Reversed subtraction

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heike hoffmann 6 years, 5 months ago

Created this as a copy of Reversed subtraction.

Name	Status	Author	Last Modified
Reversed subtraction	Ready to use	Ben Brawn	26/03/2025 04:23
Hannah's copy of Reversed subtraction	draft	Hannah Bartholomew	04/07/2018 13:34
Hannah's copy of Hannah's copy of Reversed subtraction	draft	Hannah Bartholomew	11/07/2018 09:41
Reversed subtraction	draft	heike hoffmann	14/11/2018 21:10

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This question asks you to compared different looking answers, and determine if they are equivalent.

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			Name	Type	Generated Value

a	integer	9
b	integer	5
c	integer	11
d	integer	5
p	integer	6
q	integer	8
r	integer	5
s	integer	3
sym1	string	+
sym2	string	-
seed	integer	0

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Generated value: `integer`

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Suppose you do a maths question and your answer is

$y=\var{a*b}+\frac{\var{c}-x}{\var{d}}.$

However, your friend has an answer of

$y=\var{a*b} \var{sym1}\frac{x-\var{c}}{\var{d}}.$

These answers are...

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Advice

Consider doing the subtraction $11-25$ . Often people do the easier subtraction $25-11$ , get $14$ , and then they put a negative in front of it to conclude $11-25=-14$ . This works because

$11-25=-(25-11)=-14.$

So if we swap the order of subtraction, we need to put a negative out the front, but this is the same as just multiplying by $-1$ since $-(25-11)=-1\times(25-11)$ , which is also the same as dividing by $-1$ .

Therefore, reversing the order of a subtraction is the same as multiplying (or dividing) by $-1$ .

To determine if $\frac{\var{c}-x}{\var{d}}$ is equal to $\frac{x-\var{c}}{\var{d}}$ , notice the only difference is the subtraction in the numerator is reversed. But $\var{c}-x\ne x-\var{c}$ . So these answers are not the same!

To determine if $\frac{\var{c}-x}{\var{d}}$ is equal to $-\frac{x-\var{c}}{\var{d}}$ , notice

$\begin{align}-\frac{x-\var{c}}{\var{d}}&=\frac{-(x-\var{c})}{\var{d}}\\&=\frac{-x+\var{c}}{\var{d}}\\&=\frac{\var{c}-x}{\var{d}}\end{align}$

So the negative out the front and the reversing of the subtraction cancelled each other out, and these answers are actually the same.

You should notice that these fractions are very similar except that the order of subtraction is reversed in the numerator and the denominator. We should know that reversing the order of subtraction introduces a negative out the front, if we do this twice we will have two negatives out the front, which of course means a positive! That is,

$\begin{align}\frac{\var{a}x^\var{p}-\var{b}y^\var{q}}{\var{c}xy-\var{d}x^\var{r}y^\var{s}}&=\frac{-(\var{b}y^\var{q}-\var{a}x^\var{p})}{\var{c}xy-\var{d}x^\var{r}y^\var{s}}\\&=\frac{-(\var{b}y^\var{q}-\var{a}x^\var{p})}{-(\var{d}x^\var{r}y^\var{s}-\var{c}xy)}\\&=\frac{\var{b}y^\var{q}-\var{a}x^\var{p}}{\var{d}x^\var{r}y^\var{s}-\var{c}xy}\end{align}$

So the answers are the same!

You should notice that in the numerator the order of subtraction has been swapped and in the denominator a $-\var{d}x^\var{r}y^\var{s}$ has been replaced with $+\var{d}x^\var{r}y^\var{s}$ . These are not the same answers. If you require further proof, set them to be equal and see what happens, or even easier, substitute a value for $x$ and $y$ into both of them:

Let $x=1$ and $y=1$ and we will compare the fractions. For 'your' answer we get $\simplify[fractionNumbers,simplifyFractions]{{(a-b)/(c-d)}}$ but for 'your friends' answer we get $\simplify[fractionNumbers,simplifyFractions]{{(a-b)/(c+d)}}$ and therefore the fractions are not equal!

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heike hoffmann 6 years, 5 months ago

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Reversed subtraction

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

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Comment

Checkpoint description

heike hoffmann 6 years, 5 months ago

Error in variable testing condition

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Generated value: integer

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Generated value: `integer`