Non-Traditional Solution: 15 divided by 2/2x + 1/2x equals 15 divided by 3/2 which equals 15 * 2/3 equals 10

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Explain this!

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I can solve the following by using basic algebra: x * (1/2 * x) = 15

15 divided by 2/2x + 1/2x equals 15 divided by 3/2 which equals 15 * 2/3 equals 10.

10 * (1/2 *10) = 10 + 5 = 15

Is there a different method or calculation that can be used? I like ad hoc calculations, that is, non-traditional methods for determining answers.

Are there any that can be used for my above example?
 
Apparently you didn't mean what you typed. Did you mean to solve x + (1/2 * x) = 15 ?

What you did (with a little interpretation) looks more or less like what I did in my head, once I saw what you were really solving. Why do you want something different, other than just for novelty?
 
Dr.Peterson said:
Apparently you didn't mean what you typed. Did you mean to solve x + (1/2 * x) = 15 ?

What you did (with a little interpretation) looks more or less like what I did in my head, once I saw what you were really solving. Why do you want something different, other than just for novelty?
Click to expand...

Yes, a silly error! It is supposed to be x + (1/2 * x) = 15.

There is not other reason other than just out of curiosity and having a different way of determining the answer.
 
Any different way I can think of is essentially the same thing.

It's true, though, that for many problems there is a standard method, and also various special ways of thinking tailored to the specific problem. And it's fun to find them, especially when they are easier.
 
Dr.Peterson said:
Any different way I can think of is essentially the same thing.

It's true, though, that for many problems there is a standard method, and also various special ways of thinking tailored to the specific problem. And it's fun to find them, especially when they are easier.
Click to expand...

Would some type of a weighted average calculation work? 7.50 + 7.50 = 15, 15 divided by 2. This is a regular average. However the two addends that are the solution are 10 + 5 = 15 or 10 + (1/2 * 10). This is a weighted average since one addend (10) is weighted more than the other.
 
You're inventing the method; tell me how you would decide what to do!

Ultimately, as I said, any method I can see will in some way involve adding 1 + 1/2 and then dividing by 3/2 (multiplying by 2/3). Some of them may be more natural than others in terms of how you think about it. Maybe that's what you're looking for.

Here's one that you might consider different: The addends x and 1/2 x have a 2:1 ratio, which means x is 2/3 of the total (2 parts out of 3). I've often done this kind of thing visually with kids who aren't ready for algebra yet: Make a bar representing x, and another half as long; mark the longer bar as two pieces each equal to the second; together there are 3 pieces making 15, so each is 5; then x is 2 times that.
Code: 
+-------+-------+-------+
|       X       | 1/2 X |
+-------+-------+-------+
\_______________________/
           15
 
Thank you for that explanation! It is a different way of thinking about the solution. I have no additional questions.
 
Dr.Peterson said:
You're inventing the method; tell me how you would decide what to do!

Ultimately, as I said, any method I can see will in some way involve adding 1 + 1/2 and then dividing by 3/2 (multiplying by 2/3). Some of them may be more natural than others in terms of how you think about it. Maybe that's what you're looking for.

Here's one that you might consider different: The addends x and 1/2 x have a 2:1 ratio, which means x is 2/3 of the total (2 parts out of 3). I've often done this kind of thing visually with kids who aren't ready for algebra yet: Make a bar representing x, and another half as long; mark the longer bar as two pieces each equal to the second; together there are 3 pieces making 15, so each is 5; then x is 2 times that.
Code: 
+-------+-------+-------+
|       X       | 1/2 X |
+-------+-------+-------+
\_______________________/
           15
Click to expand...

I thought that I understood this solution, but I am having some difficulty with the following examples.
How would your alternative solution be used with the following?

x + (2/3 * x) = 10, x equals 6

x + (1 3/4 * x) = 22, x equals 8 (1 3/4 is a mixed number 1 & 3/4 not 13/4.)

x + (4 * x) = 30, x equals 6
 
You're asking about applying the diagram to these problems?

Let's take the first, x + (2/3 * x) = 10 . Here's the diagram:

Code: 
+-----+-----+-----+-----+-----+
|        x        |   2/3 x   |
+-----+-----+-----+-----+-----+
\_____________________________/
              10

In order to find 2/3 of x, I made x of three pieces, so that two of them make 2/3 x. So there are 5 equal pieces that make up 10, and each of them is 2. So x = 3*2 = 6.

You can try the others.

By the way, I should have mentioned that these can often be solved easily by trial and error, especially if you assume that the answer will be a nice number because this was assigned to you before you learn algebra. Here, you want x to be a multiple of 3 that is less than 10, so the first thing you try might be 6.
 
Dr.Peterson said:
You're asking about applying the diagram to these problems?

Let's take the first, x + (2/3 * x) = 10 . Here's the diagram:

Code: 
+-----+-----+-----+-----+-----+
|        x        |   2/3 x   |
+-----+-----+-----+-----+-----+
\_____________________________/
              10

In order to find 2/3 of x, I made x of three pieces, so that two of them make 2/3 x. So there are 5 equal pieces that make up 10, and each of them is 2. So x = 3*2 = 6.

You can try the others.

By the way, I should have mentioned that these can often be solved easily by trial and error, especially if you assume that the answer will be a nice number because this was assigned to you before you learn algebra. Here, you want x to be a multiple of 3 that is less than 10, so the first thing you try might be 6.
Click to expand...

Are these correct?

x + (1 3/4 * x) = 22

I converted 1 3/4 to 7/4. 11 pieces making 22, so each is 2. x is 4 times that (2). 4 * 2 = 8. Answer 8

x + (4 * x) = 30, x equals 6

This was a tricky one. I used 4/1. 5 pieces making 30, each is 6. x is 6 times that (1). 1 * 6 = 6. Answer 6

I hope that they are correct. I want to thank you for this alternative solution.
 
Yes, those are correct. In the second, though, I wouldn't have said "x is 6 times that (1). 1 * 6 = 6", though; "that" is already 6! The unknown is one piece, so there's no multiplication to do.
 
Can this solution work with subtraction instead of addition? I am having difficulty using the method for the following:

x - (1/2 * x) = 3, x = 6

3 sections equaling 3, so each is 1. x is not 2 times 1.
 
You could do this with my picture method, though it's a little more awkward to draw. Make a bar representing x (with two parts), and cross off half of it (or otherwise mark it as removed. What's left, one piece, is 3; x is two of those pieces, i.e. 6.

Code: 
+-------+-------+
|       x       |
+-------+XXXXXXX+
\_______/
    3

But I think it's easier to use the method (equivalent to the algebra) commonly used to teach prealgebra students to solve percent increase or decrease problems. Think of your problem as this: An item is on sale at 1/2 off, and now costs $3. How much does it normally cost?

Taking half off means that the amount you are paying is 1 - 1/2 = 1/2 of the original; if 1/2 of the original is 3, then the original is twice that, 6.

In percentage problems, we would say that 20% off means paying 80%, so you divide by 80% to find the original price.

As you can see, the work is exactly what you would do in algebra, but the thinking doesn't require willingness to do algebra, much less knowledge of it.
 
Dr.Peterson said:
You could do this with my picture method, though it's a little more awkward to draw. Make a bar representing x (with two parts), and cross off half of it (or otherwise mark it as removed. What's left, one piece, is 3; x is two of those pieces, i.e. 6.

Code: 
+-------+-------+
|       x       |
+-------+XXXXXXX+
\_______/
    3

But I think it's easier to use the method (equivalent to the algebra) commonly used to teach prealgebra students to solve percent increase or decrease problems. Think of your problem as this: An item is on sale at 1/2 off, and now costs $3. How much does it normally cost?

Taking half off means that the amount you are paying is 1 - 1/2 = 1/2 of the original; if 1/2 of the original is 3, then the original is twice that, 6.

In percentage problems, we would say that 20% off means paying 80%, so you divide by 80% to find the original price.

As you can see, the work is exactly what you would do in algebra, but the thinking doesn't require willingness to do algebra, much less knowledge of it.
Click to expand...


I want to thank you for the reply and explanation. I'm somewhat confused on this one. Is the bar in two parts because of the 1/2 x and then cross off half because 1/2 of the bar is used? This 1/2 represents 3 pieces and x is 2 of 3 pieces (6). Why use 2, as in 2 of 3 pieces?

I thank you for your assistance!
 
Explain this! said:
I want to thank you for the reply and explanation. I'm somewhat confused on this one. Is the bar in two parts because of the 1/2 x and then cross off half because 1/2 of the bar is used? This 1/2 represents 3 pieces and x is 2 of 3 pieces (6). Why use 2, as in 2 of 3 pieces?

I thank you for your assistance!
Click to expand...

There are no 3 pieces in this problem! You'd get that if you added 1+2. But we are subtracting 1 part from 2, leaving 1 piece.. Where do you get 3 pieces? The only 3 here is the length of what remains.
 
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