Ratios and Proportions

A ratio compares two quantities by division. A proportion is a statement that two ratios are equal. Together, they let you scale quantities up or down while keeping relationships consistent — a skill used constantly in cooking, construction, and healthcare.

Writing and Simplifying Ratios

A ratio can be written three ways, all meaning the same thing:

$3 \text{ to } 5 \qquad 3:5 \qquad \frac{3}{5}$

The fraction form is the most useful for calculations because it lets you simplify just like a fraction.

To simplify a ratio, divide both numbers by their greatest common factor (GCF).

Example 1: Simplify the ratio 12:8

Find the GCF of 12 and 8, which is 4.

$\frac{12}{8} = \frac{12 \div 4}{8 \div 4} = \frac{3}{2}$

Answer: The simplified ratio is 3:2.

Example 2: A recipe calls for 6 cups of flour and 4 cups of sugar. What is the ratio of flour to sugar?

$\frac{6}{4} = \frac{6 \div 2}{4 \div 2} = \frac{3}{2}$

Answer: The ratio of flour to sugar is 3:2. For every 3 cups of flour, you need 2 cups of sugar.

Important: The order of a ratio matters. A ratio of flour to sugar (3:2) is different from sugar to flour (2:3).

Setting Up Proportions

A proportion states that two ratios are equal:

$\frac{a}{b} = \frac{c}{d}$

You use proportions when you know three of the four values and need to find the missing one. The key is to set up corresponding quantities in the same positions.

Example 3: If 3 gallons of paint cover 900 square feet, how many gallons do you need for 1,500 square feet?

Set up the proportion with gallons on top and square feet on the bottom:

$\frac{3 \text{ gallons}}{900 \text{ sq ft}} = \frac{x \text{ gallons}}{1{,}500 \text{ sq ft}}$

Solving with Cross-Multiplication

To solve a proportion, cross-multiply — multiply each numerator by the opposite denominator — and then solve for the unknown.

$\frac{a}{b} = \frac{c}{d} \quad \Longrightarrow \quad a \times d = b \times c$

Continuing Example 3:

$3 \times 1{,}500 = 900 \times x$

$4{,}500 = 900x$

$x = \frac{4{,}500}{900} = 5$

Answer: You need 5 gallons of paint for 1,500 square feet.

Example 4: A map scale shows 2 cm = 15 km. If two cities are 9 cm apart on the map, what is the real distance?

$\frac{2 \text{ cm}}{15 \text{ km}} = \frac{9 \text{ cm}}{x \text{ km}}$

Cross-multiply:

$2 \times x = 15 \times 9$

$2x = 135$

$x = \frac{135}{2} = 67.5$

Answer: The real distance is 67.5 km.

Real-World Application: Nursing Dosage Scaling

A doctor orders a medication at a dosage of 5 mg per kilogram of body weight. A patient weighs 72 kg. The medication comes in a liquid form with a concentration of 25 mg per mL. How many mL should the nurse administer?

Step 1: Find the total dosage needed.

$\frac{5 \text{ mg}}{1 \text{ kg}} = \frac{x \text{ mg}}{72 \text{ kg}}$

$x = 5 \times 72 = 360 \text{ mg}$

Step 2: Find the volume of liquid needed.

$\frac{25 \text{ mg}}{1 \text{ mL}} = \frac{360 \text{ mg}}{y \text{ mL}}$

Cross-multiply:

$25 \times y = 360 \times 1$

$y = \frac{360}{25} = 14.4 \text{ mL}$

Answer: The nurse should administer 14.4 mL of the medication.

Common Ratios and Their Uses

Tips for Avoiding Mistakes

1. Keep units consistent. Always put the same type of quantity in the same position (numerator or denominator) on both sides of the proportion.
2. Label everything. Writing units next to each number helps you catch setup errors before you solve.
3. Check your answer. Substitute your answer back into the original proportion and verify both sides are equal.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A ratio compares two quantities; simplify it by dividing both values by their GCF
• A proportion sets two ratios equal and lets you solve for an unknown quantity
• Cross-multiplication ($a \times d = b \times c$) is the standard method for solving proportions
• Always keep the same units in the same position on both sides of a proportion
• Proportions are essential in dosage calculations, recipe scaling, and blueprint reading

Return to Arithmetic for more foundational topics.