Times A simple visual introduction to mathematical principles for the QTS Numeracy Skills Test
Content
This book teachers you everything you need to know for mental arithmatic section of the QTS Numeracy Skills Test.
Whole Numbers
Fractions
Ratio
Place-value
Simple Fractions
Finding the Ratio
Addition
Numerator and Denominator
Simplifying Ratios
Subraction
Adding Fractions
Multiplication
Common Denominator
Division
Subracting Fractions
Multiplication Table
Common Denominator
Prime Numbers
Multiplying Fractions
Dividing Fractions
Simplifying Fractions
This section of the test is 12 minutes long worth 12 marks and no calculator is allowed to be used.
Short-cut techniques are reviewed in this section because traditional methods are too slow.
Decimals
Percent
Coverting
Place-value
What is a Percent?
Percent to Decimal to Fraction
Decimal Fractions
Percent as a Decimal
Adding Decimals
Percent of something
Subracting Decimals
Multiplying a Decimal by 10
Multiplying a Decimal by 100
Multiplying a Decimal by 1000
Multiplying a Decimal
Dividing a Decimal by 10
Dividing a Decimal by 100
Dividing a Decimal by 1000
Dividing a Decimal
Whole numbers
Place-value
Units
0 0 07
Tens
0030
Hundreds
0400
Thousands
8000
Whole numbers have a place-value based on the decimal system of units, tens, hundredds and thousands.
Thousands
Hundreds
Tens
Units
8437
Addition
First add the tens
50+30=80
Then add the units
04+02=06
Finally add the results together
54+32=86
Calculations with large numbers can be made easier by splitting them up into a series of smaller sums based on place value.
54+
32=86
Subtraction
First subtract the tens
50–30=20
Then subtract the units
04–02=02
Finally add the results together
20+02=22
Calculations with large numbers can be made easier by splitting them up into a series of smaller sums based on place value.
54–
32=22
Multiplication
First multiply the tens
050×3=15 0
Then multiply the units
003×3=009
Finally add the results together
150+9= 159
Calculations with large numbers can be made easier by splitting them up into a series of smaller sums based on place value.
53×
3=159
Division
First divide the tens
60รท4=15
Then divide the units
08รท4=02
Finally add the results together
15+2=17
Calculations with large numbers can be made easier by splitting them up into a series of smaller sums based on place value.
68รท
4=17
Multiplication Table
Memorise all your times tables. The black squares are square numbers which occure when two of the same number are multiplied together.
1
2
3
4
5
1
1
2
3
4
5
2
2
4
6
8
10
3
3
6
9
12
15
4
4
8
12
16
20
5
5
10
15
20
25
6
6
12
18
24
30
7
7
14
21
28
35
8
8
16
24
32
40
9
9
18
27
36
45
10
10
20
30
40
50
Ă—
6
7
8
9
10
6
7
8
9
10
12
14
16
18
20
18
21
24
27
30
24
28
32
36
40
30
35
40
45
50
36
42
48
54
60
42
49
56
63
70
48
56
64
72
80
54
63
72
81
90
60
70
80
90 100
Prime Numbers
A prime number is a number that is divisible by only itself and 1.
1
2
3
4
5
11
12
13
14
15
21
22
23
24
25
31
32
33
34
35
41
42
43
44
45
51
52
53
54
55
61
62
63
64
65
71
72
73
74
75
81
82
83
84
85
91
92
93
94
95
6
7
8
9
10
16
17
18
19
20
26
27
28
29
30
36
37
38
39
40
46
47
48
49
50
56
57
58
59
60
66
67
68
69
70
76
77
78
79
80
86
87
88
89
90
96
97
98
99 100
Fractions
Simple Fractions
A fraction is a way of representing division of a whole into parts.
4/4
3/4
One whole
Three quarters
1/2
1/4
One half
One quarter
Numerator and Denominator
Names given to the top and bottom number of a fraction.
Numerator
1/
/2 Denominator
Adding Fractions
This is an example of a simple addition of fractions.
The original fractions
1/2 + 1/4
1/2 + 1/4
One half is equal to two quarters.
1/2
= 2/4
So all you need to do is add on another quarter.
1/2 + 1/4 = 3/4
= 3/4
Adding Fractions
Adding fractions that already have the same denominator is easy you just add the numerators together.
Add the numerators together and leave to denominator the same.
3/
/6
+ 2/
= 5/
/6
/6
3/6 + 2/6
= 5/6
3/6 + 2/6
= 5/6
Common Denominator
When the denominators is different we have to find the common denominator which is a multiple of both denomintors.
The original fractions
1/2 + 1/3
1/2 + 1/3
Times the denominators together to find the common denominator.
/2 Ă—
/3 =
/6
You must do the same denominator and numerator.
1/
Ă—
3 = 2/
1/
Ă—
2 = 2/
So now you have to add the numerators.
3/6 + 2/6 = 5/6
=
3/6 + 2/6 = 5/6
Subtracting Fractions
This is an example of a simple subtraction of fractions.
The original fractions
1/2 – 1/4
1/2 – 1/4
One half is equal to two quarters.
1/2
= 1/4
So all you need to do is subtract the another quarter.
1/2 – 1/4
= 1/4
= 1/4
Subtracting Fractions
Subtracting fractions that already have the same denominator is easy you just subtracted the numerators.
Subtract the numerators and leave to denominator the same.
3/
– 2/
/6 –
/6
3/6 – 2/6
= 1/
=
/6
= 1/6
3/6 – 2/6
= 1/6
Common Denominator
When the denominators is different we have to find the common denominator which is a multiple of both denomintors.
The original fractions
1/2 – 1/3 =
1/2 – 1/3
Times the denominators together to find the common denominator.
/2 ×
/3 =
/6
You must do the same denominator and numerator.
1/
×
3 = 2/
1/
×
2 = 2/
So now you have to subtract the numerators.
3/6 – 2/6 = 1/6
3/6 – 2/6 = 1/6
Multiplying Fractions
To multiply a fraction all you have to times the numerators together and then the denominators.
First multiply the numerators
1/
× 3/
= 3/
Then multiply the denominator
/6 ×
/8 =
/48
So
1 / 6 × 3 / 8 = 3/48
1/6 × 3/8
= 3/48 0
5
10
15
20
25
30
35
40
45
Dividing Fractions
To divid a fraction you have to flip the right-hand fraction upside down and then multiply it by the left-hand fraction.
Original sum
1/6 รท 3/8
Firstly flip the right hand fraction upside down so it becomes
8/3
Then multiply by the left-hand side
1 / 6 รท 8 / 3 = 8/18
1/6 รท 3/8
= 8/18 0
2
4
6
8
10
12
14
16
18
Simplifying Fractions
To simplify fractions you divide the numerator and the denominator by the same prime factors to give the lowest eqivalent fractions.
Find a common prime factor of both the numerator and denominator. In this case it would be 2.
4/6
2
Divide this number by the numerator.
4/
รท
2 = 2/18
The divide this number by the denominator.
/6 รท
2 =
/3
Making the numerator and denominator decrease in size.
4/6 = 2/3
=
2/3
Ratio
Finding the Ratio
First add thogether the two numbers in the ratio to get the whole about.
0 1+03= 04
Then times each part of the ratio by the answer
0 1 × 04 = 0 4
0 3×0 4= 1 2 Then divide 60 by each answer
60÷04=15
60÷12=45
So the new answer is
15:45
Ratios are simular to fractions. They show how a whole is divided into parts.
÷60 1:3 in the ratio
=15:45 0
5
10
15
20
25
30
35
40
45
Simplifying Ratios
Find a common factor that each side can be cancelled down by.
16รท 2=8
1 2รท 2= 6
Ratios can be simplified in a similar way to fractions by cancelling both sides by a common factor.
16:12 0
2 Cancelled down again by 2.
8:6 4
6
Divide each side of the ratio by 2 again.
8
8 รท 2=4 10
6รท 2= 3
12
14
So the new ratio is.
4:3 16
=8:6=4:3
Decimals
Place Value
Thousandths
Decimals have a place-value based on the decimal system.
Hundreds
Tens
Units
Decimal Point
. 1 3 7 5
Tenths
Hundredths
Ten Thousandths Hundred Thousandths
Thousandths
5 9 6 3 1 0
0
0
0
0
1
10
100
1000
10000
2
20
200
2000
20000
3
30
300
3000
30000
4
40
400
4000
40000
5
50
5 00
5 00 0
50000
6
60
6 00
6000
60000
7
70
7 00
7000
70000
8
80
8 00
8000
80000
9
90
9 00
9000
90000
10
100
1 0 00
1 0 00 0
100000
Decimals Fractions
Fractions have an equivalent decimal. This examples are worth remembering.
0.5 0
1
2
3
4
5
6
7
8
9
10
=1/2 0
1
2
Decimals Fractions
Fractions have an equivalent decimal. This examples are worth remebering.
0.25 0
10
20
30
40
50
60
70
80
90
100
=1/4 0
1
2
3
4
Decimals Fractions
Fractions have an equivalent decimal. This examples are worth remebering.
0.125 0
100
200
300
400
5 00
6 00
7 00
8 00
9 00
1000
=1/8 0
1
2
3
4
5
6
7
8
Adding Decimals
Set out your sum so that the decimals are aligned. Add a 0 where there is a gap.
0.062
0.200 +
Then add the numbers up column by column.
0.062
0.200 + 0.262
The key when adding decimals together is to make sure that the decimal points are aligned.
0.062+
0.20=0.262
Subtracting Decimals
Set out your sum so that the decimals are aligned. Add a 0 where there is a gap.
0.062
0.200 –
Then subtract the numbers column by column.
0.062
0.200 – 0.242
The key when subtracting decimals is to make sure that the decimal points are aligned.
0.062–
0.20=0.242
Multiplying a Decimal by 10
When multiplying a decimal by 10 all you have to do is move the decimal point one place to the right.
0.75Ă— 0
10
20
30
40
50
60
70
80
90
100
10=7.5 0
0
0
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
7
7
7
8
8
9
9
10
10
Multiplying a Decimal by 100
When multiplying a decimal by 100 all you have to do is move the decimal point two places to the right.
0.75Ă— 0
10
20
30
40
50
60
70
80
90
100
100=75 0
0
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
90
100
Multiplying a Decimal by 1000
When multiplying a decimal by 1000 all you have to do is move the decimal point three place to the right.
0.75Ă— 0
10
20
30
40
50
60
70
80
90
100
1000=750 0
0
100
10
200
20
300
30
400
40
5 00
50
6 00
60
7 00
70
8 00
9 00
1000
Multiplying a Decimal
First of all remove the decimal point from the question so it becomes
8 × 1 02 4
To multiply decimals you ignore the decimal point and then add it back in at the end.
8×10.
and then begin to multiply as you would normally splitting it up into smaller sums based on place value. Frist multiply the thousands.
8×1000=8000
Then multiply the hundreds.
8×0000=0000
Then multiply the tens.
8×0020=0160
Then multiply the units.
8×0004=0032
Then add the results together and put back to two decimal places from the question into the answer.
8000+000+160+32=8192=81.92
24=81.92
Dividing a Decimal by 10
When dividing a decimal by 10 all you have to do is move the decimal point one place to the left.
0.75รท 0
10
20
30
40
50
60
70
80
90
100
10=0.075 0
0
1
100
2
200
3
300
4
400
5
5 00
6
6 00
7
7 00
8
8 00
9
9 00
10
1000
Dividing a Decimal by 100
When dividing a decimal by 100 all you have to do is move the decimal point two places to the left.
0.75รท 0
10
20
30
40
50
60
70
80
90
100
100=0.0075 0
0
10
1000
20
2000
30
3000
40
4000
50
5000
60
6000
70
7000
80
8000
90
9000
100
10000
Dividing a Decimal by 1000
When dividing a decimal by 1000 all you have to do is move the decimal point three place to the left.
0.75รท1000 0
0
10
100
20
200
30
300
40
400
50
5 00
60
6 00
70
7 00
80
8 00
90
9 00
100
1000
=0.00075 0
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000
Dividing a Decimal
For this kind of sum we set it out like this, to make it clear for working out.
. 5 75.25
With this method you divide from left to right carrying over remaining (r).
07รท5=1r2
25รท5=5 . 02รท5=0r2 25รท5=5 Which would look like this.
15.05 5 75.25 2
2
When dividing a decimal you keep decimal point in the same place.
75.25รท
5=15.05
Percent
What is a Percent?
A percent (%) is a special case of a fraction where the denominator is always 100.
60%= 0
10
20
30
40
50
60
70
80
90
100
6/100 0
10
20
30
40
50
60
70
80
90
100
Percent as a Decimal
All you have to do is divide the numerator by 100.
60รท100= 0.6
A per cent can be expressed as a decimal by dividing the numerator by 100 by moving the decimal point of the numerator two places to the left.
60%=60 0
10
20
30
40
50
60
70
80
90
100
/100=0.6 0
0
10
1
20
2
30
3
40
4
50
5
60
6
70
7
80
8
90
9
100
10
Percent of Something
To find the per cent of something you must turn the percentage into a decimal and time it by the amount.
First divide 25 by 100.
25÷100= 0.25
Then multiply the answer by 120. Ignore the decimal point and then add it back into the answer.
25×120
2 5 % of 0
10
20
20×120=2400
30
40
05×100=0500
50
60
05×020=0100
70
80
2400+ 500+ 100=3000=30
90
100
120=30
Converting
Percent to Decimal to Fraction
You should learn the following percentage and it’s equivalent decimal and fraction.
10%=0.1 0
0
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
100
100
=1/10 0
1
2
3
4
5
6
7
8
9
10
Percent to Decimal to Fraction
You should learn the following percentage and it’s equivalent decimal and fraction.
25%=0.25 0
0
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
100
100
=1/4 0
1
2
3
4
Percent to Decimal to Fraction
You should learn the following percentage and it’s equivalent decimal and fraction.
50%=0.5 0
0
10
1
20
2
30
3
40
4
50
5
60
6
70
7
80
8
90
9
100
10
=1/2 0
1
2
Percent to Decimal to Fraction
You should learn the following per centage and it’s equivalent decimal and fraction.
75%=0.75 0
0
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
100
100
=3/4 0
1
2
3
4
Percent to Decimal to Fraction
You should learn the following percentage and it’s equivalent decimal and fraction.
20%=0.20 0
0
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
100
100
=1/5 0
1
2
3
4
5
Percent to Decimal to Fraction
You should learn the following percentage and it’s equivalent decimal and fraction.
40%=0.40 0
0
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
100
100
=2/5 0
1
2
3
4
5
Percent to Decimal to Fraction
You should learn the following percentage and it’s equivalent decimal and fraction.
60%=0.60 0
0
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
100
100
=3/5 0
1
2
3
4
5
Percent to Decimal to Fraction
You should learn the following percentage and it’s equivalent decimal and fraction.
80%=0.80 0
0
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
100
100
=4/5 0
1
2
3
4
5
Percent to Decimal to Fraction
Firstly you need to express the denominator as a factor of 100.
025×4=100
There are other conversions that don’t need to be learnt by heart but you must know how to convert them.
9/25 0
What ever you do to the denominator you must do to the numerator.
009×4= 36 5
Which means
10
36/1 00 =0.3 6=36%
15
20
25
=0.36=36% 0
0
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
100
100