How to explain word problem that requires iteration to 7 year old

Question

My 7-year-old has a "Stretch your thinking" question in her homework that baffles me in terms of explaining how to solve it.

Up to this point her class has only studied addition and subtraction.

Fifteen children voted for their favorite color. The votes for red and blue together were double the votes for green and yellow together. How did the children vote?

There is also a table that looks like this:

Color | Votes
Red
Blue
Green
Yellow

I don't know how I would explain this except to say that she should start with the green + yellow set having 1, then the red + blue set would have 2, and the total is 3. That isn't fifteen so she should try again, starting with green + yellow set having 2 (adding one to initial set) until she gets to a total that equals 15.

Is there a better way to explain it?

Answer 1

There’s nothing wrong with intelligently directed trial and error. Point out that every vote for green or yellow can be matched up with two votes for red or blue, as in the following chart:

$$\begin{array}{rcc} \text{green or yellow}:&*&*&*&*&*\\ \text{red or blue}:&*\atop*&*\atop*&*\atop*&*\atop*&*\atop*\\ \text{total votes so far}:&3&6&9&12&\color{red}{15} \end{array}$$

When the magic number $15$ is reached we can go back and count the green or yellow votes ($5$) and the red or blue votes ($10$). You can point out that as a check, $5+5=10$, so there really are twice as many votes for red or blue as for green or yellow.

Answer 2

Iteration is not necessary. Let $x$ be the total number of red and blue votes. Let $y$ be the total number of green and yellow votes. Then we have $$x+y=15\\ x=2y\\ 3y=15$$ So $y=5$ and $x=10$. The actual number of red, blue, green, and yellow votes can't be determined.

Your method seems to be a guess and check method, starting with the assumption that $y=1$ and adding $1$ to $y$ for each new guess.

Answer 3

I'd start by having her imagine the number of kids in the yellow+green group (maybe imagine them all wearing yellow and green striped shirts, say). Next, we need two groups that size to comprise the red+blue voters. So, that's three groups that are all the same size, and each of the kids are in one of the groups, so those three groups have $15$ kids altogether. That lets us see how many kids voted for yellow or green. From there, we can see how many kids voted for red or blue.

Since that's all the problem told us, we can then give kids shirts of the color they actually voted for, with a little bit of freedom. There turn out to be $66$ different viable vote breakdowns, but I suspect they don't expect her to come up with all of them.