Chapter 1 · Algebraic Proofs · Concept 3
1-8 Compound Fractions
A fraction with another fraction inside it looks much worse than it is. There is one move that clears it, and it is the same move every time.
Guiding question
How do you turn a fraction that has fractions inside it into an ordinary one?
Learning outcome. Simplifies compound fractions and converts them to their simplest form.
What they are, and the one move
An expression with rational expressions in its top or its bottom is called a compound fraction.
(1/y) / (2/x) · (1 − y/x) / (1 + y/x)
Both of these are compound fractions.
Multiply the top and the bottom by the same expression.
Choose that expression so the little fractions vanish. It is always the little bottoms multiplied together.
You are allowed to multiply the top and the bottom of any fraction by the same thing, because that is the same as multiplying by 1.
With numbers
3/4 = (3 × 5)/(4 × 5) = 15/20
Same value, written differently.
Here
multiply by (x − 1) to clear a 1/(x − 1)
Same value — but now there is no little fraction left.
1 / (1 − 1/(x−1))
- 1. Find the little bottoms. (x − 1)
- 2. Multiply the top and bottom by all of them. × (x − 1)
The whole top and the whole bottom. Multiplying both by the same thing does not change the value.
- 3. The little fractions disappear.top: 1 × (x − 1) = x − 1bottom: (1 − 1/(x−1)) × (x − 1) = (x − 1) − 1 = x − 2
- 4. Now it is an ordinary fraction. (x − 1)/(x − 2)
2. Simplify [1/(1−x) + 1/(1+x)] / [1/(1−x) − 1/(1+x)]
- 1
Find every little bottom.
(1 − x) and (1 + x)They appear on the top and on the bottom, but you only need the list of them once.
Try 3
1 / (1 − 1/(1 − 1/x))
There are three fractions inside each other. You cannot start on the outside — the outside does not become a single fraction until everything inside it has.
Work with a partner. Do question 3 together, then talk about the answer. It is the nested one, so decide where the inside is before you write anything.
Simplify.
When a signal passes through a chain of booster nodes, it weakens a little at each one. An engineer models a 3-stage booster with
E(x) = 1 / (1 + 1/(1 + 1/(1 + x)))
where x > 0. Turn this into a single rational expression A(x)/B(x).
Multiply by ALL the little bottoms.
If (x+2) and (x−2) both appear, multiply by (x+2)(x−2). Using only one of them clears half the small fractions and leaves a mess.
The top AND the bottom.
Multiplying only one of them changes the value of the whole thing. Both, always — that is what makes it the same as multiplying by 1.
A nested one is peeled from the inside.
In 1/(1 − 1/(1 − 1/x)) you cannot touch the outside first. Work out the innermost fraction, then the next, then the next.
Watch a negative bottom.
In Try 2 the bottom comes to −4, so the answer is −x/2. Losing that minus sign is the single commonest slip on page 20.
That is Concept 3 finished. The review quiz covers all three lessons.
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