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Chapter 7: Problem 14

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

Short Answer

Expert verified
The ratio \(x:(x+y):(y+z)\) in simplest form is \(6:11:17\).

Step by step solution

01

Substitution

Substitute the given values of \(x\), \(y\), and \(z\) into the ratio. This yields the ratio \(12: (12+10): (10+24)\) which simplifies to \(12:22:34\).
02

Find the GCD

The next step is to find the greatest common divisor (GCD) of the three numbers in our ratio. This involves breaking down each number into its prime factors and then finding the highest power of each prime number common to all three numbers. Doing this, we find that the GCD of 12, 22, and 34 is 2.
03

Simplify the Ratio

Divide each number in the ratio by the GCD from step 2. Thus \(12:22:34\) is simplified to \((12/2):(22/2):(34/2)\) which yields the ratio \(6:11:17\) in its simplest form.

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Key Concepts

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

Greatest Common Divisor
The greatest common divisor (GCD) is a key concept when dealing with ratios. It is the largest positive integer that divides each of the numbers in a set without leaving a remainder. To simplify a ratio, finding the GCD is crucial since it helps reduce each number in the ratio by the same amount.

The process of calculating the GCD involves
  • breaking each number down into its prime factors, and
  • identifying the smallest power of all common prime factors.
Let's consider numbers 12, 22, and 34. Breaking them down:
  • 12 = 2虏 脳 3
  • 22 = 2 脳 11
  • 34 = 2 脳 17
All three numbers share the prime factor 2. Since 2 is the only common prime factor, it is the GCD. Therefore, the GCD of the numbers 12, 22, and 34 is 2.

Finding the GCD is essential before simplifying any given ratio.
Prime Factorization
Prime factorization is a method used to express a number as a product of its prime numbers. Understanding this method is vital for simplifying ratios and finding the greatest common divisor.

To perform prime factorization:
  • Start with the smallest prime number, 2, and divide the number by 2 as long as it is divisible.
  • If a number is no longer divisible by 2, move to the next smallest prime, which is 3, and continue.
  • Repeat the process with subsequent primes like 5, 7, etc., until the number is fully decomposed into prime factors.
For example, applying prime factorization to 12:
  • Divide by 2: 12 梅 2 = 6
  • Divide by 2 again: 6 梅 2 = 3
  • 3 is a prime number
Thus, the prime factorization of 12 is 2虏 脳 3. Repeat similar steps for other numbers like 22 and 34, which result in 2 脳 11 and 2 脳 17 respectively.

Prime factorization helps in identifying common prime factors, which are then used to determine the GCD.
Simplifying Ratios
Simplifying ratios involves reducing the ratio to its lowest terms while maintaining the same relationship of the quantities. This process makes the comparison of parts or quantities more manageable and understandable.

Here's how you can simplify a ratio:
  • First, find the GCD of the quantities in the ratio.
  • Next, divide each quantity by the GCD.
  • The resulting numbers form the simplified ratio.
For example, consider the ratio 12:22:34:
  • First, we determined the GCD as 2.
  • Then, divide each number by 2: 12 梅 2 = 6, 22 梅 2 = 11, 34 梅 2 = 17.
Thus, the simplest form of the ratio is 6:11:17.

By simplifying ratios, we ensure clarity and accuracy in conveying the proportionality of the quantities involved.

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