Ratios compare quantities. Rates compare quantities with different units. Both are essential for scale, speed, and pricing.

Part-to-part and part-to-whole ratios

A ratio \(2:3\) means two equal parts compared with three equal parts. Total parts are \(2+3=5\).

If a class has 12 boys and 18 girls, boys:girls is \(12:18=2:3\), and boys:total is \(12:30=2:5\).

Equivalent ratios

Multiply or divide both parts by the same nonzero number.

\(3:5 = 6:10 = 12:20\)

Unit rates

A unit rate is “per one.” If 180 km takes 3 hours, speed is \(180\div3=60\) km/h.

Proportional relationships

For proportional relationships, \(y=kx\). Graphs are straight lines through the origin where slope is the constant of proportionality \(k\).

Checkpoint: share 56 in ratio \(3:4\)

Total parts \(=7\), one part \(=8\), shares are 24 and 32.

Checkpoint: 45 dollars for 6 kg, find unit price

\(45\div6=7.5\). Unit price is 7.5 dollars per kg.