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Chapter 7: Problem 8

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

Short Answer

Expert verified
The simplest form of the ratio of \(x + y\) to \(z\) is \(11:12\).

Step by step solution

01

Add x and y

First, add \(x\) and \(y\), which are 12 and 10 respectively. \(x + y = 12 + 10 = 22\).
02

Find the Ratio

Next, find the ratio of \((x + y)\) to \(z\) which is \((22:24)\).
03

Simplify the Ratio

To simplify the ratio, calculate the GCD (greatest common divisor) of 22 and 24. The GCD is 2. Now, divide both numbers by this GCD: \(22/2 and 24/2\), which simplifies to \(11:12\). This is the simplest form of the ratio.

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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 crucial tool in simplifying ratios and fractions. It鈥檚 the largest number that divides two or more numbers without leaving a remainder. For example, when finding the GCD of 22 and 24, you are looking for the biggest number that can evenly divide both numbers.
Here鈥檚 a simple method to find the GCD:
  • List the factors of both numbers.
  • Identify the common factors.
  • Choose the largest factor that appears in both lists.
In the case of 22 and 24, the factors of 22 are 1, 2, 11, and 22. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The common factors are 1 and 2, and the greatest one is 2. This is the GCD. Using the GCD allows for the simplification of numerical expressions, making problems easier to work with and solutions clearer.
Numerical Ratios
Numerical ratios are expressions that compare two quantities. They show how much of one thing exists relative to another and are written in the format of two numbers separated by a colon, like 22:24.
Here are some key points about numerical ratios:
  • A ratio compares parts to parts or parts to a whole.
  • It鈥檚 essential to write ratios in the same units for accurate comparison.
  • Simplifying ratios involves dividing both parts of the ratio by their GCD.
Numerical ratios can represent something tangible, like the ratio of ingredients in a recipe, or something abstract, like time spent on different activities. Understanding how to form and simplify these ratios is crucial in many areas of math and everyday life. In our exercise, simplifying the ratio means turning 22:24 into its simplest form: 11:12.
Simplifying Fractions
Simplifying fractions is about reducing them to their simplest form while retaining their value. It's similar to simplifying numerical ratios since both involve division by a common factor.
To simplify a fraction:
  • Identify the numerator (top number) and denominator (bottom number) of the fraction.
  • Find the GCD of these two numbers.
  • Divide both the numerator and the denominator by this GCD.
By doing this, you shrink the numbers down, which makes the fraction easier to interpret. In the original exercise, the fraction equivalent to the ratio 22:24 is 22/24, which simplifies to 11/12. Thus, each number in the fraction has been divided by 2, the GCD of 22 and 24. Simplifying fractions is vital for making calculations more straightforward and results more understandable in both math problems and real-life scenarios.

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Most popular questions from this chapter

Find the measure of each angle. The measures of the angles of a hexagon are in the ratio 4: 5: 5: 8: 9: 9

\(A B C D\) is a parallelogram. Find the value of each ratio. $$A B: B C$$

The perimeter of a triangle is \(132 \mathrm{cm}\) and the lengths of its sides are in the ratio 8: 11: 14 . Find the length of each side.

The measures of the consecutive angles of a quadrilateral are in the ratio \(5: 7: 11: 13 .\) Find the measure of each angle. draw a quadrilateral that satisfies the requirements, and explain why two sides must be parallel.

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