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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 fractions, 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 = 54 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 12: 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 = = 80 km/hr

  32. Note 12: Rates Example: If it costs four people a total of $5456 to stay at a hotel for 11 days, how much would it cost five people to stay at the same hotel for 10 days? Rate per person per night = $ 5456 ÷ 4 ÷ 11 = $ 124 Therefore, 5 people for 10 days would cost $ 124 × 5 × 10 = $ 6200

  33. Note 12: Rates This table shows the exchange rates to change British Pounds to foreign currency Multiply by the exchange rate to get the amount in foreign currency Eg: Ann is going to Spain on holiday and changes 200 to Euros. How many Euros does she get? 200 x 1.58 One pound = € 1.58 = € 316

  34. Note 12: Rates To change back to British Pounds we divide by the exchange rate Eg: Dave returns from Switzerland with 55 Swiss francs. He changes the swiss francs back to British Pounds. How much does he get back? One pound = ₣ 2.32 ₣55 x IWB Ex4.01 pg102-104 ₣ = ₣23.70 (round to 2 dp)

  35. Starter Lee is planning a trip to Thailand. The exchange rate is 22.5 Thai baht for $1 NZD. How many baht would Lee get for $1000 NZD? $ 1NZD = 22.5 baht $ 1000 NZD × = 22 500 baht

  36. Starter Molly took a trip to Mexico. She exchanged $125NZD and got 1125 pesos. Later in the trip she wanted to exchange 4500 pesos to dollars. How much would she get in NZD? Exchange rate = $1 NZD = 9.000 pesos 4500 pesos × = $ 500 NZD

  37. Note 13: Compounding Rates When money is borrowed from a bank the person borrowing the money must pay the money back plus an extra amount called interest The most basic form of interest is Simple Interest (interest paid on the original amount borrowed only)

  38. Note 13: Compounding Rates Example: Find the amount of simple interest paid if Neil invests $35 000 for three years at 5%per annum. P = 35 000 R = 5 T = 3 I = I = $ 5250

  39. Note 13: Compounding Rates Compound interest is calculated on both the initial amount borrowed and the accumulated interest from prior compounding periods. ExampleA person invests $ 10 000 with the bank at 8% compounded annually for three years. This person would have earned $2597.12 in interest over the 3 years A = P (1 + r%)n A = 10 000 (1 + .08)3 IWB Ex3.05 pg83-84 = $12 597.12

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