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Discovering Mathematics Week 4 Unit 3: Irrational Numbers 3. Irrational numbers • An irrational number is a number that can not be written as: Examples: 2 1.414 213 562 373 095..... = 3.141 592 653 589 793 238 46 ... 0.010 010 001 000 010 000 01 ... • The irrational numbers together with the rational numbers form the real numbers represented by a number line. Each point on the number line represents a real number, so the number line is often called the real line. • In general, a square root of a number is a number that when multiplied by itself gives the original number. Every positive number has two square roots – a positive one and a negative one. • Examples: • The square roots of 36 are: 6 and – 6 (6x6=36 and -6x(-6)=36) • The square roots of 4 are: 2 and – 2 (2x2=4 and -2x(-2)=4) • The positive square root of a positive number is denoted by the • symbol. For example: 36 6 ; 25 5 ; 81 9 ; 16 4 • A cube root of a number is a number such that if you multiply three ‘copies’ of it together, you get the original number • Examples: • The cube root of 64 is: 4 since 4x4x4=64 • The cube root of -64 is: -4 since (-4)x(-4)x(-4)=64 3 82 ; 4 3 8 2 ; 3 27 3 ; 3 27 3 • Exercise: • Find the following roots of numbers, without using your calculator (a) Two square roots of 9 (b) Two fourth roots of 16 • A square root of a product is the same as a product of square roots. ab a b • A square root of a quotient is the same as a quotient of square roots. a a b b Exercise: Solution: a) 1764 36 49 36 49 6 7 42 b) 4 4 2 ; 9 9 3 36 36 6 ; 49 49 7 1 1 1 4 4 2 The square root of any natural number that is not a perfect square (e.g 4, 9, 16, 25, 81, …) is irrational. So, for example, the following roots are irrational: 2 ; 3 ; 5; 6 Exercise 1: Solution: 18 9 2 9 2 3 2 10 5 2 ; 60 4 15 4 15 2 15 80 16 5 16 5 4 5 Exercise 2: Solution: 8 4 2 4 2 2 2 75 25 3 25 3 5 3 15 5 3 56 4 14 4 14 2 14 48 16 3 16 3 4 3 Exercise 3: Solution: ( 3) 2 3 3 3 3 9 3 2 5 4 5 8 5 5 8 25 8 5 40 6 3 6 3 18 9 2 3 2 5 2 3 10 15 2 10 15 20 15 4 5 15 2 5 30 5 4 Ratios Experiment: If you have made vinaigrette salad dressing, then you may remember that the recipe is 3 parts oil to 1 part vinegar. So, for example, you could mix 30 ml oil and 10 ml vinegar, or 120 ml oil and 40 ml vinegar, or perhaps, if you need a lot of salad dressing, 1.5 l oil and 0.5 l vinegar. We say that the ratio of oil to vinegar is 3: 1 This ratio is equivalent to: 30 : 10 120:40 1.5: 0.5 To find a ratio equivalent to a given ratio Multiply or divide each number in the ratio by the same non-zero number. Ratios Exercise 1: Express the following ratios in their simplest forms. (a) 9:12:6 (b) 0.5:1.25 Solution: (a) 3:4:2 (simplify by 3) (b) 50 : 125 (multiply by 100 to eliminate decimals, then simplify by 5) 2:5 Exercise 2: Express the following ratios in their simplest forms. (a) 18:3 (b) 12:60:18 (c) 2:0.5:1.5 (d) 6:12:7 Solution: (a) 6:1 (b) 2:10:3 (c) 20:5:15 (multiply by 10 to eliminate decimals, then simplify by 5) 4: 1:3 MU 123 Discovering Mathematics Trade and Cash Key Terms-Formulas • Suggested retail price, catalog price, list price: three common terms for the price which the manufacturer suggests an item be sold to the consumer. • Discount rate: a percent of the list price. • Trade discount: the amount of discount that the wholesaler or retailer receives off the list price or the difference between the list price and the net price • Net price: the price the manufacturer or retailer pays or the list price minus the trade discount. Trade discount = rate x list price Net Price = List Price – Trade discount Look at this example Exercise: Find the trade discount for a cd player that retails at $120 and has a trade discount rate of 35%. • Trade discount = 0.35 x $120 = $42 • What does the $42 mean? It means that the wholesaler or retailer will not pay $42 of the $120 list price. Look at this example Exercise: Find the net price of a desk that lists for $320 and has a trade discount of 30%. Trade discount = Rate x List price = 0.30 x $320 = $96 Net price = List price – Trade discount = $320 - $96 = $224 Try these examples • Find the trade discount for a rug that lists for $290 and has a trade discount rate of 30%. • $87 • Find the net price of a patio table that lists for $460 and has a trade discount of 20%. • $368 Find the net price using Complement of percent Complement of percent: 100% - single trade discount rate. The difference between 100% and the given percent Example: Find the net price of a coffee maker that lists for $20 and has a trade discount rate of 20%. Solution: 80% is the complement of 20% Net price = $20 x 0.80 = $16 Try these examples • Find the net price of a set of golf clubs that lists for $1,500 and has a trade discount rate of 15%. • $1275 • Find the net price of a bicycle that lists for $102 and has a trade discount rate of 30%. • $71.40 Trade discount series Trade discount series or chain discount: additional discounts that are deducted one after another from the list price. Exercise: An item lists for $400 and has a discount of 20%. $400 x 0.2 = $80 $400 - $80 = $320 An additional discount of 10% is taken on the previous price. $320 x 0.1 = $32 $320 - $32 = $288 An additional discount of 5% is taken on the previous price. $288 x 0.05 = $14.40 $288 - $14.40 = $273.60. $273.60 is the final price Can you add the discounts together and apply it as one? If the item has three discounts of 20%, 10% and 5%, can you add them together and apply a 35% discount? No The net decimal equivalent To find the net decimal equivalent: multiply the decimal form of the complement of each trade discount rate in a series. Net amount you pay = net decimal equivalent x list price Exercise: Find the net price of an order with a list price of $800 and a trade discount series of 20/10/5. Solution: The net decimal equivalent is 0.8 x 0.9 x 0.95 = 0.684 Apply the net decimal equivalent to the list price. NP = 0.684 x $800 = $547.20 Try these examples • A tool set lists for $325 and has a trade discount series of 20/10/10. Find the net price. • $210.60 • A dress shirt lists for $125 and has a trade discount series of 15/10/7.5. Find the net price. • $88.45 Thank you