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Chapter 7: Problem 9

\(x=12, y=10,\) and \(z=24 .\) Write each ratio in simplest form. $$\frac{x}{x+z}$$

Short Answer

Expert verified
The simplified form of the ratio \(\frac{x}{x+z}\) when x equals 12 and z equals 24 is \(\frac{1}{3}\).

Step by step solution

01

Identification

Identify the given values and the ratio. Here, \(x=12, y=10, z=24\) and the ratio is \(\frac{x}{x+z}\).
02

Substitution

Substitute the given values into the ratio. It gives us \(\frac{12}{12+24}\) or \(\frac{12}{36}\).
03

Simplification

Simplify the ratio \(\frac{12}{36}\) by dividing the numerator and the denominator by their GCD which is 12. The simplified ratio is \(\frac{12/12}{36/12}\) = \(\frac{1}{3}\).

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Key Concepts

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

Simplifying Fractions
Simplifying fractions is a common mathematical task. The goal is to make the fraction as "simple" as possible by reducing it to its smallest form. This often means finding a more manageable equivalent that expresses the same value. In this exercise, we start with the fraction \(\frac{12}{36}\). To simplify, we need to find the largest number that divides both the numerator and the denominator smoothly. This process usually begins once you have replaced any variables with their values.
  • Identifying the common factors shared by the numerator and denominator is key.
  • Dividing both by their greatest common divisor results in the simplest form of the fraction.
Once simplified, the fraction \(\frac{12}{36}\) turns into \(\frac{1}{3}\). Now, it is much more straightforward and easier to interpret or use in further calculations.
Greatest Common Divisor
The Greatest Common Divisor (GCD) is crucial for simplifying fractions. It is the largest number that can divide both the numerator and the denominator without leaving a remainder. This is why it's also known as the Greatest Common Factor (GCF). Determining the GCD of two numbers lets you reduce a fraction to the simplest form, eliminating any complexity.
  • Prime factorization is a common method: break down both numbers into prime factors and multiply the smallest powers of common primes together.
  • An alternative is using the Euclidean algorithm, a systematic approach involving division until a remainder of zero is reached.
In our example, the GCD of 12 and 36 is 12. By dividing both parts of the fraction by this number, the ratio reduces from \(\frac{12}{36}\) to \(\frac{1}{3}\), showcasing the simplicity that GCD brings to fractions.
Substitution Method
The substitution method is a helpful technique for dealing with algebraic expressions or ratios. By replacing variables with their known values, you convert theoretical expressions into concrete numbers. This transition can turn complex ratios into straightforward division problems.
  • This practice involves substituting each variable in the expression with its given numerical value.
  • Once substitution is complete, actual calculations can proceed, effectively transforming the problem.
For the ratio \(\frac{x}{x+z}\), substituting \(x=12\) and \(z=24\) converts it to \(\frac{12}{12+24}\) or \(\frac{12}{36}\). Post-substitution, simplifying follows, using methods like finding the GCD as shown earlier. Thus, the substitution method is a bridge connecting variable expressions to simpler, comprehensible calculations.

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