Please help with this clarification

J

JulianMathHelp

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Assume that two figures on a flat surface, A and B, are similar. If their linear scale factor is 4/5, then does that mean Figure B's sides are 4/5 of Figure A's or Figure A's sides are 4/5 of figure B? I'm a little confused on the topic, and thanks in advance.
 
Yes, that is just what it means. Where are you confused?
If their linear scale factor is 4/5, then that means Figure B's sides are 4/5 of Figure A's or Figure A's sides are 4/5 of figure B.
 
Let's say A is the smaller one. That is that A's sides are (4/5) of B sides.

Now someone switches the labels. So now B's sides are 4/5 of A sides. This is a bit abstract for someone who has not studied much math. It is actually a paradox.

The answer to your question is as follows. If A is the larger figure then the sides of B are 4/5 the sides of A.
If B is the larger figure then the sides of A are 4/5 the sides of B.
 
It's ambiguous, and needs to be stated more clearly. Both interpretations make sense.

On one hand, scale factor 4/5 might mean the ratio A:B = 4:5 (expressing the scale factor explicitly as a ratio is one way to make it clearer). In this case, the first is 4/5 of the second.

But sometimes when the factor expressed as a number or fraction, it is clearly meant to be what you multiply the first by to get the second, which would mean that B is 4/5 of A, making the ratio 5:4. Here we would be thinking in terms of a process, not just a comparison.

So it really depends on context. I've seen a lot of variation when I scan various sites for their usage, and some are even inconsistent from one example to the next.

It's somewhat like the word "difference". Does "the difference between A and B" mean A-B or B-A? Usually it means |A-B|, that is, the larger minus the smaller. But there is no such convention here.

If you are asking about a specific problem or two, please quote them exactly so we can see whether something in the wording clarifies what is meant. Generally when we use a scale factor, we know which order we have in mind; if we communicate it to others, we have to make that clear.
 
So I'll give a couple of examples and my thinking:
1) "Find the area and perimeter of the trapezoid that is similar to this one, but has been reduced by a linear scale factor of 1/3" The original trapezoid has bases 8 and 13, and legs 5 and 2root3.
What I think "The new trapezoid has been reduced (gotten smaller) by a scale factor of 1:3" means. I think that this means that the new trapezoid's side lengths is 1/3 of the old trapezoid as in the question, it is indicating that the new trapezoid is smaller by referring to the term "reduce". So, then I solved it.
2) If the perimeter of Figure A is p and the linear scale factor is r, what is the perimeter of Figure B? In the question, Figure A is smaller than Figure B. I'm not 100% sure, but...
What I think: The scale factor in this question is equivalent to B/A as we are increasing from A to B. So, the answer would be pr. If the question was asking me "If the perimeter of Figure B is p, and the scale factor is r, then what is the perimeter of Figure A?" I would think that the scale factor would be A/B as we are reducing. So, the answer would also be pr, but in this case, r is a different value.
3) Assume that two figures on a flat surface, A and B, are similar. If the linear scale factor is 2:5, then what is the ratio of the areas of A and B?
What I think: I know this is irrelevant to the question, but for future scenarios, is B 2/5 of A or is A 2/5 of B? I know the answer is 4:25, but is it A:B is 2:5 or is B:A 2:5. I believe it is B as B follows A in the alphabet, and it kind of acts like figure A going to figure B 9 (I know, it sounds crazy)
 
JulianMathHelp said:
… this is irrelevant to the question, but for future scenarios, is B 2/5 of A or is A 2/5 of B? …
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Hi Julian. In a future scenario where you need to know which name corresponds to the which figure, the exercise would either tell you or give you enough information to figure it out. Otherwise, the names don't matter, and you're free to pick whatever names you desire. Your example exercise (3) could have omitted the names A and B:

3) Assume that two figures on a flat surface are similar. If the linear scale factor is 2:5, then what is the ratio of their areas?

?
 
Otis said:
Hi Julian. In a future scenario where you need to know which name corresponds to the which figure, the exercise would either tell you or give you enough information to figure it out. Otherwise, the names don't matter, and you're free to pick whatever names you desire. Your example exercise (3) could have omitted the names A and B:

3) Assume that two figures on a flat surface are similar. If the linear scale factor is 2:5, then what is the ratio of their areas?

?
Click to expand...
So it doesn't matter in this problem whether or not B is 2/5 of A or if A is 2/5 of B, and that's why it is not specified?
 
JulianMathHelp said:
So it doesn't matter in this problem whether or not B is 2/5 of A or if A is 2/5 of B, and that's why it is not specified?
Click to expand...
Yes and yes. That's what I meant, when I said the names A and B could be omitted. The exercise doesn't need to specify because it doesn't ask anything about the names. The important thing is to report the area ratio in the same order as the given linear ratio (small:large).

?
 
So if we were given the linear ratio in a different problem from large:small, the area ratio would also be large:small as that is what they want us to order it in correct? (unless the problem specifies otherwise)
 
I would agree. All your examples give some clue as to the order expected, either (1) by mentioning increase or decrease, or (2) implying relative size in a picture, or (3) implicitly asking you to answer in the same order as a given ratio. That was a well-chosen set of examples!
 
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