Chapter 1 · Algebraic Proofs · Concept 4
1-12 Proving Equalities (3)
The condition this time is a ratio. In lesson 1-11 you wrote one letter in terms of the others — here you write all of them in terms of one new letter, and then it cancels.
Guiding question
How do you use a ratio as a condition, and why does the answer never depend on the size of the numbers?
Learning outcome. Prove the validity of algebraic equalities and identities in proportional cases.
Name the ratio k
An equation saying that two ratios are equal — like a/b = c/d or x : y : z = a : b : c — is called a proportion.
For a/b = c/d
let a/b = c/d = k
then a = bk, c = dk
For x : y : z = a : b : c
let k be a constant
then x = ak, y = bk, z = ck
Then substitute, tidy each side on its own — and the k will cancel.
A ratio does not tell you how big the numbers are, only how they compare. So the answer cannot depend on how big they are — and it does not.
If x : y : z = 2 : 3 : 5, find
(2xy + yz + 3zx) / (x² + y² + z²)
x = 2k
4
y = 3k
6
z = 5k
10
the answer is 3/2
Drag k as far as you like. The top and the bottom both change — and the answer never does.
Why? Every term on the top has two letters multiplied, and so does every term on the bottom. So both carry a k², and it cancels. The ratio was all the question ever gave you, and the ratio is all the answer ever needed.
1. If a/b = c/d, prove that ab/(a2 + b2) = cd/(c2 + d2)
- 1
Give the ratio a name.
let a/b = c/d = k, so a = bk and c = dk …… (i)This is the only new idea in the lesson. Everything after it is lesson 1-10 again.
Work with a partner. Do question 3 together, then talk about the answer. Write down what k is before you touch anything else.
Two containers of salt water. The first has a grams of salt in b millilitres of water; the second has c grams in d millilitres. They have exactly the same concentration, so a/b = c/d.
Prove that the ratio of their combined mass to their separated difference is also equal:
(a + b)/(a − b) = (c + d)/(c − d)
Name the ratio k before anything else.
Write "let a/b = c/d = k, so a = bk and c = dk". Without that line there is nothing to substitute and the proof cannot start.
(bk)² is b²k², not bk².
The whole bracket is squared, so the b is squared too. This one step decides the Warm Up, and it is the commonest slip on page 29.
Take the common factor out, then cancel.
b²k over b²k² + b² only cancels once you write the bottom as b²(k² + 1). Cancelling before factorising cancels across a plus, which is not allowed.
A ratio with a negative in it is still a ratio.
x : y : z = 2 : −3 : 4 means y = −3k. Dropping that minus turns Exercise 8 from −29/10 into 29/10.
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