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Numeric Reasoning 1.1

Numeric Reasoning 1.1. Year 11. 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 2.01 pg 47.

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Numeric Reasoning 1.1

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  1. Numeric Reasoning 1.1 Year 11

  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 2.01 pg 47

  3. Note 4: Fractions (Revision) Rules for multiplying two fractions: • multiply the two numerators • multiply the two denominators • simplify if possible Examples: x = x = =

  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 2.03 pg 51 Ex 2.04 pg 54 = × = 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 2.02 pg 49-50 =

  6. Starter Fractions (Applications) Let x represent the capacity of the tank ×x = 64 L 96 × = 84 L x = 64 × 84 L – 64 L = 20 L should be added x = 96 L

  7. Note 5: 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 %

  8. Note 5: 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 Ex 3.01 pg 64-65

  9. Note 5: Decimals -> Fractions -> % Last season = x 100 % = 36.2% This season = x 100 % IWB Ex 3.02 pg 68-72 = 46.3%

  10. Note 5: Decimals -> Fractions -> % White Chocolate = 200 g x 0.21 = 42 + 42 x 100% = 42 g 350 = 24 % Dark Chocolate = 150 g x 0.28 IWB Ex 3.02 pg 68 = 42 g

  11. Note 6: 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 6: 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

  13. Note 6: Calculating Percentages and Fractions of Quantities Examples: Karl received a $190 cash rebate on a purchase of $3800. What percentage is this? = × 100 % = 5 % IWB Ex 3.02 pg 68-69

  14. Note 7: 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 IWB Ex 3.03 pg 77 #5 Ex 3.02 pg 70-72 = 150

  15. Starter 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 = x = 40 students

  16. Note 8: 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 %

  17. Note 8: 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 3.03 pg 77 #6 x = x 100% = 10%

  18. Note 9: 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)

  19. Note 9: 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

  20. Note 9: Increasing & Decreasing by Percentages × ( 1 ± r) Original (old) Quantity New (inc or dec) Quantity 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? ÷ ( 1 ± r) Old = 80 g/L ÷ (1 – 0.1) r = 0.1 = 80 g/L ÷ 0.9 = 88.9 g/L

  21. Note 9: Increasing & Decreasing by Percentages × ( 1 ± r) Original (old) Quantity New (inc or dec) Quantity 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? ÷ ( 1 ± r) New = $15000 × (1 – 0.15) r = 0.15 = $15000 × 0.85 = $12750

  22. Note 9: 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) IWB Ex 3.03 pg 77 #1-4 Ex 3.04 pg 78 - 80 r = 0.21 = $170000 ÷ 1.21 = $140496

  23. Starter × ( 1 ± r) Original (old) Quantity New (inc or dec) Quantity Examples: The bill for a meal came to $65.40 plus a 15% GST. What was the total bill? ÷ ( 1 ± r) New = $65.40 × (1 + 0.15) r = 0.15 = $65.40 × 1.15 = $75.21

  24. 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

  25. 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

  26. 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 3.09 pg 96-98 GST charged = $69.95 – 60.83 = $9.12

  27. Note 11: Ratios When two quantities measured in the same units are compared they give a ratio. Example: There are 32 lambs to 21 ewes. Write as a ratio. 32 : 21 Ratios can be simplified, just like fractions, by finding a factor that goes into both ratios. Examples:5 : 15 = 12 : 28 = 1 : 3 3 : 7

  28. Note 11: Ratios If the numbers are already fraction, multiply the two denominators and then multiply by the answer. Example: Simply the ratio 2 x 3 = 6 Multiply both fractions by 6 3 : 2

  29. Note 11: Ratios Equivalent ratios can find missing parts of a ratio. Example: The ratio of teachers at a primary school is 1:18. If there are 3 teachers how many students are there? # of students = 3 × 18 1 : 18 3 : x = 56 IWB Ex 4.02 pg 107-109 Ex 4.05 pg 122

  30. Note 11: Ratios To share a given ratio, work out fractions of the quantity. Example:Jack and Jill have $240 to split into the ratio of 5:7. How much does each person get? Total quantity = 5 + 7 = 12 = $100 Jack’s share = x $240 = $140 Jill’s share = x $240 IWB Ex 4.03 pg 114-117

  31. Note 10: Rates Rates compare quantities that are measured in different units. Example: The distance from Invercargill to Dunedin is 200 km. If it takes 2½ hours to cover this distance, what is my average speed in km/hr? Speed = IWB Ex4.01 pg 102-104 = 80 km/hr

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