Download Approximating Answers

More on the Number System
Including BODMAS and
Rounding
Try the following sum:
3+2x5
Try the sum on a non-scientific calculator.
What answer does it give?
Try the sum on your scientific calculator.
What answer does it give?
25
13
Which answer is correct?
To avoid confusion, there are rules to follow when a
sum involves more than one operation. This method
must always be followed even if it is not specified in
the question. Scientific calculators are programmed
to automatically follow the order of operations.
2
BODMAS
•
•
•
•
•
•
B
O
D
M
A
S
BRACKETS
()
ORDER or POWER OF
DIVISION

MULTIPLICATION
ADDITION
eg 23
x
+
SUBTRACTION
-
3
•
•
•
•
•
•
B
O
D
M
A
S
When a sum involves only division and multiplication,
the order is not important. Simply work from left to
right.
When a sum involves only addition and subtraction,
the order is not important. Simply work from left to
right.
• 4x6÷8÷2x5=
• 12 – 5 + 9 + 6 – 10 =
4
Examples (BODMAS)
• 5x3–2x3+84=?
11
• 5 x (3 – 2) x 3 + 8  4 = ?
17
• (5 x 3 – 2 x 3 + 8)  4 = ?
4.25
• (5 x (3 – 2) x 3 + 8)  4 = ?
5.75
• 9x9+5x5=?
106
• 60  5 + 42 = ?
28
5
Exercises
1. 4 x (12 – 6) + 6 ÷ 3
3. 2 x (20 – 2)  9
2. 36  (5 + 4) + 3 x 6
4. 52 x (8  2) + 3 x 8
Add in brackets to make the following correct:
5.
6.
7.
24 + (7 – 3) x 2 = 32
(9
+ 5 )x (9 – 5) = 56
4 x (3 + 2 ) + 32 = 29
6
Rounding off to decimal places
Work out 12971 ÷ 23.3 on your calculator.
The display shows 556.695279
We can round this off in several ways according to the
degree of accuracy we need to work to.
Remember, when rounding off the rules are as
follows:
 Always look at the digit to the right of the
number of places you are rounding off to.
 If that digit is 5 or more, round up.
 If the digit is 4 or less, leave the number as it is.
7
Round off the following to the degree of accuracy
specified.
a) 37.481 (2 d.p.)
b) 113.471 (1 d.p.)
c) 30.698 (2 d.p.)
d) 27.387 (2 d.p.)
e) 28.997 (1 d.p.)
f) 39.912 (1 d.p.)
g) 456.34567 (3 d.p.)
8
Rounding off to significant figures
This is another way of rounding numbers off.
NWHC News
October 2005
237 956 attend Concert.
kdjiejllk kljaldj ijlkdjiowl lijasdjf laudlfj lksajf;lk jalkjdfl ;alsjd
jfie hl;hpoijhd lakjdkjf a;lkjd jkf;alj djflajlsdj flasjldfj a;lsdjf
ja;lsjd jkd;alkjd kdk;lajsljd kmdls;sdkuriewxl ujuhqsicj
lsjhpeoiriujlknwe98osen;l dlj ldfliub 3; kadj kajn’piqa jpsidfpiup
joiajspo oijsldjfl lskdjfl /lkdjocsdlrjiuyuhgfgvnbhvt.ljvosy oewjrfl
asdf’l josdjnfglsdfo alsdfo uha’sdfjo hsdfn;lajdso jalsdj nawoif
lskdjc;o i\js ljfa\shf ‘lksjdfi kdjuef ;oisdlf ;sidf sa;l id lasdjfoiw
‘dpiuf ldjf;osdf jdf’sasijd flaksdjf ashdf;ljsdlif a.lsdjf fjdls;aldh
pasjd ;lskdjf jfkdl;saljsd fjkd’a
kdjiejllk kljaldj ijlkdjiowl lijasdjf laudlfj lksajf;lk jalkjdfl ;alsjd
jfie hl;hpoijhd lakjdkjf a;lkjd jkf;alj djflajlsdj flasjldfj a;lsdjf
ja;lsjd jkd;alkjd kdk;lajsljd kmdls;sdkuriewxl ujuhqsicj
lsjhpeoiriujlknwe98osen;l dlj ldfliub 3; kadj kajn’piqa jpsidfpiup
joiajspo oijsldjfl lskdjfl /lkdjoc
9
The most significant figure in a number is the figure
which has the greatest place value in that number, when
reading from left to right.
237 →
2 has the greatest place value, therefore is
the first significant figure
0.00328 → 3 has the greatest place value and is the first
significant figure. The zeros are simply used to indicate
place value and are not considered to be significant.
Zeros must be used when rounding off to significant
figures to preserve place value.
Round off 4 500 732 to 2 significant figures (2 s.f)
Round off 0.000 364 907 to 1 s.f.
10
Round off the following to the correct number of
significant figures.
a) 37.48 (3 s.f.)
b) 2991 (2 s.f.)
c) 0.00892 (2 s.f.)
d) 20.374 (3 s.f.)
e) 375.21 (2 s.f.)
f) 0.000748 (2 s.f.)
g) 0.0340078 (3 s.f.)
h) 456792 (2 s.f.)
i) 2334089 (1 s.f.)
11
Approximating Answers
Significant figures can be used to approximate answers.
This is a good way of checking that answers are
reasonable, especially when using calculators. Numbers
are usually rounded off to one significant figure.
Approximate
12971 ÷ 23.3
Approximate
89.6 x 10.3
19.7 + 9.8
Approximate
9.672
0.398
12
By rounding each of the numbers to one significant
figure, find approximate answers for:
a) 39.1 x 18.4
78.1
e) 61.4 x 1.87
49.2 – 28.8
b) 79.2 x 20.9
49.2
c) 42.1 x 2.97
2.017 x 31
d) 218 x 48.1
19.32
13
Sensible Answers
You will often arrive at an answer that does not
really make sense such as ½ a bus, or ¾ of a person,
or 35.46 pence.
In the exam you will be expected to look at the
problem and decide what is a sensible answer.
A college arranges a university trip for 165 students.
Each bus can carry 48 passengers. How many buses
are needed?
Bottles of pop can be bought in packs of six for
£2.49. How much does one bottle cost?
14
Using Examples to Show that a
Statement is True
By means of an example show that the sum of two
consecutive numbers is always odd.
Show, by means of an example, that the sum of four
consecutive numbers in always even.
Show, using an example, that the product of two
consecutive numbers is always even.
Ronnie says that the sum of any two prime numbers is
always even. Use an example to show she is incorrect.
Harriet says that the product of two even numbers is
never divisible by 25. Use an example to show she is
wrong.
15
Using statements to calculate similar answers
without using a calculator.
Given that 227.5 ÷ 35 = 6.5
Find the value of:
a) 6.5 x 3.5
b) 227.5 ÷ 350
c) 6.5 x 0.35
Use the calculation
58.5 x 27 = 1579.5 to find:
a) 585 x 27
b) 1579.5 ÷ 27
c) 585 x 0.027
16
Given that 487 x 3.53 = 1719.11
Find the value of:
a) 487 x 0.0353
b) 48700 x 0.00353
c) 1719.11  487
17