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Chapter 5: Problem 35

Find the ratio of the first number to the second and simplify. $$ 128 \text { to } 256 $$

Short Answer

Expert verified
The simplified ratio is 1 to 2.

Step by step solution

01

Write the Ratio as a Fraction

Express the given numbers as a fraction. The ratio of 128 to 256 can be written as \ \ \( \frac{128}{256} \) \.
02

Simplify the Fraction

Simplify the fraction by finding the greatest common divisor (GCD) of 128 and 256. The GCD of 128 and 256 is 128. Divide both the numerator and the denominator by their GCD: \ \ \( \frac{128 \div 128}{256 \div 128} = \frac{1}{2} \)
03

Write the Simplified Ratio

Write the simplified fraction as a ratio. The simplified form of \( \frac{1}{2} \) is 1 to 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

finding greatest common divisor
The Greatest Common Divisor (GCD) is a key concept in simplifying ratios and fractions. Often, we're asked to find it to simplify complex fractions. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. For example, in the ratio 128 to 256:
  • First, start with the two numbers: 128 and 256.
  • Find the factors of each number. For 128, the factors are 1, 2, 4, 8, 16, 32, 64, and 128. For 256, the factors are 1, 2, 4, 8, 16, 32, 64, 128, and 256.
  • The GCD is the largest factor they both share. Here, it's 128.
Finding the GCD helps you in reducing fractions. Instead of guessing, you break it down systematically. Practice with other numbers to get a strong grasp of this concept.
fraction simplification
Simplifying fractions means making the numerator and denominator as small as possible while keeping the same value. Here's a simple guide to fraction simplification:
  • Write the fraction. For our example, it's \(\frac{128}{256}\).
  • Determine the GCD of the numerator and denominator.
  • Divide both the numerator and the denominator by the GCD.

For instance, \(\frac{128}{256}\) simplifies as follows:
  • Both 128 and 256 share a GCD of 128.
  • Divide each by 128. So, it becomes \(\frac{128/128}{256/128} = \frac{1}{2}\).

This fraction represents the same value but is simpler and easier to understand. Simplifying fractions is essential in solving many mathematical problems.
ratios and proportions
Ratios are comparisons between two numbers, showing how many times one value contains or is contained within the other. Simplified ratios help make these relationships more understandable. To simplify the ratio:
  • Express the ratio as a fraction. For 128 to 256, it is \(\frac{128}{256}\).
  • Simplify the fraction. From our previous steps, we know \(\frac{128}{256}\) simplifies to \(\frac{1}{2}\).
  • Write the simplified fraction as a ratio. Hence, 128 to 256 simplifies to 1 to 2.

Ratios can be used in various contexts such as scale models, maps, and cooking recipes. Proportions, on the other hand, are equations that express two ratios are equivalent.
For example, if a recipe needs 2 cups of flour for every 1 cup of sugar, and you have 4 cups of flour, you proportionally need 2 cups of sugar. Mastering ratios and proportions will give you valuable skills for solving many practical problems.

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