How To Do Equivalent Ratios | Mastering Proportions

Understanding equivalent ratios is a foundational concept in mathematics, crucial for everything from scaling recipes to interpreting scientific data. It’s about recognizing when different expressions describe the same relative amounts, a skill that deepens our numerical fluency and problem-solving abilities.

What a Ratio Truly Is

A ratio is a mathematical comparison of two or more quantities. It shows how much of one quantity exists in relation to another quantity. Ratios provide a relative measure, not an absolute one.

For example, a ratio of 1:2 describes a relationship where for every one unit of the first quantity, there are two units of the second quantity. This relationship holds true regardless of the actual size of the units.

Components of a Ratio

Ratios consist of two or more numbers, called terms, separated by a colon (:), the word “to,” or a fraction bar (/).

• In the ratio 3:5, 3 is the first term and 5 is the second term.
• The order of the terms is significant. A ratio of 3:5 is distinct from a ratio of 5:3.
• The terms of a ratio must represent quantities of the same type or related types, such as apples to oranges, or parts of a mixture.

Expressing Ratios

Ratios can be written in several forms, each conveying the same proportional relationship.

• Colon form: 3:5
• “To” form: 3 to 5
• Fraction form: 3/5

The fraction form highlights the division aspect inherent in ratios, where the first term is divided by the second term. This representation connects ratios directly to fractions, which also represent parts of a whole or comparisons.

The Core Principle of Equivalence

Equivalent ratios share the same proportional value. They are different numerical expressions that represent the identical relative relationship between quantities. Think of it like different ways to write the same fraction, such as 1/2, 2/4, or 5/10; all represent the same amount.

The principle behind finding equivalent ratios relies on the multiplicative identity property. Multiplying or dividing a number by one does not change its value. In the context of ratios, this means multiplying or dividing both terms of a ratio by the same non-zero number maintains the ratio’s underlying proportion.

This operation scales the ratio up or down proportionally. For instance, if a recipe calls for a 1:2 ratio of sugar to flour, doubling the recipe means using 2:4, which is still the same proportional relationship. The Department of Education emphasizes proportional reasoning as a key mathematical skill.

Method 1: Multiplication to Find Equivalent Ratios

Multiplication is a direct method for generating equivalent ratios by scaling up the quantities involved. This process is useful when you need to expand a ratio while preserving its original proportion.

Step-by-Step Multiplication

1. Start with your initial ratio: Express it in any form, such as A:B or A/B.
2. Choose a non-zero whole number: This number will be your common multiplier. It can be any integer greater than zero.
3. Multiply both terms of the ratio by this number: If your ratio is A:B, the new equivalent ratio will be (A multiplier) : (B multiplier).

Example: Find an equivalent ratio for 2:3.

• Choose a multiplier, say 4.
• Multiply the first term: 2 4 = 8.
• Multiply the second term: 3 4 = 12.
• The equivalent ratio is 8:12.

Both 2:3 and 8:12 represent the same proportional relationship. For every 2 units of the first quantity, there are 3 units of the second, and for every 8 units of the first, there are 12 units of the second. This maintains the relative amounts.

Original Ratio	Multiplier	Equivalent Ratio
1:5	3	3:15
4:7	2	8:14
2/9	5	10/45

Method 2: Division to Find Equivalent Ratios (Simplifying Ratios)

Division is used to find equivalent ratios by scaling down the quantities. This method is particularly useful for simplifying ratios to their simplest form, also known as their lowest terms. A ratio is in its simplest form when its terms have no common factors other than 1.

Step-by-Step Division

1. Start with your initial ratio: Express it as A:B or A/B.
2. Find the Greatest Common Divisor (GCD) of both terms: The GCD is the largest number that divides evenly into both A and B.
3. Divide both terms of the ratio by their GCD: If your ratio is A:B, the simplified equivalent ratio will be (A / GCD) : (B / GCD).

Example: Simplify the ratio 18:24.

• Identify the terms: 18 and 24.
• Find the GCD of 18 and 24. Factors of 18 are 1, 2, 3, 6, 9, 18. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The GCD is 6.
• Divide the first term: 18 / 6 = 3.
• Divide the second term: 24 / 6 = 4.
• The equivalent (simplified) ratio is 3:4.

The ratio 3:4 is equivalent to 18:24, representing the same proportional relationship in a more concise form. This simplification aids in easier comparison and understanding of the ratio.

Original Ratio	Divisor (GCD)	Equivalent (Simplified) Ratio
12:18	6	2:3
25:35	5	5:7
40/100	20	2/5

Using Equivalent Ratios in Real-World Contexts

Equivalent ratios are fundamental across various practical applications, extending beyond the classroom into daily life and professional fields. They provide a framework for scaling quantities while maintaining their relative balance.

Consider a chef adapting a recipe. If a recipe calls for 2 cups of flour for every 1 cup of sugar (a 2:1 ratio), and the chef needs to triple the recipe, they use equivalent ratios. The new ratio becomes 6 cups of flour to 3 cups of sugar (6:3), maintaining the original proportion for a successful dish.

In mapping, a scale provides a ratio comparing a distance on the map to the actual distance on the ground. A map scale of 1:10,000 means 1 unit on the map represents 10,000 units in reality. To determine a real distance from a map measurement, one uses equivalent ratios. Khan Academy provides extensive resources on proportional reasoning and its applications.

Mixing solutions in chemistry or gardening also relies on equivalent ratios. If a fertilizer requires a 1:5 ratio of concentrate to water, preparing a larger batch requires multiplying both terms by the same factor to maintain the correct strength.

The Unit Rate Connection

A unit rate is a ratio where the second term is 1. It expresses how much of the first quantity corresponds to one unit of the second quantity. Unit rates are a specific form of equivalent ratio, always simplified to have a denominator of one.

For example, if a car travels 120 miles in 2 hours, the ratio of miles to hours is 120:2. To find the unit rate, divide both terms by 2, resulting in 60:1, or 60 miles per hour. This unit rate simplifies the comparison of speed.

All equivalent ratios share the same unit rate. If you have ratios 2:4, 4:8, and 10:20, dividing each by its second term (or simplifying to a common denominator of 1) reveals a consistent unit rate of 0.5:1, or 0.5. This consistency confirms their equivalence.

Calculating the unit rate offers a quick way to compare different ratios. A product costing $5 for 2 units (5:2) has a unit rate of $2.50 per unit. A product costing $7 for 3 units (7:3) has a unit rate of approximately $2.33 per unit. The lower unit rate indicates a better value.

Verifying Equivalence: The Cross-Multiplication Test

When you have two ratios and need to determine if they are equivalent, the cross-multiplication test provides a reliable method. This test is particularly useful when the ratios are expressed in fraction form.

Consider two ratios, A/B and C/D. These ratios are equivalent if and only if their cross-products are equal. The cross-products are found by multiplying the numerator of one ratio by the denominator of the other.

• Multiply A by D (A D).
• Multiply B by C (B C).
• If (A D) equals (B C), then the ratios A/B and C/D are equivalent.

Example: Are the ratios 2/3 and 8/12 equivalent?

• Cross-multiply: 2 12 = 24.
• Cross-multiply: 3 8 = 24.
• Since 24 = 24, the ratios 2/3 and 8/12 are equivalent.

This method works because it essentially clears the denominators, allowing a direct comparison of the proportional values. It is an algebraic manipulation that confirms whether the underlying relationships are identical. The cross-multiplication test applies to any two ratios, regardless of their complexity, providing a definitive check for equivalence.