O-Level Additional Math

[ADoc]

Here’s a quick guide to understanding proportionality & solving formulating / solving proportion statements/equations (E Math Paper I)


By definition, a proportion is a mathematical statement or equation stating that two ratios are equal, i.e. a/b = c/d

But this is definition isn’t really useful at first glance to understand and solve proportionality.

Let’s understand these first:

(1) Direct Proportion
when we say A is proportional to B, we mean as A increases, B increases as well, vice versa.

Example: The more money I have, the more I-phone apps I can buy.

(2) Inverse Proportion
when we say A is inversely proportional to B, we mean as A increase, B decreases.
Or, as A decreases, B increases.

Example: The more time I spend playing on my I-phone, the lesser time I have for revision.

Now that we have sorted out definitions (1) & (2), next is about the proportionality constant (usually denoted as k).

Example:
y is directly proportional to x,
this means y ∝ x
to formulate a proportion statement, we rewrite as y = kx
Why is there a need for the proportionality constant k?
We only know that as y increases, x increases, however we do not know the exact magnitude of increment (or decrease). The constant k is the unknown that will allow us to equate y to x. So now we can happily remove the ∝ symbol and replace it by =

Example:
if y is inversely proportional to x³
we rewrite as y ∝ (1/x³). Convince yourself that as x increases, we are dividing by a larger number, hence y decreases. Thus an inverse proportional relationship.

similarly, we can arbitrarily insert the constant k to formulate an equation such that, y = k(1/x³)

Solving for K usually requires the question to provide a pair of values of x & y for example.

Example:
given that y = 2 when x = 4, and y is ∝ x
y = kx –> 2 = k(4) –> k = 1/2

given that y = 2 when x = 4, and y is ∝ 1/x
y = k(1/x) –> 2 = k(1/2) –> 2 = k/2 –> k = 4 (cross-multiplication is a common technique for solving inverses)

Hope these basics are useful. Cheers! There are many more variations and techniques, and styles of questions. I find it difficult to explain thru typing. Besides, the post will get incredibly long and boring! ha! Tks for the understanding.

[ADoc]

Just an add-on. The understanding of proportional is exceptionally useful for Physics (and other subjects of cos).


For example, the all-time favourite of D-S-T.
we know that D = S x T
with a constant S, the longer the D, the more T is required.
or, S = D / T, with a constant D, the longer T one takes to travel, it translate to a lower average S.

Density = Mass / Volume

For a constant M, the bigger the V, the lower the D, for example.

Pressure = Force / Area
The larger the A with a constant F, the lower the P.

Essentially, students must learn to appreciate the physics formula in this manner to be effective and efficient in their learning.

[leeven]

Hi GuanHui


how to find the lowest postive integer of x and X+1 . both divisible by 7

[blitz]

http://img.photobucket.com/albums/v257/tbctbc/DSC00960.jpg\">


The upper part of a solid wooden right circular cone is cut off leaving the frustum shown in the diagram. The height of the original cone, VO is 18cm and the base radius, OA is 9cm.

The radius, BC, of the base of the upper part is 6cm.

Taking pi to be 3.142, calculate
(a) the length of OC
(b) the total surface ara of the frustum

[FrekiWang]

blitz:

http://img.photobucket.com/albums/v257/tbctbc/DSC00960.jpg\">


The upper part of a solid wooden right circular cone is cut off leaving the frustum shown in the diagram. The height of the original cone, VO is 18cm and the base radius, OA is 9cm.

The radius, BC, of the base of the upper part is 6cm.

Taking pi to be 3.142, calculate
(a) the length of OC
(b) the total surface ara of the frustum

(a) Since VCB and VOA are similar, we have
VC/VO=CB/OA
VC/18=6/9
solve, we get VC=12.
OC=VO-VC=6cm

(b) Curved surface area of a cone = pi x r x L
VB = square root of (VC square + BC square) = 13.416
VA = square root of (VO square + OA square) = 20.125
Curved surface area of the frustum
=curved surface area of larger cone - curved surface area of larger cone
= 3.142 x 9 x 20.125 - 3.142 x 6 x 13.416
= 316.18
Upper base of the frustum
= 3.142 x 6 x 6
= 113.11
Lower base of the frustum
= 3.142 x 9 x 9
= 254.50
Total = 316.18+113.11+254.50 = 684cm^2 (correct to 3s.f.)

[FrekiWang]

leeven:

Hi GuanHui


how to find the lowest postive integer of x and X+1 . both divisible by 7

It is not possible to have x and x+1 both divisible by 7, please check.

[blitz]

FrekiWang:

blitz:

http://img.photobucket.com/albums/v257/tbctbc/DSC00960.jpg\">


The upper part of a solid wooden right circular cone is cut off leaving the frustum shown in the diagram. The height of the original cone, VO is 18cm and the base radius, OA is 9cm.

The radius, BC, of the base of the upper part is 6cm.

Taking pi to be 3.142, calculate
(a) the length of OC
(b) the total surface ara of the frustum

(a) Since VCB and VOA are similar, we have
VC/VO=CB/OA
VC/18=6/9
solve, we get VC=12.
OC=VO-VC=6cm

(b) Curved surface area of a cone = pi x r x L
VB = square root of (VC square + BC square) = 13.416
VA = square root of (VO square + OA square) = 20.125
Curved surface area of the frustum
=curved surface area of larger cone - curved surface area of larger cone
= 3.142 x 9 x 20.125 - 3.142 x 6 x 13.416
= 316.18
Upper base of the frustum
= 3.142 x 6 x 6
= 113.11
Lower base of the frustum
= 3.142 x 9 x 9
= 254.50
Total = 316.18+113.11+254.50 = 684cm^2 (correct to 3s.f.)

Thanks so much! But I dont understand how to prove that the triangles are similiar

[suiyuan]

Hi


Please help.

Thank you

Secondary Two Maths.

Factorise

1) -6x^2 + x + 1

2) -5x^2 + 63x - 36

[jieheng]

suiyuan:

Hi


Please help.

Thank you

Secondary Two Maths.

Factorise

1) -6x^2 + x + 1

2) -5x^2 + 63x - 36

1) -6x^2 + x + 1 = (3x+1)(-2x+1)

2) -5x^2 + 63x - 36 = (5x-3)(-x+12)

[kittybank]

Hi Guan Hui,


Could you help solve this?

Solve the pair of simultaneous equation for the values of x and y.
xy=1/3
xy+x-y=1/2

Ans: x=-1/2, y= -2/3 or x=2/3, y=1/2

Thanks a million!