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Chapter 7: Problem 7

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

Short Answer

Expert verified
The simplified form of the ratio (x to x) when x=12 is '1 to 1'.

Step by step solution

01

Identify the ratio

The ratio mentioned in the exercise is 'x to x', which here is 12 to 12.
02

Simplify the ratio

A ratio of '12 to 12' can be simplified to '1 to 1', as essentially the same value is being compared to itself.

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Key Concepts

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

Algebra
Algebra is all about finding relationships between numbers and solving equations that involve variables. In this case, the variables are represented by letters, such as \(x\), \(y\), and \(z\).

When dealing with the variables, algebra requires understanding how to manipulate them to solve problems, such as the one involving ratios here. You are often looking to express relationships in a simplified manner. This simplification helps make sense of the variables and their values without adding unnecessary complexity.
Simplifying Ratios
Ratios are comparisons between two numbers, describing how much of one exists in relation to the other. Simplifying a ratio means reducing it to its smallest form, where the numbers no longer have any common factors, except for 1.

In our exercise, the ratio \( x \text{ to } x \) is given as \( 12 \text{ to } 12 \). Since both numbers are the same, the simplest form of this ratio is \( 1 \text{ to } 1 \). This is achieved by dividing both terms of the ratio by their greatest common divisor. In this scenario, it's simply the number itself, because \( 12 \div 12 = 1 \).
  • Always look for common factors between the numbers in a ratio to simplify them.
  • Simplified ratios are easier to understand and compare.
  • Identifying the relationship between terms becomes clearer with simplified ratios.
Mathematical Problem-Solving
Mathematical problem-solving is a methodical way of approaching and finding solutions to problems using structured techniques.

In the context of the exercise, the problem-solving process involves:
  • **Identifying the Problem**: Recognizing that we need to simplify the given ratio \( x \text{ to } x \).
  • **Breaking Down the Problem**: Understanding what the ratio represents and how it should be simplified.
  • **Executing the Solution**: Simplifying the ratio by finding the greatest common divisor and applying it to reduce the ratio.
By following these steps, problem-solving becomes a less daunting task, allowing for clear and logical conclusions. This exercise illustrates the idea of taking a step-by-step approach to find the simplest form of a mathematical expression.

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Most popular questions from this chapter

Show that the given proportions are equivalent. $$\frac{a-b}{a+b}=\frac{c-d}{c+d} \text { and } \frac{a}{b}=\frac{c}{d}$$

Find the value of \(x\). $$\frac{4}{x}=\frac{2}{5}$$

Refer to a triangle. Express the ratio of the height to the base in simplest form. $$\begin{array}{|l|c|l|l|l|l|l|} \hline \text { height } & 1 \mathrm{m} \\ \hline \text { base } & 85 \mathrm{cm} \\ \hline \end{array}$$

Find the value of \(x\). $$\frac{x+3}{2}=\frac{4}{3}$$

Find the ratio of \(x\) to \(y:\) $$\begin{aligned} &\frac{4}{y}+\frac{3}{x}=44\\\ &\frac{12}{y}-\frac{2}{x}=44 \end{aligned}$$
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