Ratios and Rates

A RATIO is a way to compare two quantities. A RATE is a ratio with DIFFERENT units. These two ideas power cooking, driving, shopping, sports statistics, science, and almost every kind of analysis you will ever do. They are the workhorses of 6th-grade math.

If a classroom has 12 boys and 15 girls, the RATIO of boys to girls is 12 to 15.\n\nThree ways to write it:\n\n- "12 to 15"\n- "12:15"\n- "12/15"\n\nAll say the same thing. Ratios compare two numbers with the SAME units (people to people here).

Just like fractions, ratios can be SIMPLIFIED:\n\n12:15 — both divide by 3 → 4:5\n\nSo the boy-to-girl ratio of 12:15 is the same as 4:5. For every 4 boys, there are 5 girls.\n\nAlways simplify when possible!

A RATE compares DIFFERENT units. Common ones:\n\n- **60 miles PER hour** (distance per time)\n- **$5 PER pound** (dollars per weight)\n- **15 pages PER day** (pages per time)\n- **40 calories PER serving** (calories per amount)\n\nWhen the rate is "per 1" of something, it is called a UNIT RATE — the most useful kind.

Ratios and rates are EVERYWHERE:\n\n- **Recipe**: 2 cups flour : 1 cup sugar. Doubling it? 4:2.\n- **Paint mixing**: 3 parts red : 2 parts blue = purple\n- **Sports**: "batting average" is a rate (hits per at-bat)\n- **Grocery shopping**: $3 for 2 cans = $1.50 per can (unit price)\n- **Medicine**: "10 mg per kg of body weight"\n\nLearning to compare numbers this way = a POWER TOOL.

If a recipe calls for 2:3 flour to sugar, and you want to use 6 cups of flour, how much sugar?\n\n- Ratio: 2:3\n- Flour: 2 → 6 (multiplied by 3)\n- Sugar: 3 × 3 = 9 cups\n\nThis is called PROPORTIONAL REASONING — scaling ratios up or down.