MENTAL CALCULATION
RE-ARRANGING
When trying to add a row of numbers, we should look for pairs that add up to make a
multiple of 10 or 100
1.
e.e.
13 + 8 + 7 + 6 + 2
20
2.
13 + 8 + 7 + 6 + 2
10
+ 10AND
+ 6 SO
= 36ON
UNITS, 20
TENS
When adding two numbers the units, tns etc can be taken separately. i.e
we are using our knowledge of addition number bonds (0-9) and an
understanding of place value.
Example
47 + 26
Adding 6 + 7
Then adding 20
OR
47 + 26
= 53 + 20
= 73
47 + 26
Adding 40 + 20
Adding 7 + 6
23
= 67 + 6
= 73
1
3.
COMPENSATION
We can sometimes add or subtract more than we should and then compensate.
We usually round the number to the nearest 10.
e.e.
37 + 19
We can round up the 19 to 20 and then compensate
by subtracting 1, because we have added 1 too much
= (37 + 20) - 1 = 57 - 1 = 56
e.e. 6.7 + 3.9
We can round the 3.9 up to 4.0 and then compensate by
subtracting 0.1, because we have added 0.1 too much.
= (6.7 + 4.0 ) – 0.1 = 10.7 - 0.1 = 10.6
A similar method can be used in subtraction:
e.e. 137 - 28
We can round the 28 up to 30 and compensate by adding
2, because we have subtracted 2 too many
= (137 - 30) + 2 = 107 + 2 = 109
4.
NEAR DOUBLES
If we are adding two numbers that are near to each other,
we can double one number and then compensate. We can
double the smaller number and add or double the larger
number and subtract.
e.e.
13+14
This can be considered as double 13 add one or double
14 subtract one
13 + 13 + 1 = 26 + 1 = 27
neu
14 + 14 - 1 = 28 - 1 = 27
2
MULTIPLICATION AND DIVISION
Most mental strategies for multiplication and division depend on a knowledge
of tables. This must be extended to the multiplication and division of larger
numbers:
x2
x3
x4
x5
x6
double
2 x 56 = (2 x 50) + (2 x 6)
= 100 + 12 = 112
double, then add the number
3 x 125 = (2 x 125) + 125
= 250 + 125 = 375
double and double again
4 x 34 = (2 x 34) x 2
= 68 x 2 = 136
multiply by 10 and halve
5 x
240 = (240 x 10) / 2
= 2400 / 2 = 1200
multiply by 5 and add the number
x7
double, double and double again
and subtract the number
x8
double, double again and double
again
8 x 24 = (24 x 2) x 2 x 2
= (48 x 2) x 2
= 96 x 2 = 192
x9
multiply by 10 and subtract the
number
9 x 57 = (10 x 57) – 57
= 570 - 57
= 513
x 10
move the numbers to the left
10 x 12 = 120
10 x 3.75 = 37.5
3
1.
ADDITION
(a)
Carrying from one column to the next, starting with the units.
This is the
usual method
Example
(a)
carrying
under the line
(i)
+
7648
1486
9134
1
1
1
We can use the methods above with decimals but we must remember to place
the decimal points underneath each other and to fill every gap with ‘0’ (zero)
as required
Example
+
124.90
73.25
198.15
2.
SUBTRACTION
(a)
Counting on method
In this method we gradually add to the lower number
Example
The bill at a market stall is £6.34. How much change should be given from
£20?
£6.34
6
6.40
+
10
6.50
+
50
7.00
+ 3.00
10.00
+ 10.00
£20.00
+
adding
these
£
0.06
0.10
0.50
3.00
+ 10.00
£ 13.66
4
(b)
Written calculation: decomposition method
In this method we take from the next column.
Example
5
6 14 78 14
- 2 6 6 7
3 8 1 7
(c)
We can use the above methods with decimals but we must remember to
place the decimal points underneath each other and to fill every gap
with ‘0’ (zero) as required.
Example
-
31214.8910
7. 2 3
3 1 7. 6 7
decomposition
5
3. MULTIPLICATION
(a)
Area Method
In this method we will break down the numbers as the sides of a
rectangle. The area of the rectangle will be the answer to the
multiplication
Example
(i)
56
x
50 and 6
34
30 and 4
50
6
1500
180
200
24
30 x 6
30
50 x 30
4
4x6
4 x 50
The answer is the total area therefore:1500
180
300
+
24
1904
(ii)
236
x
27
20 & 7
200 & 30 & 6
20 x 30
20 x 200
7 x 200
200
30
6
4000
600
120
1400
210
42
20
20 x 6
7
7x6
7 x 30
6
236 x 27 = 4000
600
120
1400
210
+
42
6372
(b) Multiplying Decimals
We can adapt the previous methods of multiplication to multiply decimals.
To simplify the multiplication process we will eliminate the decimal point
and then put it back in the right place at the end, after multiplying.
Example 1 (linked to the preceding example, above)
236 x 27 = 6372
3.27 is 100 times smaller than 327, so tye answer will be 100
times smaller, that is, 150.42
4.6 is 10 times smaller than 46, so the answer will be 10 times
smaller, that is , 15.042
Example 2
3.8 x 2.5
2x3
2
0.5
3
0.8
6
1.6
1.5
0.40
2 x 0.8
0.5 x 0.8
0.5 x 3
6
1.6
1.5
+ 0.4
9.5
7
(b)
Partition Method
In this method the smaller number is partitioned.
Example
(i)
352
x 27
352 x 7
2464
352 x 20
+ 7040
352 x 27 =
9504
it is broken down
into tens and units
4. DIVISION
(a)
Method
Example
4 1 8
7 | 2 9 12 56
If the number does not divide exactly we can show the answer with a
remainder or as a mixed number.
4997 ÷ 6 = 832 g 5 neu
832
5/
6
8
This is te traditional method
(b)
Long Division method
In this method it is important that you set out work with the tens and
units columns correctly underneath each other
Example
(i)
782  34
34
-
23
782
680
102
102
000
(answer line)
(34 x 20 = 680, put 2 in the tens column on the answer line)
(34 x 3 = 102, put 3 in the units column on the answer line)
So, 782  34 = 23
(i)
977  36
36
-
27
977
720
257
252
5
(answer ;ine)
(36 x 20 = 720, put 2 in the tens column on the answer line)
(36 x 7 = 252, put 7 in the units column on the answer line)
So, 782  36 = 27 remainder 5
or 782  36 = 27 5/36
(ch)
Dividing Decimals
Example
9.72  3.6
The traditional method can be used to calculate 972  36 = 27
9.72 is 100 times smaller than 972, so the answer will be
100 times smaller, that is, 0.27.
3.6 is 10 times smaller than 36 but the answer will be 10 times
more, that is, 2.7
9
3. FRACTION, PRCENTAGES AND DECIMALS
3.1
FRACTIONS
A fraction is one whole number divided by another
whole number.
this fraction is ¾, that is three shaded parts and four equal
parts in total.
Here are other examples
Equal/Equivalent Fractions
The shaded parts are the same size in the three diagrams so ½ = 2/4 = 4/8.
10
We can create equal fractions by multiplying or dividing.
We must treat the “top” number [the numerator] and the “bottom” number
[the denominator] in exactly the same way.
3.1
PERCENTAGES
A percentage means OUT of 100.
% is the percentag symbol.
We say that one whole is 100%.
3.2 CHANGING A FRACTION TO A PERCENTAGE
’Convenient’ numbers’ e.g
7
20
Try to change the number on the “bottom” (dnominator) to 100
and changing the “top” number in excatly the sam way.
In this example we can change 20 to 100 by multiplying by 5.
So, we also multiply the ‘7’ by 5 to make 35.
So
7 changes into 35
20
100
which is 35%
11
Less ‘convenient’ numbers e.g.
23
36
Here we use a calculator : 23 ÷ 36 = 0.638888…
.
= 0.638
neu 0.64 i ddau le degol
So, 23 is equivalent to 0.64 which is worth 64 which is 64%
36
100
PLACE VALUE: understanding place value thoroughly is
a cornerstone when working with fractions, decimals
and percentages.
Place Value
M
Million
Hundred Thousand
Ten Thousand
HTh TTh Th H T U . t h th
Thousand
thousandth
Hundred
hundredth
Ten
Units
tenth
Decimal Point
12
3.3
FINDING A PERCENTAGE OF A NUMBER
(a)
What is 10% of £40?
10% is 10 or 0.1
100
So 10% makes a sum 0.1 smaller which is 10 times smaller.
10 times smaller than £40 is £4
So 10% of £40.00 is £4
(b)
What is 5% of 50kg?
10% is 50kg  10 = 5kg
5%
= 2.5kg (5% is half of 10%)
(c)
What is 20% of £80?
10% is £8
20% is £16
(ch)
What is 8% of 250kg
10% is 25kg
1% is 2.5kg
2% is 5kg
8% is 4 x 2% sef 4 x 5kg = 20kg
3.4
FRACTIONS AND DECIMALS
To change a fraction to a decimal we must divide
the “top” (the numerator) by the “bottom” (the
denominator) using a calculator if necessary.
(a)
Expressing 3/8 as a decimal
3  8 = 0.375
(b)
Expressing 4 2/5 as a decimal
2  5 = 0.4
so 4 2/5 = 4 + 0.4
= 4.4
13
To cahnge a decimal to a fraction we must create a fraction over 10, 100,
1000 and so on, and then cancel if necessary.
(c)
Expressing 0.54 as al fraction
M
HTh TTh
0.54
=
Th
H
54
100
=
T
U
.
t
h
0
.
5
4
th
27
50
(ch) Expressing 3.6 as a fraction
3.6 = 3 + 0.6
0.6 = 6
10
=
3
5
3.6 = 3 + 3/5 = 3 3/5
3.5
(a)
DECIMALS AND PERCENTAGES
To understand changing a decimal to a percentage one must
understand “place value”.
M
Million
Hundred Thousand
Ten Thousand
HTh TTh Th H T U . t h th
Thousand
thousandth
Hundred
hundredth
Ten
tenth
Unit
Decimal Point
14
Expressing 0.35 as a percentage
M
HTh TTh
Th
H
T
U
.
t
h
0
.
3
5
th
That is 35 per cent = 35 which is 35%
100
Expressingi 1.275 as a percentage
M
HTh TTh
Th
H
T
U
.
t
h
th
1
.
2
7
5
That is 1 whole and 27.5 hundredth = 127.5%
(b) The same method is usedto change a percenmtyage to a decimal.
Expressing 45% as a decimal
45% means 45 which is 45 hundredth
100
M
HTh TTh
Th
H
T
U
.
t
h
th
0
.
4
5
U
.
t
h
th
0
.
1
7
5
So, 45 % = 0.45
Expressing 17½% as a decimal
17½% = 17.5% which is 17.5
100
M
HTh TTh
Th
H
T
So 17½% = 0.175
15
3.6
Exchanging common fractions, decimals and percentages
A table of common fractions, decimals and percentages.
FRACTION
½
¼
¾
1
/8
1
/10
1
/5
3
/10
3
/5
DECIMAL
0.5
0.25
0.75
0.125
0.1
0.2
0.3
0.6
PERCENTAGE
50%
25%
75%
12.5%
10%
20%
30%
60%
16