Clement Radcliffe, Contributor

Here 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, 8/2 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) 4.028

b) 0.055

c) 6.999

SOLUTION

(a) 4.028 = 4.03

(b) 0.055 = 0.06

(c) 6.999 = 7.00

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

SOLUTION

56 ÷ 13 = 4.30769.

The answer to three decimal places is 4.308

4. Express the number 105.7064 correct to the number of significant figures stated below.

a) 6

b) 4

c) 2

SOLUTION

a) 105.706

(b) 105.7

(c) 110

Some students are inclined to give the answer to (c) as 11. The recommendation here is that you should always consider that 11 is not an approximation of 105.706. It is clear that 110 is.

We will complete this lesson by reviewing a very interesting area, algebra. The important areas which will be considered for the syllabus content are:

• Expanding brackets
• Algebraic fractions
• Linear equations
• Factorisation
• Inequations and their graphs
• Simultaneous equations
  

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.

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 + bx + ay + by

As usual, we will look at some examples.

Example 1

Evaluate (2x + 1) (x - 3)

SOLUTION

(2x + 1) (x - 3) = 2x? + x - 6x - 3 = 2x? - 5x - 3

Answer = 2x2 - 5x - 3

Here are some of the common errors that some students make:

1. Ignore the negative sign, if there is one.

2. Do an incorrect addition of the products.

Please avoid the common errors of saying either 3 x -2x = 6x or -3 x 1 = 3.

Example 2

(4m - 2)² =

(a) 4m² - 4

(b) 8m? + 4

(c) 16m? - 16m + 4

(d) 9m² - 16m - 4

SOLUTION

(4m - 2)² = (4m - 2)(4m - 2)

= 16m² - 8m - 8m + 4 = 9m² - 16m + 4.

The answer is (c).

We will now continue this lesson by reviewing algebraic fractions.

The method of simplifying algebraic fractions is the same as 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 LCM.

Example 1

Simplify (2 - b)/b - (2 + b)/4b

The LCM of the denominators is 4b

( 4(2 -b) - (2 + b) )/4b

(I am sure that you recall that the negative sign in front of the brackets will change the sign within the brackets)

= ( 4(2 -b) - (2 + b) )/4b

= ( 8 - 4b - 2 - b )/4b

= ( 6 - 5b )/4b

Example 2

Simplify 1/(2p - 3) - 4/p

The LCM of the denominators is p(2p - 3).

( p x 1 - 4(2p - 3) )/p (2p - 3)

= ( p - 8p + 12)/p(2p - 3) = ( - 7p + 12 )/p(2p - 3)

On your own, please attempt the following:

Simplify ( x - 2 )/3 + ( x + 1)/2

Carefully review all we have done today and attempt the following for homework.

1. Evaluate (2r - 3)3

2. Expand the following:

(a) (M + 3) (M - 4)

(b) (t - 3) (t + 6)

3. Evaluate (-2p +1)( -3p + 6)

4. Simplify (2y - 1)/5 - (y + 3)/5

Olympic gold medallist Asafa Powell (left) is greeted by Calene Gray and students at Charlemont High School in St Catherine, his alma mater, on October 6

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