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find an equivalent ratio in simplest terms 88:70

find an equivalent ratio in simplest terms 88:70

2 min read 24-01-2025
find an equivalent ratio in simplest terms 88:70

Finding equivalent ratios is a fundamental skill in mathematics with applications across many fields. This article will guide you through the process, using the ratio 88:70 as a practical example. We'll learn how to simplify ratios to their simplest form and understand the underlying concepts.

What are Equivalent Ratios?

Equivalent ratios represent the same proportional relationship. Think of it like this: if you have a recipe that calls for 2 cups of flour and 1 cup of sugar, you can double the recipe (4 cups flour, 2 cups sugar) or halve it (1 cup flour, ½ cup sugar), and the resulting ratios (2:1, 4:2, 1:½) are all equivalent. They all maintain the same proportional relationship between flour and sugar.

How to Find the Simplest Form of a Ratio (88:70)

The key to finding an equivalent ratio in simplest terms is to find the greatest common divisor (GCD) of the two numbers. The GCD is the largest number that divides both numbers without leaving a remainder.

Here's how to find the simplest form of the ratio 88:70:

  1. Find the factors of each number:

    • Factors of 88: 1, 2, 4, 8, 11, 22, 44, 88
    • Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70
  2. Identify the greatest common divisor (GCD): Looking at both lists, the largest number that appears in both is 2.

  3. Divide both parts of the ratio by the GCD:

    • 88 ÷ 2 = 44
    • 70 ÷ 2 = 35

Therefore, the simplest form of the ratio 88:70 is 44:35.

Alternative Method: Prime Factorization

Another way to find the GCD is through prime factorization.

  1. Prime factorize each number:

    • 88 = 2 x 2 x 2 x 11 (2³ x 11)
    • 70 = 2 x 5 x 7
  2. Identify common prime factors: Both numbers share one factor of 2.

  3. Multiply the common prime factors: The GCD is 2.

  4. Divide both parts of the ratio by the GCD: (as shown above) resulting in 44:35.

Checking Your Work

To ensure your simplified ratio is correct, you can check if the original ratio and the simplified ratio are equivalent by cross-multiplying:

  • Original Ratio: 88 x 35 = 3080
  • Simplified Ratio: 44 x 70 = 3080

Since the cross-products are equal, we've confirmed that 44:35 is the correct simplified form of 88:70.

Practical Applications of Equivalent Ratios

Understanding equivalent ratios is crucial in many real-world scenarios, including:

  • Scaling recipes: Adjusting ingredient amounts while maintaining the correct proportions.
  • Map scales: Interpreting distances on maps.
  • Calculating unit rates: Determining the price per unit of a product.
  • Mixing solutions: Maintaining the correct ratio of ingredients in a chemical solution.

By mastering the techniques outlined above, you can confidently simplify any ratio to its simplest form and apply this knowledge to various practical problems. Remember to always double-check your work by cross-multiplying to ensure accuracy.

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