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Chapter 4: Problem 8

In Exercises \(1-10,\) write each of the phrases as a ratio. \(3 x\) to \(x\)

Short Answer

Expert verified
The ratio is 3:1.

Step by step solution

01

Understand the problem

The exercise requires converting the phrase '3x to x' into a ratio format.
02

Identify the components

Recognize the parts of the phrase: '3x' is the first term, and 'x' is the second term.
03

Write the ratio

Express the ratio of '3x' to 'x' as a fraction: \[ \frac{3x}{x} \]
04

Simplify the ratio

Simplify the fraction by canceling out the common term 'x': \[ \frac{3x}{x} = 3 \]
05

Write the final ratio

Express the simplified result as a ratio: 3:1

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Key Concepts

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

Writing Ratios
To begin with, let's explore what a ratio is. A ratio compares two quantities showing the relative size of one quantity to another. When an exercise asks you to 'write each phrase as a ratio,' it usually provides you with two elements to compare.
In our example, the phrase is '3x to x'. To write this as a ratio, we first identify both parts of the phrase: '3x' and 'x'. Once identified, we can express this relationship in different forms like '3x:x' or \[ \frac{3x}{x} \]. Thus, a ratio can be written as a fraction, using a colon, or even with the word 'to.'
Simplifying Fractions
Simplifying fractions is another essential algebra concept. It involves reducing the fraction to its simplest form. This makes it easier to understand and work with.
To simplify a fraction \[ \frac{3x}{x} \], we cancel any common factors in the numerator and denominator. In our example, the common factor is 'x'. By canceling 'x' in both the numerator and denominator, we get \[ \frac{3x}{x} = 3 \].
Lastly, this means our final ratio is simplified to just 3, often represented as 3:1 in the ratio format.
Algebraic Expressions
Algebraic expressions are at the core of many mathematical problems, providing a way to represent numbers and variables in mathematical operations.
In the provided exercise, '3x' and 'x' are algebraic expressions. An algebraic expression contains numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. For instance, '3x' means '3 times x.' Similarly, 'x' represents a variable or unknown number.
Understanding algebraic expressions helps to simplify ratios and solve equations accurately. By mastering these concepts, you get better equipped to handle more complex algebra problems in future exercises.

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Most popular questions from this chapter

Group the equivalent fractions together: (a) \(-\frac{3}{4}\) (b) \(\frac{-3}{-4}\) (c) \(\frac{3}{-4}\) (d) \(-\frac{-3}{-4}\) (e) \(\frac{-3}{4}\) (f) \(-\frac{-3}{4}\) (g) \(-\frac{3}{-4}\)

solve the equations and inequalities. \(0.05 x+0.08(x+3,000)=890\)

Solve the equations and inequalities. $$\frac{w+3}{4}+1=\frac{w+4}{3}$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. John and Susan leave their homes at 8: 00 A.M., going toward each other along the same route, which is 9 miles long. John walks at 4 mph while Susan jogs at 8 mph. At what time will they meet?

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{2}{5 x y}+\frac{1}{10 x^{2}}$$
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