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math question (ratios)

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In my dd's geo bk, they are covering a section on ratios. The question reads:


2 numbers a and b are in the ratio 3:4. If the first number is decreased by 2 and the second is decreased by 1, they are in the ratio 2:3. Find a and b.


Dd worked the problem similar to the way as the answer key. I didn't. :tongue_smilie: We both got the same answer, but I got my answer in a couple of seconds, whereas their approach is much longer.


I am wondering if there is a reason not to work the problem the way I did, b/c I certainly don't want to confuse dd.


So......here is the answer key's answer:






(a-2)/(b-1)=2/3 (then all the steps cross multiplying)



Then by solving via simultaneous equations:






When I solved the problem, I thought way more simply. I knew that the numbers were a ratio, so I thought in terms of 3x/4x


So I simply did

(3x-2)/(4x-1)= 2/3 and solved for x and got x=4, so the ratio is 12/16


Am I missing something? Is there a reason not to solve the problem the way I did?



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There is really not much difference - except that you introduced a new variable x and did several steps in your head instead of writing them out individually.

A strong math student will have no trouble understanding your approach which looks more elegant.

A struggling math student, however, would benefit from setting it up more slowly and NOT introducing yet another variable x - the main difficulty for students is operating with variables, and solving it in terms of the original variables a and b is preferable from a didactic point of view.


You know what type of student your DD is and whether seeing a more elegant solution would cause delight or confusion.

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