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### Number

Year 10

Note 4: Fractions (Revision)

To reduce fractions to their simplest form:

find the highest common factor in the numerator and denominator and divide by this factor.

Examples: 12 = 315 = 3

16 4 40 8

IWB

Ex 3.02 pg 71-72

Note 4: Fractions (Revision)

Rules for multiplying two fractions:

- multiply the two numerators
- multiply the two denominators
- simplify if possible

Examples: x =

x =

=

IWB

Ex 3.03 pg 79

Note 4: Fractions (Revision)

To get the reciprocal of a fraction, turn it upside down

Examples: The reciprocal of is

The reciprocal of 5 ( ) is

To divide by a fraction we multiply by the reciprocal of the second fraction.

= ×

=

IWB

Ex 3.05 pg 83

Ex 3.06 pg 87

Examples: ÷

Note 4: Fractions (Revision)

- To add/subtract fractions with different denominators
- change to equivalent fractions with the same denominator
- add/subtract the equivalent fractions
- simplify if possible

Examples: +

=

+

IWB

Ex 3.07 pg 88

Ex 3.08 pg 91

=

Note 5: Mixed Numbers

A mixed number is a combination of a whole number and a fraction

Note 5: Mixed Numbers

To change a mixed number to an improper fraction, multiply the denominator by the whole number and add the numerator (the denominator stays the same)

+

x

7 x 3 + 1

=

3

IWB

Ex 3.10 pg 95

Ex 3.12 pg 99

Note 6: Decimals -> Fractions -> %

To convert a decimal and fraction to a percentage multiply by 100%.

Examples:

0.6 = 0.6 x 100% 0.348 = 0.348 x 100%

= 60 %

= 34.8 %

= x 100%

= x 100%

= 32.5 %

= 20 %

Note 6: Decimals -> Fractions -> %

To convert a percentage to a decimal or fraction, divide by 100 ( and simplify if a fraction is required).

Examples:

75% 64 %

=

=

=

= 0.75

IWB – odd only

Ex 4.01 pg 110

Ex 4.02 pg 111

Ex 4.03 pg 112

Note 7: Calculating Percentages and Fractions of Quantities

To calculate a percentage/fraction of a given quantity, multiply the quantity by the percentage (as a fraction or a decimal).

Examples:

24% of 70

30% of the Year 11 pupils at JMC (90 pupils) are left handed. How many Year 11 pupils are left handed?

= x 70

= 16.8

30% of 90 = 0.3 x 90

= 27

Note 7: Calculating Percentages and Fractions of Quantities

Examples:

Jim plans to reduce his 86 kg weight by 15%. How much weight is he planning to lose?

= 0.15 × 86kg

= 12.9 kg

IWB - Beta

Ex 4.05 pg 117

Ex 4.06 pg 118-120

Note 8: Increasing & Decreasing by Percentages

× ( 1 ± r)

Original (old) Quantity

New (inc or dec) Quantity

÷( 1 ± r)

* r = percentage (use as a decimal)

* for increase use (1 + r) decreases use (1 – r)

Note 8: Increasing & Decreasing by Percentages

× ( 1 ± r)

Original (old) Quantity

New (inc or dec) Quantity

Increase $70 by 15%

= $70 × (1 + 0.15)

= $70 × 1.15

= $80.50

Increase 88 kg by 23%

= 88 kg × (1 + 0.23)

= 88 kg × 1.23

= 108.24 kg

Note 8: Increasing & Decreasing by Percentages

× ( 1 ± r)

Original (old) Quantity

New (inc or dec) Quantity

Decrease $50 by 15%

= $50 × (1 – 0.15)

= $50 × 0.85

= $42.50

Decrease 35 kg by 10%

= 35 kg × (1 – 0.1)

= 35 kg × 0.9

= 31.5 kg

Note 8: Increasing & Decreasing by Percentages

× ( 1 ± r)

Original (old) Quantity

New (inc or dec) Quantity

÷ ( 1 ± r)

Examples:

The price of a computer currently selling for $2500 increases by 5%. Calculate the new selling price.

r = 0.05

New price = $2500 × (1 +0.05)

= $2625

Note 8: Increasing & Decreasing by Percentages

× ( 1 ± r)

Original (old) Quantity

New (inc or dec) Quantity

÷ ( 1 ± r)

Examples:

A car depreciates 15% over a year. It was worth $15000 at the start of the year. What was it worth at the end of the year?

New = $15000 × (1 – 0.15)

r = 0.15

= $15000 × 0.85

IWB

Ex 4.07 pg 123-124

= $12750

Note 8: Increasing & Decreasing by Percentages

× ( 1 ± r)

Original (old) Quantity

New (inc or dec) Quantity

÷ ( 1 ± r)

Examples:

Coca Cola reduced the caffine content of their cola drink by 10%. They now contain 80g/L of caffine. How much did they contain before the reduction?

Old = 80 g/L ÷ (1 – 0.1)

r = 0.1

= 80 g/L ÷ 0.9

= 88.9 g/L

Note 8: Increasing & Decreasing by Percentages

× ( 1 ± r)

Original (old) Quantity

New (inc or dec) Quantity

Examples:

House prices have risen 21% over the last 3 years. The market value of a house today is $170000. What was the value of the house 3 years ago?

÷ ( 1 ± r)

Old = $170000 ÷ (1 + 0.21)

r = 0.21

= $170000 ÷ 1.21

IWB

Ex 4.07 pg 123-124

= $140496

Note 9: Percentage Changes

To calculate a percentage increase or decrease:

Percentage = difference in values x 100 %

original amount

Examples:

The number of junior boys boarding at JMC hostel increases from 70 to 84 boys. What percentage increase is this?

x = x 100%

x = 20 %

Note 9: Percentage Changes

To calculate a percentage increase or decrease:

Percentage = difference in values x 100 %

original amount

Examples:

The population of a town decreased from 600 to 540 people. What percentage decrease is this?

x = x 100%

IWB

Ex 4.08 pg 129-131

x = x 100%

= 10%

Starter

× ( 1 ± r)

Original (old) Quantity

New (inc or dec) Quantity

÷ ( 1 ± r)

Examples:

The bill for a meal came to $65.40 plus a 15% GST. What was the total bill?

New = $65.40 × (1 + 0.15)

r = 0.15

= $65.40 × 1.15

= $75.21

Note 10: Goods & Services Tax (GST)

GST is a tax on spending (15 %)

× ( 1.15)

Price including GST (inclusive)

Price before GST

(exclusive)

÷ ( 1.15)

r = 0.15

Note 10: Goods & Services Tax (GST)

× ( 1.15)

Price including GST (inclusive)

Price before GST

(exclusive)

- Examples:
- A filing cabinet is advertised for $199 plus GST.
- a.) Calculate the GST inclusive price.
- b.) How much is the GST component?

÷ ( 1.15)

r = 0.15

New = $199 × (1.15)

= $228.85

GST = Price inclusive – Price exclusive

= $228.85 – $199

= $29.85

Note 10: Goods & Services Tax (GST)

× ( 1.15)

Price including GST (inclusive)

Price before GST

(exclusive)

- Examples:
- A company meal costs $69.95 including GST. Calculate the price before GST was added, and the amount of GST charged.

÷ ( 1.15)

Price excluding GST = $69.95 ÷ 1.15

= $60.83

IWB

Ex 4.09 pg 135-136

GST charged = $69.95 – 60.83

= $9.12

Note 12: Calculating ‘Original’ Quantities

To calculate the original quantity we reverse the process of working out percentages of quantities. We express the percentage as a decimal and write an algebraic equation to solve.

Examples:

30 is 20% of some amount. What is this amount?

20% of x = 30

0.2 x x= 30

= 150

Note 12: Calculating ‘Original’ Quantities

Examples:

15% of the students in a class are left handed. If there are 6 students who are left handed, how many students are in the class?

15% of x = 6

0.15 x x= 6

x =

IWB

Ex 4.11 pg 143-144

x = 40 students

Note 11: Ratios

When two quantities measured in the same units are compared they give a ratio.

IWB

Ex 5.01 pg 147-150

Ex 5.02 pg 153

Note 12: Rates

Rates compare quantities that are measured in different units.

IWB

Ex 5.04 pg 160-161

= $9.12

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