- Translate the verbal ratio "x is to y" as \(\frac{x}{y}\).
- Form the two ratios from the given text: \(\frac{3 \text{ printers}}{27 \text{ computers}}\) and \(\frac{1 \text{ printer}}{9 \text{ computers}}\).
- Equate the ratios to form the proportion: \(\frac{3 \text{ printers}}{27 \text{ computers}} = \frac{1 \text{ printer}}{9 \text{ computers}}\).
- Select the matching option, which is \(\boxed{\text{B}}\).
Explanation
- Analyzing the Problem Statement
The problem asks us to express the verbal statement "3 printers is to 27 computers as 1 printer is to 9 computers" as a mathematical proportion. A proportion is a statement that two ratios are equal, which is typically written in the form \(\frac{a}{b} = \frac{c}{d}\). In this format, "a is to b as c is to d" directly maps to the equation \(\frac{a}{b} = \frac{c}{d}\).
- Setting up the Proportion
We identify the components of the statement:
- The first ratio is "3 printers is to 27 computers", which translates to \(\frac{3 \text{ printers}}{27 \text{ computers}}\).
- The second ratio is "1 printer is to 9 computers", which translates to \(\frac{1 \text{ printer}}{9 \text{ computers}}\).
By setting these two ratios equal to each other, we obtain the proportion:
\[\frac{3 \text{ printers}}{27 \text{ computers}} = \frac{1 \text{ printer}}{9 \text{ computers}}\]
- Identifying the Correct Option
Now we compare our derived proportion with the given options:
- Option A: \(\frac{9 \text{ computers}}{1 \text{ printer}} = \frac{3 \text{ printers}}{27 \text{ computers}}\)
- Option B: \(\frac{3 \text{ printers}}{27 \text{ computers}} = \frac{1 \text{ printer}}{9 \text{ computers}}\)
- Option C: \(\frac{3 \text{ printers}}{9 \text{ computers}} = \frac{1 \text{ printer}}{27 \text{ computers}}\)
The proportion we derived matches Option B perfectly. We can also verify that both ratios are equivalent to \(\frac{1}{9}\). Since \(\frac{3}{27} = \frac{1}{9}\) and \(\frac{1}{9} = \frac{1}{9}\), the statement is mathematically sound. Thus, option B is the correct representation.
Examples
Proportions are fundamental in everyday decision-making, such as scaling recipes or calculating unit prices at the grocery store. For example, if you know that 3 apples cost \$2, you can use a proportion to determine the cost of 12 apples by setting up the ratio \(\frac{3 \text{ apples}}{\$2} = \frac{12 \text{ apples}}{\$x}\) and solving for \(x\). Understanding how to set up these ratios allows for efficient comparison and resource management in various life scenarios.