Now Johnny Can Do Arithmetic
2 Sharing in Proportion Here is a game that Johnny can easily play but which will later enable him to solve many difficult problems.
1 We begin by taking a red and a yellow rod which, end to end, equal the black. We see that the red is of the yellow and that the yellow is (or ) of the red. When Johnny understands this, he can tackle such questions as these: If we have nuts, can we share them between two boys so that one has of what the other has? Johnny will see that the answer is to give nuts to one and to the other. If we had nuts to be shared in the same way, what would the shares then be? Johnny finds no difficulty, for with twice as many nuts to share each would get twice what he did before but the proportion will still be nuts for the first to every nuts for the second. What if there were nuts or nuts, and the shares were to be in the same proportion?
2
We now take the length of , using any two rods to form it, and share nuts, first in the proportion of the two rods we have chosen. 52