Maths Olympiad

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Simplification

The process of cancelling (reducing) e.g. ,

**e.g.1 remember that exponents are subtracted
Clearing one of the most common errors, an important “cancellation” rule:

only factors may be cancelled

Factors multiply, terms add and subtract

First count terms … > 1 term means that there is only 1 factor, the whole expression.

**e.g.21. 1 term, 3 factors
2. 2 terms, 1 factor
3. 2 terms, 1 factor, BUT
… 1 term, 2 factors
4. 1 term, 2 factors
5. 2 terms, 1 factor i.e. cannot cancel
cannot cancel
cannot cancel

Factorisation is usually a necessary step for simplification of a fraction

**e.g.31. In simplest form, no cancelling
2.

3. 4.

§Exercise 1

1.1 1.2 1.3
1.4 1.5 1.6
1.7 1.8 1.9
1.10 1.11

Multiplication

**e.g.4

Divisionby fractions is the same as multiplying by the inverse fraction.

**e.g.5
**e.g.6
**e.g.7
**e.g.8
**e.g.9

NOTE: Only the fractions FOLLOWING the sign invert.

§Exercise 2

2.1 2.2 2.3
2.4 2.5 2.6

§Exercise 3

3.1 3.2 3.3
3.4 3.5 3.6
3.7 3.8

Addition and subtraction

In this process 2 important principles are used:

1.The numerator (top) of the fraction indicates how many? and the denominator
indicates what?
e.g. : what? … quarters how many? … three
i.e. in the same way that 3 dogs + 7 dogs 10 dogs,
e.g. ; ;
BUT, as with 7 dogs + 3 cats ???, cannot be added in this form,
being unlike terms.

When working with fractions, we get around this by using the second principle:

2.The numerator and denominator can be multiplied by the same factor without
the value of the fraction being changed.
e.g.

This is summarised by writing the equal denominators as a common
denominator:



Finding the common denominator

1.Find the lowest number which is divisible by each of the denominators.
e.g. 6 ; 10 ; 15
15 30 45 60

2.For variables (letters) … each factor is needed once, the highest power
of each.
e.g. …must have
highest powers:

§Exercise 4Determine the common denominator for the following given denominators:

4.1 4.2 4.3 4.4
4.5 4.6 4.7

We will not do examples without variables as these can be done by calculator.

**e.g.101.
2.

§Exercise 5Simplify:

3.1 3.2
3.3 3.4
3.5 3.6
3.7 3.8
3.9. 3.10
3.11 3.12.
3.13 3.14

3.15 3.16
3.17 3.18
3.19 3.20

**e.g.12(Extension work)
1. 2.

§Exercise 6Simplify:

6.1 6.2 6.3
6.4 6.5 6.6

Mixed fractions

Division and multiplication must be done before addition and subtraction. **e.g.131. 2. 3.

§Exercise 7Simplify:

7.1 7.2 7.3
7.4 7.5
7.6 7.7 7.8
7.9 7.10
7.11 7.12 7.13
7.14 7.15 7.16
7.17 7.18 7.19
7.20 7.21
7.22

ALGEBRAIC FRACTIONS: Answers to exercises

Exercise 1

1.1 1.2 1.3 1.4 1.5 1.6
1.7 1.8 1.9 1.10 1.11

Exercise 2

2.1 2.2 2.3 2.4 2.5 2.6

Exercise 3

3.1 3.2 3.3 3.4
3.5 3.6 3.7 3.8
3.9 3.10 3.11 3.12
3.13 3.14 3.15 3.16
3.17 3.18 3.19 3.20

Exercise 4

4.1304.2 4.3 4.4 4.5 4.6 4.7

Exercise 5

5.1 5.2 5.3 5.4 5.5 5.6
5.7 5.8 5.9 5.10
5.11 5.12 5.13 5.14

Exercise 6

6.1 6.2 6.3
6.4 6.5 6.6

Exercise 7

7.1 7.2 7.3 7.4 7.5
7.6 7.7 7.8 7.9 7.10
7.11 7.12 7.13 7.14 7.15
7.16 7.17 7.18 7.19 7.20
7.21 7.22
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