Guide 1 Real #s

[Ceej05]

2 libraries plan to combine a portion of their collections in one new space. 1/3 of the books from Library A and 1/4 of the books from Library B will be in the new space. If there are twice as man books in Library B as in A, what proportion of the books in the new space will have come from Library A?

The book shows the steps using Real #s, but I couldn't make it work algebraically.

I tried 1/3 A + (1/4 of 2a) to get the combined proportion of books. I then simplified the 1/4 of 2a to a/2. With 1/3 a + 1/2 a, I got 5/6 A. Where is this equation incorrect?

[RonPurewal]

2 libraries plan to combine a portion of their collections in one new space. 1/3 of the books from Library A and 1/4 of the books from Library B will be in the new space. If there are twice as man books in Library B as in A, what proportion of the books in the new space will have come from Library A?

The book shows the steps using Real #s, but I couldn't make it work algebraically.

I tried 1/3 A + (1/4 of 2a) to get the combined proportion of books. I then simplified the 1/4 of 2a to a/2. With 1/3 a + 1/2 a, I got 5/6 A. Where is this equation incorrect?

is that where you stopped? because that's not where you stop.

5a/6 represents the total number of books in the new space. exactly a/3 of those books come from library A. both 5a/6 and a/3 represent integers-- they are numbers of books. neither of them is a proportion!

to find the proportion, you need to do the same thing you'd do with smart numbers: divide the number of books from library A by the total number of books.
if you divide a/3 by 5a/6, you get 2/5, the same proportion you should get by using smart numbers.

[RonPurewal]

by the way, your query is an absolutely perfect illustration of one of the key advantages of smart numbers: it's MUCH easier to keep track of what those numbers mean.

e.g., let's say library A has 30 books and library B has 60 books.
then the new space contains 10 books from library A and 15 books from library B, for a total of 25 books... and it's obvious that 25 is not the answer to the problem.

when you're doing algebra, you should ALWAYS LABEL the algebraic quantities, because otherwise you're at high risk of forgetting what they represent (as you did here).

e.g., do not just write "5a/6"; write "5a/6 books total". and so on.