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Reviewing
standard form
Clement
Radcliffe, Contributor
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, therefore, 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 2 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
below. a)
5
b) 3
c) 1 SOLUTION
a)
5 significant figures: 105.81
(b) 3 Significant figures: 106
(c) 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 moved the decimal point 3 places to the right, then n = -3.
- If
you moved the decimal point 5 places to the left, then n = 5.
Example
2 0.00971
expressed in standard form is: 9.71 x 10-5 N.B
The decimal point is moved three places to the right. Example
3 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
4 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: - 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) = 2x²
+ x - 6x - 3 = 2x²
- 5x - 3 Answer
= 2x²
- 5x - 3 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 3 multiplied by
-2x = 6x or -3 x
1 = 3. Example
2 (3m
- 2)²
= (a)
9m² -
4 (b) 9m²
+ 4 (c) 9m²
- 12m + 4 (d) 9m²
- 12m - 4 SOLUTION
(3m
- 2)²
= (3m - 2)(3m - 2) =
9m? - 6m - 6m + 4 = 9m? - 12m + 4. The
answer is (c). Please
review all we have done today and 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 + 2) (M - 2) (b) (K - 4) (K + 1) Enjoy
the rest of the week. Clement
Radcliffe teaches at Glenmuir High School. | 
