Fractions, comparing numbers, indices, standard form

1. FRACTIONS, COMPARING NUMBERS,
INDICES, STANDARD FORM

2. FRACTIONS – ADDITION AND SUBTRACTION
Same denominators: if the fractions have the same denominator then add the two
numerators and leave the denominator unchanged. Example:
𝟐
𝟕
+
𝟑
𝟕
=
𝟓
𝟕
Different denominators: if the fractions have different denominators then rewrite
them so that they have the same denominator (the LCM of the denominators). Example:
1
6
+
1
10
=? LCM of 6 and 10 is 30 so write:
1
6
as
5
30
and
1
10
as
3
30
.
Hence
1
6
+
1
10
=
5
30
+
3
30
=
8
30
=
4
15
Mixed numbers: Example: 5
7
12
+ 7
2
3
=? Two methods:
Method 1: 5
7
12
+ 7
2
3
= 12 +
7
12
+
2
3
= 12 +
7
12
+
8
12
= 12 +
15
12
= 12 + 1
3
12
= 13
3
12
= 13
1
4
Method 2: write both fractions as improper fractions. So: 5
7
12
= 5 +
7
12
=
60
12
+
7
12
=
67
12
and:
7
2
3
= 7 +
2
3
=
21
3
+
2
3
=
23
3
. So: 5
7
12
+ 7
2
3
=
67
12
+
23
3
=
67
12
+
92
12
=
159
12
= 13
3
12
= 13
1
4

3. FRACTIONS – MULTIPLICATION AND DIVISION
Multiplication: multiply the numerators together and the denominators together.
2
7
×
3
5
=
2×3
7×5
=
6
35
NOTE: remember to cancel out common factors!!
For mixed fractions, write both fractions as improper fractions.
Division: “flip over” the second fraction and change the división sign into a
multiplication sign.
3
1
2
÷ 9
1
3
=
7
2
÷
28
3
=
7
2
×
3
28
=
21
56
=
3
8

4. NOTATION FOR COMPARING THE SIZES OF NUMBERS
> Greater than. X>5 means that X is greater than 5
< Less than. X< 3 means that X is less than 3
≥ Greater than or equal to. X≥6 means that X is greater than or equal
to 6.
≤ Less than or equal to. X≤-2 means that X is less than or equal to -2.

5. EXAMPLE:
Choose the one of the symbols: <, > or =, to complete each of the following
statements:
When X=6 and Y=-7
a) X … Y b)X² … Y² c)Y-X … X-Y
So:
a) 6 > -7, so X > Y
b) X²=6²=36, Y²=(-7)²=49 and 36<49, so X²<Y²
c) Y-X = -7-6 = -13, X-Y = 6-(-7) = 6+7 = 13 and -13<13, so Y-X<X-Y

6. INDICES
To multiply powers if the same base ADD the indices.
Example: 93
× 94
= 97
To divide powers of the same base SUBTRACT the indices:
Example:
29
24 = 29−4
= 25
To find a power of a power, MULTIPLY the indices:
Example: (53)4= 53×4 = 512
Zero and negative indices:
𝑎0 = 1 FOR ALL VALUES OF a
𝑎−1
=
1
𝑎
𝑎−𝑛
=
1
𝑎 𝑛

7. EXAMPLES:
1. Calculate the following:
a) 5−3
b) (
3
2
)−2
c)110
a) 5−3 =
1
53 =
1
125
b) (
3
2
)−2
=
1
(
3
2
)2
=
1
9
4
=
4
9
c) 110
= 1
2. Simplify:
(3𝑥2 𝑦)2 × (2𝑥3 𝑦2)3
(2𝑥5 𝑦7)2
(3𝑥2 𝑦)2 × (2𝑥3 𝑦2)3
(2𝑥5 𝑦7)2 =
9𝑥4 𝑦2 × 8𝑥9 𝑦6
4𝑥10 𝑦14 =
72𝑥13 𝑦8
4𝑥10 𝑦14 = 18𝑥3 𝑦−6

8. NUMBERS IN STANDARD FORM
Standard form is useful when writing very large and very small numbers.
To write a number in standard form express it as a number between 1 and 10
multiplied by the appropiate power of 10.
𝑎 × 10 𝑛
Example:
Write 456 000 000 000 in standard form:
456 000 000 000 = 4.56 × 100 000 000 000 = 4.56 × 1011
Write 0.000372 in standard form:
0.000372 = 3.72 ×
1
10 000
= 3.72 × 10−4
Number between
1 and 10
Whole number, positive for
large numbers, negative for
small numbers

9. EXAMPLES:
1. There are 565 sheets of paper in a book.
a) How many sheets of paper are there in 2000 of these books? Give your answer in standard form.
b) A pile of 565 sheets of paper is 25mm high. Calculate the thickness of 1 sheet of paper. Give your
answer in standard form.
a) If there are 565 sheets in 1 book, in 2000 books we have:
1 book 565 sheets
2000 books X sheets = 2000booksx565sheets÷1book = 1 130 000 = 1.13 × 106
sheets
b) If 565 sheets of paper are 25mm high, for 1 sheet of paper we have:
565 sheets 25mm
1 sheet X mm = 1sheetx25mm÷565sheets = 0.044247 = 4.42 × 10−2
mm

10. 2. The density of water is 1x103 kg/m3. Find the following:
a) The mass of water (in kg) in a cuboid measuring 2m by 3m by 5m.
b) The volume (in m3) of water whose mass is 5x108 tonnes (one tonne is 1000kg)
NOTE: 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 =
𝑚𝑎𝑠𝑠
𝑣𝑜𝑙𝑢𝑚𝑒
a) First we calculate the volume of the cuboid: 2mx3mx5m=30m3
Then since dens= mass/vol, we have the density and the volume, so we are going to find the mass:
Mass= density x volume mass= 1x103 kg/m3 x 30m3
= 30 000 = 3x104kg
b) Since 1 tonne is 1000kg= 1x103kg, we write the mass of water in kg:
5x108 x 1x103 =5x 1011kg
Now, we find the volume: volume= mass/density volume=5x 1011kg÷1x103 kg/m3
=500 000 000 = 5x108m3

11. ADDITIONAL: SIGNIFICANT FIGURES
https://www.youtube.com/watch?time_continue=28&v=eCJ76hz7jPM
Useful explanation if you dont have very clear what significant figures are.