math question (ratios)

[8filltheheart]

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/b=3/4

4a=3b

4a-3b=0

 

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

3b-2b=4

 

Then by solving via simultaneous equations:

4a-3b=0

3a-2b=4,

a=12

b=16

 

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?

 

Thanks!

[regentrude]

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.

[8filltheheart]

Thanks, Regentrude. I was pretty sure there was no reason not to do it my way, but I wanted to make sure.