Ratios and Unit Rates
Write ratios in multiple forms (a:b, a/b, "a to b") and simplify them. Define unit rate as a ratio with denominator 1 and apply to speed, price per unit, and other real-world comparisons.
A lesson of Pre-Algebra, in Ratios, Rates & Proportional Reasoning, about 10 minutes.
By the end of this lesson, you will be able to write ratios in multiple forms, find unit rates, and use unit rates to solve real-world comparison problems.
What Is a Ratio?
A ratio compares two quantities. It can be written three ways:
- Fraction form: $\dfrac{3}{4}$
- Colon form: $3:4$
- Word form: 3 to 4
Example: A bag has 6 red marbles and 9 blue marbles.
- Red to blue: $6:9 = 2:3$ (simplified)
- Blue to red: $9:6 = 3:2$
- Red to total: $6:15 = 2:5$
Ratios compare parts to parts or parts to wholes. Always identify which comparison is being made.
Simplifying Ratios
Simplify by dividing both terms by their GCF.
$18:24 \to \frac{18}{24} \to$ GCF = 6 $\to 3:4$
Unit Rates
A unit rate is a ratio with a denominator of 1 — it tells you "per one unit."
Finding a unit rate: divide both terms so the second term equals 1.
Example: A car travels 240 miles in 4 hours.
$$\text{Unit rate} = \frac{240 \text{ miles}}{4 \text{ hours}} = \frac{60 \text{ miles}}{1 \text{ hour}} = 60 \text{ mph}$$
Example: A 12 oz box of cereal costs $3.60. Find the cost per ounce.
$$\frac{$3.60}{12 \text{ oz}} = \frac{$0.30}{1 \text{ oz}} = $0.30 \text{ per oz}$$
Using Unit Rates to Compare
Unit rates make comparison easy — just compare the per-one values.
Example: Brand A: 15 oz for $4.50; Brand B: 20 oz for $5.80. Which is the better deal?
Brand A: $$4.50 \div 15 = $0.30$/oz
Brand B: $$5.80 \div 20 = $0.29$/oz
Brand B is cheaper per ounce → better deal.
Summary
Ratio: comparison of two quantities in three forms (fraction, colon, words).
Unit rate: rate with denominator 1; found by dividing.
Comparison: convert both to unit rates, then compare the per-one values.