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Number

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 = 3 15 = 3 16 4 40 8. IWB Ex 3.02 pg 71-72.

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Number

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  1. Number Year 10

  2. 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

  3. 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

  4. 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: ÷

  5. 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 =

  6. Note 5: Mixed Numbers A mixed number is a combination of a whole number and a fraction

  7. 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

  8. 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 %

  9. 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

  10. Starter Last season = x 100 % = 36.2% This season = x 100 % = 46.3%

  11. 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

  12. 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

  13. 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)

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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 %

  21. 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%

  22. 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

  23. 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

  24. 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

  25. 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

  26. Note 11: Earning Interest from a bank Examples:

  27. Note 11: Earning Interest from a bank Examples: x = x 100% IWB Ex 4.10 pg 129-131 x = x 100% = 10%

  28. 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

  29. 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

  30. 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

  31. 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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