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Chapter 4: Problem 7

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

Short Answer

Expert verified
The ratio is \(1:3\).

Step by step solution

01

Identify the Terms

The given terms in the ratio are expressed as variables. The first term is \(x\) and the second term is \(3x\).
02

Write the Ratio

A ratio can be expressed as 'a to b', 'a:b', or as a fraction \(\frac{a}{b}\). In this case, we write the ratio of \(x\) to \(3x\) as \(\frac{x}{3x}\).
03

Simplify the Ratio

Simplify the fraction \(\frac{x}{3x}\) by canceling out the common factor \(x\) from the numerator and the denominator, which leaves \(\frac{1}{3}\).

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Key Concepts

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

Variables
A variable is a symbol, often a letter, that represents a number that can change or vary. In mathematics, variables allow us to write expressions and equations that generalize problems and solutions. For instance, in our original exercise, the variable is given as 'x'. This 'x' can stand for any number.

Variables help us see patterns and relationships in math. They make it easier to solve problems by letting us work with general terms instead of specific numbers. In our exercise, we have two terms involving the same variable: one is 'x' and the other is '3x'. Identifying variables in terms like these is important for setting up ratios correctly.

So, remember: a variable is just a placeholder for numbers, which makes it easier to see how things change and work together.
Simplification
Simplification in mathematics means making an expression or ratio easier to work with. It often involves reducing fractions or canceling out common terms.

In our exercise, we started with the ratio \(\frac{x}{3x}\). To simplify this, we notice that both the numerator and the denominator have a common factor which is 'x'. By canceling out this common variable, we simplify the ratio to \(\frac{1}{3}\).

This simplification makes the ratio straightforward and easy to understand. Simplifying ratios and fractions can greatly help in understanding the core relationship between two terms.
Fraction
A fraction represents a part of a whole. It consists of a numerator (the top part) and a denominator (the bottom part). In our exercise, the ratio of 'x' to '3x' was written as the fraction \(\frac{x}{3x}\).

Fractions can often be simplified by dividing both the numerator and the denominator by their greatest common factor. In our case, both the numerator and the denominator were divisible by 'x', simplifying the fraction to \(\frac{1}{3}\). Understanding how to work with fractions is essential in solving many mathematical problems, especially in ratios and proportions.

Fractions are a fundamental concept that show up in many areas of math, making it important to practice and get comfortable with them.

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Most popular questions from this chapter

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. How much pure antifreeze should be added to a radiator that contains 10 gallons of a \(30 \%\) antifreeze solution to produce a \(50 \%\) antifreeze solution? [Hint: Pure antifreeze is \(100 \% \text { antifreeze. }]\)

Solve the equations and inequalities. $$z+\frac{z+5}{3}-\frac{z-2}{6}=\frac{z+4}{4}+1$$

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. How much water should be added to a radiator that contains 10 gallons of an \(80 \%\) antifreeze solution to dilute it to a \(50 \%\) antifreeze solution? [Hint: Pure water is \(0 \% \text { antifreeze. }]\)

Solve the equations and inequalities. $$\frac{x+1}{2}+x=6$$

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. If the temperature range on a certain day was between \(25^{\circ} \mathrm{C}\) and \(40^{\circ} \mathrm{C},\) what was the temperature range in degrees Fahrenheit? Use the formula \(C=\frac{5}{9}(F-32)\)
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