 = 9m? - 12m + 4. The answer is (c). Please proceed to practise some on your own.
PRACTICE
Expand the following: i) 5y (3y - 7) ii) ( 2M - 5N) ( M + 3N) iii) (3x2 - 5)2 We will now continue this lesson by reviewing ALGEBRAIC FRACTIONS.
ALGEBRAIC FRACTIONS
The method of simplifying algebraic fractions is similar to that used for vulgar fractions. This is also true for addition or subtraction of algebraic fractions. It follows, then that you must know the method used to find least common multiple (LCM). For example:
The LCM of 3, 4 and 6 is 12. That is, 12 is the smallest number for which 2, 4, and 6 are factors.
LCM of 2, 3 and 5 is 30. As 2, 3 and 5 are prime factors, then the LCM is the product of all three numbers.
LCM of 3 and 7 is 21, while 3, 6 and 5 is 30 (3 is a factor of 6 so the LCM is 6 x 5 and not the product of 3, 6 and 5). Please note the pattern well.
EXAMPLE 1
Simplify 2a/3 + a/6 . As the LCM of 3 and 6 is 6 then converting 2a/3 to 4a/6,
The sum = 4a/6 + a/6 = 5a/6 Answer is 5a/6 Alternatively: 2a/3 + a/6 2x2a + 1xa /6 = 4a + a /6 = 5a/6
EXAMPLE 2
Simplify 1 - b /b - 3 + b /4b