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Algebra
Clement Radcliffe,Contributor
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 2 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 3 decimal places.
SOLUTION
56
÷ 13 = 4.30769.
The
answer to three decimal places is,
therefore, 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
Note
Some
students are inclined to give the
answer to (c) as 11. The recommendation
here is that while 11 is correct to
two significant figures, you should
always note 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
- Factorization
- 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
(4x -1) (x + 7)
SOLUTION
(4x
- 1) (x +7) = 4x2 - x +
28x - 7 = 4x2 + 27x - 7
Answer
= 4x2 + 27x - 7
Here
are some of the 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 errors of saying
either 7 x -1 = 7 or -1 x x = x.
Example
2
(4m
- 2)2 =
(a)
4m2 - 4
(b)
8m2 + 4
(c)
16m2 - 16m + 4 (d) 9m2
- 16m - 4
SOLUTION
(4m
- 2)2 = (4m - 2)(4m - 2)
=
16m2 - 8m - 8m + 4 = 16m2
- 16m + 4.
The
answer is (c).
We
will now continue this lesson by reviewing
algebraic fractions.
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.
For
example:
- The
LCM of 2, 4 and 6 is 12
- LCM
of 2, 3 and 5 is 30
Please
note the pattern well.
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
Let
us consider the solution.
The
LCM of 3 and 2 is 6
The
sum is ( 2 (x - 2) + 3 (x + 1) )/6
=
( 2x - 4 + 3x + 3 )/6
The
answer is ( 5x - 1 )/6
Carefully
review all we have done this week
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 )/6 - ( y + 3 )/5
5.
Express the following in Standard
Form:
-
276843
-
0.0005624
-
493.3785
Clement
Radcliffe is an independent contributor.
Send questions and comments to kerry-ann.hepburn@gleanerjm.com
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