Ratios & Rates

6.A.1Write ratios01/04

Write and interpret ratios.

Step01of 04

A ratio compares two quantities. It can be written THREE ways.

Word form	→3 to 5
Colon	→3 : 5
Fraction	→3/5

6.A.1 — Write and interpret ratios.

Step 01 of 04

A ratio compares two quantities. It can be written THREE ways.

Word form	→3 to 5
Colon	→3 : 5
Fraction	→3/5

Step 02 of 04

Worked example. A bag has 12 red marbles and 8 blue marbles. Ratio of red to blue?

12 : 8 or 12 to 8 or \frac{12}{8}

Simplify like a fraction: $12 : 8 = 3 : 2$ .

Step 03 of 04

Order matters. "Red to blue" is 12:8 = 3:2; "blue to red" is 8:12 = 2:3. Different ratios for the same situation.

Step 04 of 04

Part-to-part vs. part-to-whole.

Part to part	→12 red : 8 blue = 3 : 2
Part to whole	→12 red : 20 total = 3 : 5
Other part to whole	→8 blue : 20 total = 2 : 5

Key insight

Three notations for the same ratio. Always simplify when possible. Order matters: "A to B" and "B to A" are different. Distinguish part-to-part from part-to-whole.

6.A.2 — Equivalent ratios (tables, tape diagrams).

Step 01 of 04

Two ratios are equivalent if they reduce to the same simplified form. Multiply (or divide) BOTH sides of a ratio by the same number to get an equivalent ratio.

3 : 5 = 6 : 10 = 9 : 15 = 12 : 20 = 30 : 50

Step 02 of 04

Ratio table. Useful for solving "scale up" problems.

Cups water	→2
Cups syrup	→5
Cups water	→4
Cups syrup	→10
Cups water	→6
Cups syrup	→15
Cups water	→10
Cups syrup	→25

Each row preserves the 2:5 ratio.

Step 03 of 04

Tape diagram = visual model. Draw boxes representing the parts of the ratio. For 3:2, draw 3 boxes for the first quantity and 2 boxes for the second. Each box represents the same UNIT amount.

Step 04 of 04

If a tape diagram for 3:2 ratio totals 25, each box represents $25/5 = 5$ . So the parts are $3 \times 5 = 15$ and $2 \times 5 = 10$ .

Key insight

Equivalent ratios = same ratio scaled up or down. Tables and tape diagrams help compute and visualize. Each "box" in a tape diagram is one UNIT of the ratio.

6.A.3 — Unit rates.

Step 01 of 04

A rate is a ratio comparing two DIFFERENT units. A unit rate = "per ONE."

unit rate = \frac{total quantity}{number of units}

Step 02 of 04

Examples.

$15 for 3 lbs of apples	→$5 per lb
180 miles in 3 hours	→60 miles per hour
12 cookies for 4 friends	→3 cookies per friend

Step 03 of 04

Compare rates by converting both to "per one."

Brand A: $9 for 3 lbs = $3/lb. Brand B: $15 for 4 lbs = $3.75/lb. Brand A is cheaper per pound.

Step 04 of 04

Speed. Distance / time = speed. 240 km in 4 hours → 60 km/h. Reverse: at 60 km/h for 6 hours → 360 km.

Key insight

Unit rate = total ÷ count. Useful for comparing prices, speeds, anything "per one." Used as a constant multiplier to scale up or down.

6.A.4 — Ratio word problems.

Step 01 of 04

Identify the ratio, set up an equivalent ratio with one missing piece, solve.

Step 02 of 04

Worked example. A recipe uses 2 cups flour for every 5 cups water. How much water for 7 cups flour?

Ratio 2:5. Flour scales 2 → 7 = ×3.5. Water also scales: 5 × 3.5 = 17.5 cups.

Step 03 of 04

Cross-multiplication shortcut.

\frac{2}{5} = \frac{7}{x} \Rightarrow 2 x = 35 \Rightarrow x = 17.5

Step 04 of 04

Tape diagram example. A class has boys and girls in a 4:3 ratio. Total 28 students. How many boys?

4 + 3 = 7 parts total. Each part = 28/7 = 4. Boys = 4 × 4 = 16.

Key insight

Set up a proportion (or ratio table). Cross-multiply or scale. For "split a total in ratio X:Y," divide total by (X+Y) to get one "part," then multiply.

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