Question

In Citizen of the Galaxy by Robert Heinlein, Thorby learns that he owns “an interest in a company . . . through a chain of six companies—18% of 31% of 43% of 19% of 44% of 27%, a share so microscopic that he lost track.” Let x represent the total worth of the company in which Thorby owns a share. Suppose Thorby's share of the company is worth 1000 credits. Write and solve an equation to find the total worth of the company.

Short answer

The total worth of the company is approximately 17953197.48 credits.

Step-by-step solution

Understanding the Problem

If the percentages are denoted by their decimal equivalents, (18% as 0.18, 31% as 0.31, 43% as 0.43, 19% as 0.19, 44% as 0.44 and 27% as 0.27), the fraction of the total worth of the company that Thorby owns can be written as \(0.18 \times 0.31 \times 0.43 \times 0.19 \times 0.44 \times 0.27 = 0.0000557\). This means that Thorby owns \(0.0000557x\) of the company.

Formulate the Equation

If Thorby's share of the company is worth 1000 credits, we can set our equation as \(0.0000557x=1000\). This represents the total worth of the company based on the value of Thorby's shares.

Solve the Equation

Now we can solve the equation for \(x\). Divide both sides of the equation by \(0.0000557\). \(x= \frac{1000}{0.0000557} = 17953197.48 \) approximated. This means the total worth of the company is approximately 17953197.48 credits.

Key concepts

Solving Equations

When it comes to solving equations, it’s essential to understand that an equation represents a balance between two expressions. In the context of percentage word problems, the equation is a way to quantify relationships. In Thorby's scenario from 'Citizen of the Galaxy', we form an equation to represent the proportion of the company he owns. Setting up the equation correctly is crucial to finding the solution.

Starting with identifying Thorby’s ownership as a product of percentages, we translate these into decimals to simplify calculations. The equation \(0.0000557x = 1000\) is then created, with \(x\) representing the total worth. Here, the aim is to isolate \(x\) to find its value by performing balance-maintaining operations – in this case, division. This step is an application of the fundamental principle of algebra that what you do to one side of the equation, you must do to the other. Solving for \(x\) gives us the total worth in credits.

Percentages in Algebra

The concept of percentages in algebra often involves interpreting a percentage as a part of a whole in equation form. A percentage can be thought of as a fraction with a denominator of 100, which is particularly useful when it needs to be used in calculations.

For instance, 18% represents 18 out of 100, or \(\frac{18}{100}\) which simplifies to 0.18. In algebra, this can help with creating equations that involve percentages by converting them into decimal form, facilitating multiplication and division. In our example, compounding percentages (percentages of percentages) can be a bit tricky, but converting to decimals makes it much easier to find Thorby’s fractional share by a simple multiplication of the decimal equivalents. Understanding how to manipulate these figures is fundamental in algebraic expressions and equations involving percentages.

Decimal Representation of Percentages

Understanding the decimal representation of percentages is a key skill in both algebra and everyday mathematics. Converting a percentage to a decimal involves dividing by 100. This is because 'percent' means 'per hundred.' So, 27% converts to 0.27, and this indicates 27 parts out of 100.

When dealing with word problems, converting percentages to decimals not only simplifies multiplication and division but also helps prevent common mistakes that can occur when manipulating percentages. It's easier to multiply or divide with decimals than with actual percentages because they are in a form readily accepted in mathematical operations. In our textbook problem, each of the six percentages is converted into decimal form before being multiplied together, showcasing the practical application of this concept.

Here’s a tip: always remember to move the decimal point two places to the left to convert a percentage to a decimal. For instance, converting 44% to decimal gives us 0.44, which then fits neatly into algebraic equations and solutions.