MathIsland YouTubeYouTube

PROOF 117

Prove.

The grounds of percentage computation

On the rational numbers ℚ.

What you may use

A1(ab)c = a(bc)
associativity of multiplication
A2ab = ba
commutativity of multiplication
A3a × 1 = a
multiplicative identity
A4if a ≠ 0, then a·a⁻¹ = 1
existence of multiplicative inverses
A5a(b + c) = ab + ac
distributive law
A6(a+b)+c = a+(b+c)
associativity of addition
A7a + b = b + a
commutativity of addition
A8a + 0 = a
additive identity
A9a + (−a) = 0
additive inverse
A10a(−b) = −(ab)
sign rule
A11a ≠ 0 ⟹ 0 < a²
squares are positive
A12a < b ⟹ a + c < b + c
addition and order
A130 < c, a < b ⟹ ca < cb
multiplication and order
A1420/100 = 1/5
computation rule
A151 + 1/5 = 6/5 = 1.2
computation rule
A161 − 1/5 = 4/5 = 0.8
computation rule
A17(6/5)(4/5)=24/25=0.96
computation rule
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

For b ≠ 0, a/b means a · b⁻¹.

Proof.

Related labs

Ratios lab →Direct & Inverse Proportion lab →