Joe spent 1/3 of his allowance on a model airplane. He spent 2/3 of the remainder on a fishing rod. He had $20 left over. How much did he start with?

>> I'm starting my son off with a picture that is a rectangle divided into 3 parts (thirds), the first part representing the model airplane, but then I'm confusing myself with representing the 2/3 of the remaining 2 parts to arrive at the total. I know the answer is 90, but I can't explain how I got there except in the most manual, illogical way. I need a better mental template to approach these word problems with. Can I see a good step-by-step solution?

Thank you!:confused:

Joe spent 1/3 of his allowance on a model airplane. He spent 2/3 of the remainder on a fishing rod. He had $20 left over. How much did he start off with?

>> I'm starting my son off with a picture that is a rectangle divided into 3 parts (thirds), the first part representing the model airplane, but then I'm confusing myself with representing the 2/3 of the remaining 2 parts to arrive at the total. I know the answer is 90, but I can't explain how I got there except in the most manual, illogical way. I need a better mental template to approach these word problems with. Can I see a good step-by-step solution?

Thank you!:confused:

So here's what we do.

Let A = the original amount

Joe spent 1/3 if his original allowance, so your first term will be

\frac{1}{3}*A = \frac{A}{3}

The remaining amount would be
A-\frac{A}{3} = \frac{2A}{3}

Joe then goes and spends 2/3 of what's left (2A/3) and has 20 dollars left over

\frac{2}{3}*\frac{2A}{3} = \frac{4A}{9} This is what was spent.

So it must be true that what he spent plus 20 should give us what he had

20+\frac{4A}{9}=\frac{2A}{3}

From here you only have 1 unknown

Thank you all - that helped a lot!

I think you should try to stick to the picture. You can apply the math used, but showing it on the picture helps to understand what it means, and helps people to make pictures of their own so they can apply it to future problems. Using the math only is usually pretty abstract to most people - like just random numbers flying about. With the picture, it is no longer abstract.

So I think you were on the right track there. I was going to attach an example, but I discovered this one was not easy to do looking at it after the fact. Much easier to watch it happen "live."

I think you should try to stick to the picture. You can apply the math used, but showing it on the picture helps to understand what it means, and helps people to make pictures of their own so they can apply it to future problems. Using the math only is usually pretty abstract to most people - like just random numbers flying about. With the picture, it is no longer abstract.

So I think you were on the right track there. I was going to attach an example, but I discovered this one was not easy to do looking at it after the fact. Much easier to watch it happen "live."
His method would have worked if he had divided the rectangle into 9 parts instead of 3.

That method can still work. Start with 3 parts. They can be divided down further as you go. That's the whole point of showing it "live." Which obviously I can't do here.

Besides, I can do it without dividing into 9 parts. It just takes 2 equations instead of trying to make it all one big equation. It's only elementary school - I wonder how much they're expecting. I would never heard learned this method in elementary school.

That method can still work. Start with 3 parts. They can be divided down further as you go. That's the whole point of showing it "live." Which obviously I can't do here.

Besides, I can do it without dividing into 9 parts. It just takes 2 equations instead of trying to make it all one big equation. It's only elementary school - I wonder how much they're expecting. I would never heard learned this method in elementary school.

Well it would be quiet difficult to show 2/3 of 2/3 using 3 pieces to a child.

Going forward with the rectangle idea, that's how I'd done it:

The grey area the first 1/3 removed. Then, you consider the remaining area as a whole, and divide it into 3 parts, 2/3 of which will be spent (blue). The third part (white) will be $20.

25787

The finely dotted lines represent the first time you divide into three parts, and the longer dotted lines, the second time you divide the rectangle.

Going forward with the rectangle idea, that's how I'd done it:

The grey area the first 1/3 removed. Then, you consider the remaining area as a whole, and divide it into 3 parts, 2/3 of which will be spent (blue). The third part (white) will be $20.

25787

The finely dotted lines represent the first time you divide into three parts, and the longer dotted lines, the second time you divide the rectangle.

The color however is hard to notice unless you highlight the picture :rolleyes:

Lol! I don't think that every person coming around would select the picture!

Well it would be quiet difficult to show 2/3 of 2/3 using 3 pieces to a child.

No harder than explaining your equation to a child? And Unky just did quite a nice picture, a bit like I had in mind. Except it doesn't actually need to be divided into 9 parts to solve it. Trust me, it doesn't. Give me a 6th grader and I'll prove it to you.

No harder than explaining your equation to a child?? And Unky just did quite a nice picture, a bit like I had in mind. Except it doesn't actually need to be divided into 9 parts to solve it. Trust me, it doesn't. Give me a 6th grader and I'll prove it to you.

Who said that formulaic explanation was for the child? The guy asked it merely for himself.

As requested by OP "I need a better mental template to approach these word problems with. Can I see a good step-by-step solution?"

Well, that's true... but I still see no issue with my suggestion.