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. Write and simplify an expression to show Thorby's share of the company.

Short answer

The simplified expression for Thorby's share of the company is \((0.18 \times 0.31 \times 0.43 \times 0.19 \times 0.44 \times 0.27) \times x \)

Step-by-step solution

Conversion of percentages to decimals

First, convert all the given percentages into their decimal form. This is done by dividing each percentage by 100. Therefore, 18% becomes 0.18, 31% becomes 0.31, 43% becomes 0.43, 19% becomes 0.19, 44% becomes 0.44 and 27% becomes 0.27.

Multiply the percentages

The next step is to multiply the decimal forms of the percentages together in order to find the net percentage that Thorby owns. Therefore, multiply \(0.18 \times 0.31 \times 0.43 \times 0.19 \times 0.44 \times 0.27 \) together.

Multiply by total worth

Lastly, multiply the resulting decimal by the total worth of the company, represented by the variable \(x\). Therefore, the final expression for Thorby’s share of the company is: \((0.18 \times 0.31 \times 0.43 \times 0.19 \times 0.44 \times 0.27) \times x \)

Key concepts

Percent to Decimal Conversion

When working with percentages in algebra, one of the first skills needed is to convert percentages into decimals. This is a crucial step because it transforms the more abstract concept of a percentage into a more concrete decimal form which can then be used in various mathematical operations. Converting from percent to decimal simply involves dividing the percentage value by 100. For example, 18% becomes 0.18, 31% becomes 0.31, and so on.
To make this process even easier, you can just move the decimal point two places to the left. So for 43%, you would move the decimal point from the end (43.0% since percent means per hundred) to two places left, yielding 0.43. It is essential to remember this conversion method because it allows you to accurately perform subsequent multiplication or division with other decimals or algebraic expressions.

Multiplying Decimals

Once percentages have been converted to decimals, the process of multiplication can be used to combine them, as they often represent parts of a whole in context. Multiplying decimals is similar to multiplying whole numbers, but extra caution is needed to correctly place the decimal point in the final answer.
To multiply decimals, place the numbers vertically and multiply as if they were whole numbers, ignoring the decimal points for a moment. After calculating the product, count the total number of decimal places in all of the factors you've multiplied. The final product should have the same number of decimal places. For instance, when multiplying 0.18 and 0.31, you'll first get 558 from the whole number multiplication. Since there are four decimal places in total in the factors (two in each), the final answer must have four decimal places as well, giving you 0.0558.

Algebraic Expressions

An algebraic expression is like a sentence in the language of mathematics, where numbers, variables, and operation symbols come together to convey a certain mathematical idea. These expressions can represent quantities that vary, which is why we commonly see variables such as 'x' or 'y' in algebra.
In the context of our example with Thorby's share, the algebraic expression embodies his portion of the company's worth. Once the percentage-to-decimal conversion and multiplication are complete, the resulting decimal is combined with the variable representing the total worth of the company. This is done by simply placing the variable alongside the decimal. For example, if Thorby's share is calculated as a decimal, and the total worth of the company is 'x', the expression would be written as the decimal multiplied by 'x'. These expressions are incredibly useful for modeling real-world situations and solving problems symbolically before specific values are known.