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Approximation:
Standard form
Clement
Radcliffe, Contributor
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| Jamaican
WNBA Basketball Star Simone Edwards,
give some pointers to young basketballers
and netballers attending a Netball
Rally at St. Andrew High School
on October 7, 2005. - Ian Allen
Photo |
THE
FOLLOWING is the solution to the homework
given last week.
1.
An estate valued at $75,000 is divided
among three daughters in the ratio
5 : 8 : 2 respectively. Calculate
the amount each received.
SOLUTION
As
$75,000 is divided in the ratio 5
:8 :2, then the total is represented
by 5 + 8 + 2 = 15.
ie.
The respective fractions are 5/15
= 1/3, 8/15 and 2/15
The
answers are:
(a)
1/3 x $75,000 = $25,000
(b)
8/15 x $75,000 = $40,000
(c)
2/15 x $75,000 = $10,000
2.
Find the following numbers correct
to two decimal places.
a)
5.126
b) 0.085
c)3.999
SOLUTION
a)
5.126 = 5.13
b) 0.085 = 0.09
c)3.999
= 4.00
3.
Divide 41 by 15. Give your answer
to 3 decimal places.
SOLUTION
41
÷ 15 = 2.7333. The answer to
three decimal places is therefore
2.733.
4.
Express the number 105.8054 correct
to the number of Significant Figures
stated.
a)
6 Significant Figures: 105.805
b)
4 Significant Figures: 105.8
c)
2 Significant Figures: 110
d)
5 Significant Figures: 105.81
e)
3 Significant Figures: 106
f)
1 Significant Figure: 100
Let
us complete APPROXIMATION by
reviewing Standard Form.
Standard
Form is another means of approximating
the value of a measurement. It is
an effective means, especially, for
very large or very small values. The
Standard Form is A x 10n,
where A is a number between
1 and 10 and n, the power of
10, is an integer (positive or negative
whole number, or zero.)
Example
1
Express
6,715 in Standard Form.
a)
672 x 103
b) 6.72 x 10-3
c)
6.72 x 103
d) 6.72 x 104
Since
the Standard Form is A x 10n,
using the above definition of A
and n, then A is 6.72.
It should be noted that CXC accepts
values expressed correct to three
significant figures. The other figures
are simply ignored.
In
the number 6,715, since the decimal
place is after the 5, then we divide
by 1,000 or 103, that is, the decimal
point is moved 3 places to the left,
to between 6 and 7. This is consistent
with the definition of A, therefore
n = 3. The Standard Form is
therefore 6.72 x 103. The answer is
(c).
Finding
the correct value of n is the most
difficult aspect of this problem.
Just remember that:
*
If you move the decimal point 3 places
to the right, then n = -3
Example
0.00971
expressed in Standard Form is 9.71
x 10-³
*
If you move the decimal point 5 places
to the left, then n = 5
Example
463105.6
expressed in Standard Form is 4.63
x 105
In
the former case, A is greater than
the number given, while in the latter
case, A is less.
Example
2
Express
0.000651 in Standard Form.
Answer
is 6.51 x 10-4
N.B.:
The decimal point is moved four
places to the right.
It
is recommended that you check your
answer by reverting the answer in
Standard Form to a single number,
for example, 6.51 x 10-4
= 0.000651
The
number is the SAME VALUE whether
expressed in the original or Standard
Form.
We
will complete this lesson by reviewing
a very interesting area, ALGEBRA,
The
important areas which will be considered
for the syllabus content are:
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Expanding brackets
*
Algebraic fractions
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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
(x -2) (3x + 1)
= 3x²
-6x + x -2 = 3x²
-5x -2
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
3x multiplied by -2 = 6x or 1 x -
2 = 2.
Example
2
(3m
- 2)²
=
(a)
9m²
- 4
(b) 9m²
+ 4
(c)
9m²
-12m + 4
(d) 9m²
-12m - 4.
(3m
- 2)²
= (3m - 2)(3m - 2)
=
9m²
- 6m - 6m + 4 = 9m²
-12m + 4.
The
answer is (c).
ACTIVITY
Please
attempt the following for homework.
1.
State each of the following numbers
correct to the number of decimal places
given in brackets.
(a)
5.05 (i d.p.)
(b) 286.598 (2 d.p.)
(c)
0.0088 (3 d.p.)
2.
Write each of the following numbers
correct to the number of Significant
figures indicated in each bracket.
(a)
46.93106 (2 s.f.)
(b) 45.37 (1 s.f.)
(c) 37.8567 (3 s.f.)
3.
Write in Standard Form:
(a)
0.003921
(b) 49,376.82
4.
Expand the following:
(a)
(M + 1) (M - 1)
(b) (K - 3) (K + 1)
(c) (x + y) (x - y)
Enjoy
the rest of the week.
*
Clement Radcliffe is principal
of Glenmuir High School in Clarendon.
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