Simplifying Ratios and Comparing Quantities

Introduction

Key Concepts

What are Ratios?

A ratio is a relationship between two numbers indicating how many times the first number contains the second. It compares quantities of the same kind and is expressed in the form A:B or A/B, where A and B are the quantities being compared.

Simplifying Ratios

Simplifying ratios involves reducing them to their simplest form by dividing both terms by their greatest common divisor (GCD). Simplification makes ratios easier to understand and compare.

Example: Simplify the ratio 20:60.

1. Find the GCD of 20 and 60, which is 20.
2. Divide both terms by 20: $\frac{20}{20} : \frac{60}{20}$.
3. The simplified ratio is 1:3.

Equivalent Ratios

Equivalent ratios are different pairs of numbers that express the same relationship. They are obtained by multiplying or dividing both terms of the ratio by the same non-zero number.

Example: The ratios 2:4, 3:6, and 4:8 are equivalent because they all simplify to 1:2.

Comparing Ratios

Comparing ratios involves determining whether two ratios are equivalent, or understanding which ratio is greater or smaller based on their simplified forms.

Methods:

• Simplification: Simplify both ratios and compare their simplest forms.
• Cross Multiplication: Multiply the means by the extremes and compare the products.

Example: Compare the ratios 3:4 and 6:8.

1. Simplify both ratios:
   • 3:4 (already simplified)
   • 6:8 simplifies to 3:4
2. Since both simplify to 3:4, the ratios are equivalent.

Applications of Ratios in Real Life

Ratios are used in various real-life scenarios, including:

• Cooking: Adjusting recipe ingredients based on the number of servings.
• Finance: Comparing price-to-earnings ratios in stock analysis.
• Geometry: Understanding scale ratios in maps and models.

Understanding Proportions

A proportion states that two ratios are equal. It is an equation that shows the equality of two ratios, typically written as A:B = C:D or $\frac{A}{B} = \frac{C}{D}$.

Solving Proportions: To find an unknown value in a proportion, use cross multiplication:

Example: Solve for x in the proportion 2:3 = x:9.

1. Set up the proportion: $\frac{2}{3} = \frac{x}{9}$.
2. Cross multiply: $2 \times 9 = 3 \times x$.
3. Solve for x: $18 = 3x \Rightarrow x = 6$.

Percentages and Ratios

Percentages are a way of expressing ratios as parts per hundred. Understanding the relationship between percentages and ratios is crucial for converting and comparing different forms of numerical relationships.

Conversion: To convert a ratio to a percentage, divide the first term by the second and multiply by 100.

Example: Convert the ratio 1:4 to a percentage.

1. Divide 1 by 4: $\frac{1}{4} = 0.25$.
2. Multiply by 100: $0.25 \times 100 = 25\%$.

Scaling Ratios

Scaling ratios involves increasing or decreasing a ratio by multiplying or dividing both terms by the same number. This is useful in various applications such as resizing models or adjusting quantities in recipes.

Example: Scale the ratio 3:5 by a factor of 2.

1. Multiply both terms by 2: $3 \times 2 = 6$ and $5 \times 2 = 10$.
2. The scaled ratio is 6:10, which simplifies to 3:5.

Ratio Units

Ratios can involve different units, and it's essential to ensure consistency when comparing or simplifying them.

Example: Comparing miles to hours (speed) versus miles to gallons (fuel efficiency).

Types of Ratios

• Part-to-Part Ratio: Compares one part of a whole to another part.
• Part-to-Whole Ratio: Compares one part to the entire whole.

Common Mistakes in Simplifying Ratios

• Failing to identify the correct greatest common divisor.
• Incorrectly simplifying only one term of the ratio.
• Misinterpreting the order of terms, leading to incorrect comparisons.

Exercises for Practice

To solidify understanding, students should practice simplifying ratios and comparing quantities through various exercises.

Exercise 1: Simplify the ratio 45:60.

1. Find GCD of 45 and 60, which is 15.
2. Divide both terms by 15: $\frac{45}{15} : \frac{60}{15}$.
3. Simplified ratio is 3:4.

Exercise 2: Compare the ratios 5:8 and 10:16.

1. Simplify both ratios:
   • 5:8 (already simplified)
   • 10:16 simplifies to 5:8
2. Since both simplify to 5:8, the ratios are equivalent.

Exercise 3: Determine if the proportion 7:14 = 2:4 is true.

1. Simplify both ratios:
   • 7:14 simplifies to 1:2
   • 2:4 simplifies to 1:2
2. Since both simplify to 1:2, the proportion is true.

Comparison Table

Summary and Key Takeaways