Adding two of them
A wide tap fills a tank in x hours. A narrow tap takes one hour more. What share do the two fill in one hour? Slide x and compare two routes: the taps, and the expression.
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1/x + 1/(x + 1).1/x + 1/(x + 1).- Shared bottom:
x(x + 1). - Rename both tops to match it.
- Add the tops:
(x + 1) + x = 2x + 1. (2x + 1) / [x(x + 1)].- Check at x = 2: both routes give 5/6.
The rule
In one hour the two taps fill (2x + 1)/(x² + x) of the tank.
An algebraic fraction has a letter in it, like 1/x. Two of them add the way number fractions add. Build one shared bottom by multiplying the two bottoms. Rewrite both tops to match the shared bottom. Add the tops. Keep the bottom.
The wide tap fills 1/x of the tank in an hour. The narrow one fills 1/(x + 1). The shared bottom is x(x + 1). Multiply the top and bottom of 1/x by x + 1. Do the same to 1/(x + 1), multiplying by x. This does not change either fraction's value. The two tops become (x + 1) and x. They add to 2x + 1.
At x = 2 the taps take 2 and 3 hours. With the taps: 1/2 + 1/3 = 5/6. With the expression: (2 × 2 + 1)/(2 × 3) = 5/6.
| The two | Shared bottom | Added | Both give |
|---|---|---|---|
1/x + 1/(x + 1) | x² + x | 2x + 1 | 5/6 |
2/x + 3/(x + 2) | x² + 2x | 5x + 4 | 7/4 |
3/x + 4/x | already matches | 7/x | 7/2 |
Watch out!
Adding the tops and the bottoms gives 1/2 + 1/3 = 2/5. That is less than one half. Two taps cannot fill less than the wide tap alone.