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Solution: the animal problem

Solution: the animal problem
So let’s see if we can do something about this task. So let’s just see what we’ve got. We’ve got let’s say we’ve got a Cow, which we’ll depict as a ‘C’. We’ve got a dog which we’ll depict with the letter ‘D’ and we’ve got a goat which is the ‘G’. And let’s now assume that they all have a certain quantity so let’s do a first bar, which we’ll call ‘G’. Let’s do another bar, which we’ll call ‘C’ and you can do this in any order if you want. And what have we got in this information over here.
We’ve got a cow weighs 150 kilos more than a dog, so what we know is that there is a little bit more for the cow, the ‘C’, and this bit more is 150 kilos more, as you can see. The cow is 150 kilos more than the dog. We know something more. We also know that a goat weighs 130 kilos less than a cow, so we’ve got the cow over here and we know that the goat is a little bit more but only 130 kilos less. So I hope you can see that all the information or almost all the information now is in this, in this diagram.
We’ve got 150 kilos because we can see that the ‘C’, the cow, is 150 kilos more than the dog. But we also can see that the cow is 130 kilos more than the goat. Or the goat 130 kilos less than the cow. And then there’s one more snippet of information that is important. We know that all together, all together the three animals weigh 410 kilos. So all together. So what do we now know? Well what we all now know is that there
are three units: there is a ‘G’, the goat. There is a cow ‘C’ and there is a dog ‘D’. And all together they are 410 plus 150 plus 130, which is 690. So what do we know about one unit? We know that one unit is 690 divided by three equals 230 kilos. So let’s do a quick check now, so we’re saying that the cow is 230 kilos we know that dog then must be 230 - 150, because it was less which is 80 kilos. And we also know that the goat should be 230-130 which is a 100 kilos. So in total we’ve got 230, 80 and 100 which makes 410 kilos in total.
So we’ve checked it now and we see that it is correct. We can also use another approach, which also uses the bar model and actually the mechanism behind it is not that different but it just looks different. Again, I’ve experienced people who thought that one or the other method perhaps was a bit easier or more insightful but I saw people also thought that one perhaps was a bit more difficult than the other so let me explain.
Let’s again start with, say, we’ve got a ‘C’ [well let’s use some black], the cow and we’ve got a ‘D’, a dog and we’ve got a ‘G’, a goat, and let’s again start with a couple of bars but now we’re really going to only first start with these bars. One is for a cow, one is for a dog and one for a goat. And now rather than look at the differences we’re going to look what is a little bit more so one thing that we know for example is that the cow is 150 kilos more. Then, in this case the dog, as you can see. And we also know that this little snippet must be 20 kilos.
And why is that the case? Well because we know that in total 130 kilos, a goat weighs 130 kilos less than a cow so we know that this is 130 kilos, so then if we look at the totality of the bars we know that this small piece needs to be 20 kilos. And again what we also know is that altogether there are 410 kilos. Okay, so what does this tell us? What we can now do is… just look at the total of the bars. What do we know about this red?
We know that the red is 410 kilos: that’s all of it together. But we also know that there are two parts over here which are, could say ‘surplus’, which are extra on top of these three bars on the left. So what we can do is, we can take 410 kilos and we can subtract the yellow bits which is 150, which is 20 as well and what are we left with? We’re left with 240 kilos. These 240 kilos are divided over three units, as we know. So one unit, one of these red bars that say the dog, because the dog is only the red bar. It doesn’t have any surplus then must be 240 divided by three which is 80 kilos.
So what do we now know? We now know that the dog is 80 kilos. We know that the goat is 20 more and we know that the cow is 150 more. What’s that all together? 410 kilos, which is a final check that it is correct. So what is the weight of the cow? 230 kilos.

This is a solution video of the animal problem, which is a famous problem from the literature about bar modelling.

The animal problem goes like this:

A cow weighs 150 kg more than a dog. A goat weighs 130 kg less than a cow. Altogether the three animals weigh 410 kg. What is the weight of the cow.

There also is a document with visual solutions to download here.

It is a well-known task from Bar Model literature, and you can read more about it in this article:

Ng, S.F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40(3), 282-313.

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