For today’s lesson, we all learn more about standard form, ratio, rate, uniform speed and average speed, decimals and fractions, *direct and inverse proportion. In addition, we will also be going through percentages and its applications such as changing the fractions to percentages and calculating % of quantity to find out the discount, taxation, increase and decrease, profits and losses.
Lesson time:
Standard form is a scientific notation which is used when using large numbers to find out great
values such as 1.5x1015 is the standard form of 1,500,000,000,000,000. Why standard form?
Standard form is used as it is a much simpler way of writing very large or small number. It
makes these numbers easier to read and less liable to make errors. A standard form is written in
this form: If the number is large, such as: 9,500,000,000,000= 9.5x1012 (move 12 decimal
places to meet the requirements for standard form which the coefficient must be from 1 to 9
with only one decimal place) If the number is very small, such as: 0.00000000005=5.0x1011
(move 11 decimal places in order to have 1 decimal place like the standard form where the
coefficient must be from 1 to 9
Problem Solving: Solve every problem and convert them to standard form.
1. Earth is actually 6,000,000,000,000,000,000,000,000 kg.
Ans: 6.0x1024kg
2. The distance of Saturn from the Sun is 1,429,000,000 km. 
Ans: 1.429x109 km
3. 3x5x105=?
Ans: 1.5x106. Change the standard forms back to the original notation.
4. 6.535x10-5 =?
Ans: 0.00006535 5. (5.5x105)x(5.5x105) Ans: 30.25x1010
------------------------------------------------------------------------------------------------
Lesson Time (Fractions and decimals)
- Fractions:
The word fraction means part, so part of a quantity or a unit is called a fraction of it.
The bottom number of a fraction is denominator and it describes the number of equal-sized
parts into which the whole number has been divided. The top number of a fraction is numerator
and states how many of the equal parts are being considered. All fractions can be written as
finite/infinite, recurring/non recurring decimal numbers. Link to lesson on irrational numbers.
------------------------------------------------------------------------------------------------
-Decimals
Decimals includes finite/infinite, recurring/non recurring decimal numbers. Finite or terminating
decimals are
decimals that are exact and can be expressed easily and have not been rounded off as the
decimals can end easily. Such as: 0.125. Recurring decimals are decimals that keep repeating the
last decimal place(s) over and over again and do not end easily. The never-ending last decimal
place is represented by dots and line which are placed over the repeating digits. In addition,
some recurring decimals have a regular pattern in it. Such as 5.353535……(with a pattern of 5
and 3). Non recurring and infinite or terminating decimal numbers have decimal places that
cannot be expressed exactly and easily, their decimal places keep going on and do not repeat or
have no regular pattern in it. Such as: 0.1395934……
------------------------------------------------------------------------------------------------
Lesson Time (Ratio, rate, average speed and uniform speed)
-Ratio
Ratio is a comparison between of 2 quantities that are counted in the
same units. But always remember that it is the quantities that have units but not the ratio. A
ratio can be converted to a fraction as both of them have the same units of the 2 quantities. \
a:a2 = a/a2
Note! The quantities must be a countable number and not 0. *Always remember that the
order of the ratio is very important and the first quantity must always be the first number that
is mentioned in a statement or problem. Ratio includes equivalent ratio which is referring to 2
quantities that have the same factors and can be simplified by a same number. It is the same as
the equivalent fraction as you need to find the common factors.
------------------------------------------------------------------------------------------------
-Rate
A rate is a measurement
or degree of 2 quantities that have different units. We often use rate to show the production in a
limited or specific time(usually per minute or hour).
------------------------------------------------------------------------------------------------
-Speed(average and uniform speed)
Speed is the rate of distance travelled in per specific unit of time. Such as: A train is travelling at the 
average speed 85km/h to another country. Speed is often used when
motion occurs to see how fast or slow an object moves over a distance.
*Speed= distance travelled/ time taken to travel
= distance(km/m/cm) /Time(hours/ minutes/ seconds
Speed includes uniform and average speed. Uniform speed is a speed that remains unchanged
while travelling. It is also known as constant speed. Average Speed is used when the speed is not
constant and changes throughout the whole distance while travelling. We must get the total
distance travelled and total time as to obtain the average speed.
*Average Speed= total distance traveled/total time used
Must Revise!
------------------------------------------------------------------------------------------------
-Formula Study:
Speed:
*Speed= distance travelled/ time taken to travel
= distance(km/m/cm) /Time(hours/ minutes/ seconds)
*Average Speed= total distance travelled /total time used
Total Distance= average speed /total time used
*Total Time= total distance / average speed
------------------------------------------------------------------------------------------------
-HCF and LCM
- Highest Common Factor(HCF) is a common factor of the 2 or more different numbers. It is the
highest common factor among all the common factors. It is often used to express numbers in its
simplest form.
- Lowest Common Multiple(LCM) is a common multiple of the 2 or more different numbers. It is
the lowest common multiple among all the common multiples. It is often used to express the
same denominators in fractions.
-Worked Example
1. A:B=3:5, and B:C=3:5, what is A:B:C?
Method: A:B B:C
3 : 5 3 : 5 (lowest common multiple[LCM])
As the quantity of B is the same, we need to find out the common multiple of the number in
order to make the 2 numbers the same.
Multiple of 5: 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100
Multiple of 3: 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63
A:B
= 3:5
B:C
= 3:5
LCM of 3 and 5=15
Therefore
A:B:C
= 9:15:25
Ans: 9:15:25
2. Printer A can print up to 200 copies of worksheets in 5 minutes. It will take 10 hours to print out an order of worksheets. However, printer B is also used to speed up the production in a faster rate than printer A. After printer A had printed for 3 hours, printer B started to print. As all the worksheets are printed in another 1 hour, find the rate of printer B.
Method:
Rate of printing (printer A)= 200/5
= 40 worksheets per minute
10 hours of printing (printer A)= 40x600
= 24000 worksheets
3 hours of printing (printer A)=40x180
= 7200 worksheets
Printer A and B had to print=24000-7200
=16800 in an hour
1 hour of printing (printer A)= 40x60
= 2400
number of worksheets printed in 1 hour (Printer B)= 16800-2400
=14400
Rate of printing (printer B)= 14400/60
= 240 worksheets per minute
Ans: The rate of printer B is 240 worksheets per minute.
3. Motorists A, B and C are travelling from Town X to Town Y which is 48km apart. When motorist A reached Town Y, motorists B and C are still travelling with 12km and 15km left respectively to travel. How far will motorist C be when motorist B reached Town Y?
Method:
Distance travelled by Motorist B= 48km-12km
= 36km
Distance travelled by Motorist C= 48km-15km
= 33km
Motorist A: Motorist B: Motorist C
Divide by 3 48km : 36km : 33km
16km : 12km : 11km
Every 16km Motorist A travels, Motorist B will travel 12km and Motorist C will travel 11km.
8-7=1
1 unit=48km 12km
=4km
Ans: 4km
Exercise Time:
Express each ratio as its simplest form.
1. (12x14) : (3x7x6)
2. 0.266: 0.824
3. 50km: 0.25km
Ans:
1. (12x14) : (3x7x6)
= 168 : 126 Divide by 8
= 8 : 6 Divide by 2
= 4 : 3
Ans: 4:3
2. 0.266: 0.824
= 266/1000 : 824/1000
= 133/1000 : 412/1000
= 133:412 (Cancel away the 1000)
Ans: 133:12
3. 50km : 0.25km Convert into metres
= 50000m : 250m Divide by 10
= 5000 : 25 Divide by 25
= 200 : 1
Ans: 200 : 1
------------------------------------------------------------------------------------------------
Variables increasing (or decreasing) together need not always be in direct
proportion. For example:
(i) physical changes in human beings occur with time but not necessarily in a predetermined
ratio.
(ii) changes in weight and height among individuals are not in any known proportion and
(iii) there is no direct relationship or ratio between the height of a tree and the number
of leaves growing on its branches. Think of some more similar examples.
------------------------------------------------------------------------------------------------
No comments:
Post a Comment