|
Approximation
Clement Radcliffe,Contributor
Last
week we reviewed indices and several
points were presented for your information.
I ask that you look carefully at these
and then join me in checking the homework.
1.
Simplify the following:
(a)
5a3b x 4a2b
x 7ab3
(b)
12x-4y2 / 3x3y-5
Solution
(a)
5a3b x 4a2b
x 7ab3 = 5 x 4 x 7 x a3
x a2 x a x b x b x b3
As
multipliction of the above involves
adding indices:
...
5a3b x 4a2b
x 7ab3 = 140a6b5
(b)
12x-4y2 / 3x3y-5
= (12/3) x (x-4/x3) x (y2/y-5)
As division of the above involves
subtracting of indices:
...
= 4 x x-4-3 x y2--5
= 4x-7y7
2.
Find the values of:
(a)
64-5⁄6
(b)
27-2⁄3
Solution
(a)
Given 64-5⁄6,
we express 64 in terms of base 2.
...
64-5⁄6
= (26)-5⁄6
In
the case, you multiply indices.
...
(26)-5⁄6
= 2-5 = 1⁄25
=
1⁄32
(b)
27-2⁄3,
we initially express 27 in terms of
base 3.
=
(33)-2⁄3
= 3-2
=
1/32 = 1⁄9
(c)
Given 813⁄4
= (33) = 33
= 27
3.
Solve the following equation for x.
42x
= 1⁄32
Solution
42x
= 1⁄32
(22)2x
= 1 x 2-5
...
24x = 2-5
24x
= 2-5
Since
4x and -5 are both powers of 2 then
equating indices,
4x
= -5
x = -5⁄4
I
expect that you experienced no difficulty,
so we can now proceed to review ratio.
Ratio
If
two values are in the ratio 3:5, then
each represents, respectively, the
fraction of 3/8 and 5/8 of the whole.
Proof
In
this case, the whole is taken as 3
+ 5 = 8.
The
fractions are 3/8 and 5/8.
It
is vital for you to be able to convert
ratios to fractions in all cases.
Example
1
A
number is divided in the ratio 2:3.
What fraction does the smaller ratio
represents?
As
the number is divided into the ratio
2:3, then 2 + 3 = 5
The
fractions are 2/5 and 3/5
The
answer is 2/5
Example
2
A
sum of money is to be divided among
A, B and C in the ratio 3:4:5. The
largest portion amounts to $1,800.
Calculate:
(a)
The total sum of money to be shared
(b)
A's share
Since
the money is shared in the ratio 3:4:5
and the whole is represented by 3
+ 4 + 5 = 12, the respective portions
are as follows:
A
= 3/12 or 1/4
B
= 4/12 or 1/3
C
= 5/12
If
the largest share = $1,800, then this
represents C's share the total sum
is ( 1,800 x 12 )/5.
The
total sum is $4,320.
A's
share represents 1/4 of the total.
This is equal to 1/4 x $4 320 = $1,080
A's
share is $1,080
Finally,
we will now review briefly aspects
of approximation.
APPROXIMATION
This
topic highlights the various degrees
of accuracy to which a value may be
expressed. While counting always gives
an accurate value, it is measurement
which lends itself to approximation,
depending on the nature of the instrument
used. For example, an electronic balance
can measure the weight of a sample
to three or more decimal places while
this degree of accuracy is not always
required. You, therefore, have the
option of giving a value to the degree
of accuracy you require.
The
three methods which are usually used
at this level are:
1.
Decimal places
2.
Significant figures
3.
Standard form
Decimal
Places
Numbers
may be expressed correct to a specified
number of decimal places as in the
case of the following:
Express
46.42806
(i)
Correct to 2 decimal places.
(ii)
Correct to 3 decimal places
(iii)
Evaluate 2.732 + 1.2 Correct to decimal
place.
Solution
(i)
46.43
(Start
by looking first at the number which
is holding the third place after the
decimal point. It is 8. Since 8 is
more then 5, 1 is added to the number
2, the number which comes two places
after the decimal point.)
Note
If
it were 5 instead of 8 holding the
third place after the decimal, 1 would
also be added to the number 2.
(ii)
46.428 (The number holding the fourth
place is 0, in this case the number
8, which comes three places from the
decimal point, remains unchanged)
(iii)
2.732 + 1.2 = 7.4529 +
1.2 = 8.6529 =
8.7
Note
8.70
is incorrect as it represents
two decimal places.
Significant
Figure
The
degree of accuracy to which a value
is required may be determined by the
number of figures in the value. For
example, a value expressed correct
to two significant figures may be
in the form of 24, 1200km or 0.036
litres. All three represent the respective
value correct to two significant figures.
Please note the pattern.
Example
Express
259.163 correct to:
(i)
three significant figures
(ii)
four significant figures
Solution
(i)
259 (The number holding the fourth
place is 1 so the 9 remains unchanged.)
259.0 is incorrecet as it represents
four significant figures.
(ii)
259.2 (Since 6 > 5, then 1 is added
to the 1 similar to the method above.)
Standard
Form
This
is a very effective means of expressing
values, especially for very large
or very small numbers. 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
Express
3,715,382 in standard form.
a)
372 x 106
b)
3.72 x 10-6
c)
3.72 x 106
d)
3.72 x 107
Since
the standard form is A x 10n, using
the above definition of A and n, then
A is 3.72.
It
should be noted that CXC accepts values
expressed correct to three significant
figures. The other figures are simply
ignored.
In
the number 3,715,382, since the decimal
place is after the 2, then it is moved
six places to the left, to between
3 and 7, consistent with the definition
of A, therefore n = 6. The standard
form is therefore 3.72 x 106. The
answer is (c).
Example
Express
the following numbers in standard
form :
(i)
3846.78
(ii)
0.0007834 cm.
Answers
(i)
3846.78 = 3.85 x 103
(ii)
0.0007834 = 7.83 x 10 -4
Homework
1.$750,000
is divided among three sisters in
the ratio 5 : 8 : 2, respectively.
Calculate the amount each received.
2.
Find the following numbers correct
to 2 decimal places.
a)
4.028
b)
0.055
c)
6.999
3.
Divide 56 by 13. Give your answer
to 3 decimal places.
4.
Express the number 105.7064 correct
to the number of significant figures
stated below.
a)
6
b)
4
c)
2
d)
5
e)
3
f)
1
Clement
Radcliffe is an independent contributor.
Send questions and comments to kerry-ann.hepburn@gleanerjm.com
|
