Ratios and Rates

A ratio compares two quantities by division. Ratios appear everywhere — recipes, building plans, medication dosages, and sports statistics. Understanding ratios is your first step toward solving proportion problems, working with scale drawings, and handling real-world calculations in trades and healthcare.

What Is a Ratio?

A ratio is a comparison of two quantities that tells you how much of one thing there is relative to another. If a classroom has 12 boys and 18 girls, the ratio of boys to girls is 12 to 18.

There are three common ways to write a ratio:

All three mean the same thing. The first number is compared to the second, and order matters. The ratio of boys to girls ($12 : 18$) is different from the ratio of girls to boys ($18 : 12$).

Part-to-Part vs. Part-to-Whole

Ratios come in two flavors:

• Part-to-part compares one part of a group to another part. Example: 12 boys to 18 girls.
• Part-to-whole compares one part to the total. Example: 12 boys out of 30 total students, written as $12 : 30$.

A part-to-whole ratio can also be written as a fraction or percent. The ratio $12 : 30$ equals $\frac{12}{30} = \frac{2}{5} = 0.4$, which is $40$%.

Simplifying Ratios

Simplifying a ratio works exactly like simplifying a fraction — divide both numbers by their greatest common factor (GCF).

Example 1: Simplify the ratio 12 : 18

Step 1: Find the GCF of 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The GCF is 6.

Step 2: Divide both parts by 6.

$12 \div 6 = 2 \qquad 18 \div 6 = 3$

Answer: The simplified ratio is $2 : 3$.

Example 2: Simplify the ratio 45 : 60

Step 1: The GCF of 45 and 60 is 15.

Step 2: Divide both parts by 15.

$45 \div 15 = 3 \qquad 60 \div 15 = 4$

Answer: The simplified ratio is $3 : 4$.

Ratios with Three or More Parts

Some ratios compare three quantities. A concrete mix might use cement, sand, and gravel in the ratio $1 : 2 : 3$. To simplify, divide all parts by the GCF of all three numbers.

Example 3: Simplify 8 : 12 : 20

The GCF of 8, 12, and 20 is 4.

$8 \div 4 = 2 \qquad 12 \div 4 = 3 \qquad 20 \div 4 = 5$

Answer: The simplified ratio is $2 : 3 : 5$.

What Is a Rate?

A rate is a special type of ratio that compares two quantities measured in different units. When you say a car travels 180 miles in 3 hours, that is a rate:

$\frac{180 \text{ miles}}{3 \text{ hours}}$

Common rates include:

• Speed: miles per hour, kilometers per hour
• Price: dollars per pound, cents per ounce
• Heart rate: beats per minute
• Pay rate: dollars per hour

Ratio vs. Rate: The Key Difference

If both quantities have the same unit, you have a ratio. If the units are different, you have a rate.

Example 4: Classify each comparison

• 5 red marbles to 8 blue marbles — Ratio (both are marbles, same unit)
• 240 miles in 4 hours — Rate (miles and hours are different units)
• 3 nurses for every 12 patients — Rate (nurses and patients are different units)

Real-World Application: Nursing — Medication Mixing

A nurse needs to prepare a diluted medication. The protocol calls for a medication-to-saline ratio of $1 : 250$ (1 mL of medication concentrate for every 250 mL of saline).

Question: If the nurse uses 1,000 mL of saline, how much medication concentrate is needed?

Step 1: Write the ratio as a fraction.

$\frac{1 \text{ mL concentrate}}{250 \text{ mL saline}}$

Step 2: Set up an equivalent ratio for 1,000 mL of saline.

$\frac{1}{250} = \frac{x}{1000}$

Step 3: Since $1000 \div 250 = 4$, multiply the numerator by 4.

$x = 1 \times 4 = 4$

Answer: The nurse needs 4 mL of medication concentrate for 1,000 mL of saline.

Real-World Application: Carpentry — Wood Stain Mixing

A carpenter mixes wood stain with thinner in a ratio of $3 : 1$ (3 parts stain to 1 part thinner).

Question: If the carpenter needs 2 quarts of mixture total, how much of each component is needed?

Step 1: The total number of parts is $3 + 1 = 4$.

Step 2: Each part equals $\frac{2}{4} = 0.5$ quarts.

Step 3: Calculate each component.

$\text{Stain} = 3 \times 0.5 = 1.5 \text{ quarts}$

$\text{Thinner} = 1 \times 0.5 = 0.5 \text{ quarts}$

Answer: The carpenter needs 1.5 quarts of stain and 0.5 quarts of thinner.

Common Mistakes to Avoid

1. Mixing up the order. The ratio of boys to girls is not the same as girls to boys. Always write the quantities in the order the problem specifies.

2. Forgetting to simplify. A ratio of $15 : 25$ is correct, but $3 : 5$ is cleaner and easier to work with.

3. Confusing ratios with fractions. A part-to-part ratio of $3 : 5$ does not mean $\frac{3}{5}$ of the whole. It means 3 out of every $3 + 5 = 8$ total parts, so the fraction of the whole is $\frac{3}{8}$.

4. Using different units in a ratio without converting. If comparing 2 feet to 8 inches, first convert to the same unit: 24 inches to 8 inches, which simplifies to $3 : 1$.

Practice Problems

Key Takeaways

• A ratio compares two quantities in the same units; a rate compares quantities in different units.
• Ratios can be written with a colon, as a fraction, or with the word “to.”
• Order matters — always match the order stated in the problem.
• Simplify ratios by dividing all parts by their greatest common factor.
• Part-to-part ratios are different from part-to-whole ratios — be clear about which one a problem is asking for.

Return to Pre-Algebra for more topics in this section.