Discovering Mathematics Week 4 Unit 3: Irrational Numbers

3. Irrational numbers An irrational number is a number that can not be written as: Examples: 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: 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

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. A square root of a quotient is the same as a quotient of square roots.

Exercise: Solution:

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: Exercise 1: Solution:

Exercise 2: Solution:

Exercise 3: Solution:

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 : :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 = $ $ 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 = Apply the net decimal equivalent to the list price. NP = 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. $ 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