Transcript MU123week4

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
82 ;
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.
ab  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