i never heard of length, area and volume scale before tho

I want you to think about two cubes.

One has a length of 1 unit, the other has a length 2 unit.

What is the ratio of their heights?
What is the ratio of their base areas?
What is the ratio of their volumes?
Are all of them 1 : 2? And why?

3s.f 47.9cm²

Your teachers should have been taught the following by now :

If you have two 2-dimensional shapes/figures that are similar, and the simplest ratio of their corresponding sides is a : b,

Then the simplest ratio of of their areas is a² : b²


If you have two 3D figures that are similar, and the simplest ratio of their corresponding sides is a : b,

Then the simplest ratio of of their areas is a² : b² , and the simplest ratio of of their volumes is a³ : b³


For every extra dimension of measurement, the multiple applies.

If I recall the last time, exact figures should not be rounded off. 1 mark would be deducted for rounding off an exact figure.

"All non-exact figures are to be rounded off to 3 significant figures or to 1 decimal place in the case of angles in degrees" or something similar.

1:2
1:4
1:8
not all 1:2, not sure about the reason tho

i dont really understand part c

"1:2
1:4
1:8
not all 1:2, not sure about the reason tho"

That's the key point. In doing so, you have done this, as J has described.

------------------------------------------------------

"I see that ratio of the sides of a cube is 1 : 2. So, its height ratio is 1 : 2."

The above is known as the "length scale" or the "length ratio", because you are comparing heights, or a one-dimensional quantity.

------------------------------------------------------

"I see that the length is in the ratio 1 : 2 and the breadth is also in the ratio 1 : 2. Therefore, the area ratio is (1 x 1) : (2 x 2), or 1² : 2², or 1 : 4."

The above is also known as the "area scale" or the "area ratio", because you are comparing a two-dimensional quantity.

As we can see, the area ratio is the square of the length ratio, simply because the change is two dimensional.

-------------------------------------------------------

"I see that the length is in the ratio 1 : 2, the breadth is also in the ratio 1 : 2 and the height is also in the ratio 1 : 2. Therefore, the volume ratio is (1 x 1 x 1) : (2 x 2 x 2), or 1³ : 2³, or 1 : 8."

The above is also known as the "volume scale" or the "volume ratio", because you are comparing a three-dimensional quantity.

As we can see, the volume ratio is the cube of the length ratio, simply because the change is three dimensional.

------------------------------------------------------

So, as we can see,

Area ratio = (length ratio)²
Volume ratio = (length ratio)³

We use these ratios to make area and volume comparisons for areas and volumes, just as you would have done in the topic Maps and Scales which employ the exact same idea.

We cannot conveniently use the length ratio to compare areas, as you have seen in the cube example. You mentioned that the area ratio is 1 : 4 and not 1 : 2, because length ratios cannot be used directly to compare areas!

This concept extends to two geometrically similar objects whose shapes need not be perfect like a cube. It could be a large bottle versus a geometrically similar smaller bottle. It could be a bus versus a bus model. And so on.

So in part c, using the same idea, we cannot use the length ratio of 2 : 3 to compare areas directly. This is because area is two dimensional. All the length aspects of the triangle are different.

Any length pairs of the triangle will always be in the ratio 2 : 3.

For example,

AB : CD = 2 : 3
AX : CX = 2 : 3
BX : DX = 2 : 3

But the list does not just end here. If I draw a line from X which is perpendicular to AB, and another line from X which is perpendicular to CD, then the perpendicular heights of the triangle (with bases AB and CD) are also in the ratio 2 : 3.

So, AB : CD = 2 : 3 AND the perpendicular heights are also in the ratio 2 : 3.

Area of ABX / Area of CDX
= (0.5 x AB x perp height) : (0.5 x CD x perp height)
= (AB x perp) : (CD x perp)
= (2 x 2) : (3 x 3)
= 2² : 3²

The two dimensions involved are the base and the height of each respective triangle.

This is why the area ratio of ABX : CDX (2² : 3²) is the square of the length ratio of the triangle (2 : 3).

Now, with the area ratio ready, we can use the area ratio to compare the areas of the triangles directly.

So, you must use the appropriate ratios to compare quantities.

Length scale applies when you are comparing one-dimensional quantities like length, breadth, height, radius, circumference, slant height etc.

Area scale applies when you are comparing two-dimensional quantities like base area, surface area etc.

Volume scale applies when you are comparing three-dimensional quantities and their related quantities like volume, capacity, cost, mass etc.

In the topic Maps and Scales, you would have probably used length scales for comparing distances of roads etc and area scales for comparing areas of lakes etc. That is because a map is kind of a "geometrically similar" representation to the actual ground conditions.

does that mean in a diagram, the length scale will be the same for all sides?
like AX:CX
BX :DX
AB:CD
all of them are 2:3 so we only need to calculate the length scale of one of the sides and we can just simply use it for everything that requires length scale

"does that mean in a diagram, the length scale will be the same for all sides?
like AX:CX
BX :DX
AB:CD
all of them are 2:3 so we only need to calculate the length scale of one of the sides and we can just simply use it for everything that requires length scale"

Yes, that is correct. The ratios of the CORRESPONDING sides are always the same in geometrically similar figures.

"Corresponding" is important. Of course AX : CX = BX : DX = AB : CD = perpendicular heights = 2 : 3, but we CANNOT mix around and say that AX : CD = 2 : 3.

All we need to know are that both figures are similar and we have the ratio of one corresponding sides, for us to be able to compute the area scale and the volume scale.

tyvm for explaining to me :)

'Unless stated otherwise within a question, three-figure accuracy will be required for answers. This means that four-figure accuracy should be shown throughout the working, including cases where answers are used in subsequent parts of the question. Premature approximation will be penalised, where appropriate. Angles in degrees should be given to one decimal place.'

If your exact figure is something like 1255 or 153 then no need to 3sf.

Basically , every dimension of measurement needs to be scaled and accounted for, since the ratio holds for each of them.

Eg. 2 similar triangles. One is 1.5 times the other for base. Height also 1.5 times.

Area has 2 dimensions of measurement so scale twice.

Volume has 3 dimensions of measurement so scale three times.

Hmm... I have always thought that exact figure means rational numbers which truncate the decimal early like in this case. I would think that this is considered “exact” when referenced to numbers like sqrt 2 which is “not exact”.

I couldn’t remember the “four-figure accuracy” part at all. I don’t recall my O Level year of examination having that clause at all.

I have my old A Level exam question paper with me. It says...

Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.

(Graphic calculator parts)

You are reminded of the need for clear presentation in your answers.

At the end of the examination, fasten...

This is taken from the syllabus 4048 under the 'notes' section. They don't state in explicitly in the question paper. But this is what teachers have been telling students all along.

If the answer can be left in terms of √, e, ln then no need to evaluate. These are considered exact. π they will usually say in terms of π if needed.

Eg. e³ , ln5 + 2 , 3√3 / 7 - 4


(Even though these are irrational or transcendental numbers, leaving in this form is more exact than truncated decimals. Same logic for recurring decimals like 0.33333..... which is more exact in the form ⅓

Usually I tell them to put two/three decimal places / significant figures beyond the required accuracy, and if possible more, because a proportion of them likes to round off to 3sf, then continue to make calculations using the rounded off value.

As for exact results like 335.561784, where the decimals do not go on, usually rounding up is done for clarity despite being exact. It's things like 4358, 1.226 that can be left alone but teachers usually prefer to err on the safe side for exact figures with 5sf or more i.e round them.

LockB you can leave it as 47.925 and see your teacher's feedback

Ironically, some things like 1.05^12 (from exponential or compound interest topic), which is actually an exact value, is treated as not exact in the calculator due to spacing limitations. In these circumstances I tell my students to treat the values as not exact and to truncate the decimals.

But chemistry and physics papers require answers to be rounded off to 3 sf all the time, even for answers like 47.925. Different from maths papers for some reason, possibly for scientific consistency when it comes to measurements and precisions.

That's what they're doing but not supposed to do. Premature approximation.

For written working it's good to use4-5s.f (or 1-2 above required accuracy, 3 is still okay but anything more than that not recommended).

As for calculation, just save the full value in the calculator and use for subsequent working.

Even for exact ones like 1.1^8 = 2.14358881, round up is expected because 8s.f is just too many for clarity.

Chemistry is 3s.f final answer but mid workings 4 s.f has been the norm in my own and my students experience

i rremember getting an answer wrong for a question due to calculation error and the teacher told us to leave the working answer in 5sf before calculating the answer but i dont know the reason. was confused as it was the first time hearing this kind of stuff
sometimes she tell us to leave as 4sf, sometimes 5sf. now i dont know when to leave as 4sf and when to leave as 5sf

Stick to 5 sf is safer in that situation

As mentioned,

For written working it's good use 1-2 above required accuracy.

As for calculation, just save the full value in the calculator and use for subsequent working.

The reason is premature approximation/truncation to 3s.f for working results in a less accurate answer.

so let's say the question asks to give your answer to 4 significant figures, writr 5-6 in your working.


If question asks for 5 significant figures, write 6-7 in your workings.

Otherwise, 4-5 is fine if question never state.

Teachers cannot mark you down for using 1-2 more sf for workings (whichever you choose)

But one thing is important. Consistency. If you choose 4s.f in your workings, use it for
the working.

If you choose 5 s.f, use it all the way. Don't mix around

Teachers are humans too, sometimes they ask for 4 sf, sometimes 5 sf.

I myself, sometimes do compare coefficients, sometimes do substitution of x for identities, depending on my mood.