Ratios, proportions, and percents
A ratio compares two quantities, like 3 cups of flour to 2 cups of sugar, written 3:2 or 3/2. A unit rate reduces the second quantity to 1 — 60 miles in 2 hours is 30 miles per hour. Two equal ratios form a proportion, and we solve one by cross-multiplying. A percent is simply a ratio out of 100: 25% means 25/100 = 1/4.
Proportion by cross-multiplying 3 / 4 = x / 20 3 * 20 = 4 * x (cross-multiply) 60 = 4x x = 15 Percent as a ratio What is 25% of 80? 25/100 * 80 = (1/4)*80 = 20
The laws every later rule rests on
Arithmetic obeys a few laws so deep we usually forget to name them — yet all of algebra leans on them. The commutative property says order does not matter for addition or multiplication: a + b = b + a and ab = ba. The associative property says grouping does not matter: (a + b) + c = a + (b + c). The distributive property links the two operations: a(b + c) = ab + ac — this is the engine behind expanding, factoring, and combining like terms later on.
Two special numbers act as identities: adding 0 changes nothing (a + 0 = a) and multiplying by 1 changes nothing (a · 1 = a). And each number has inverses that undo it — the additive inverse −a gives a + (−a) = 0, while the multiplicative inverse 1/a gives a · (1/a) = 1 for a ≠ 0. Identities and inverses are exactly what let you “move things to the other side” when solving equations.
The distributive property at work
6 * 23 = 6 * (20 + 3)
= 6*20 + 6*3 (distribute)
= 120 + 18
= 138
Later, the same law expands and factors:
3(x + 4) = 3x + 12 (expand)
5x + 5y = 5(x + y) (factor out 5)