- Identify the number of students in the 100-90 range as 32 and in the 89-80 range as 65 from the table.
- Formulate the ratio as \(32:65\).
- Verify that 32 and 65 have no common factors, confirming the ratio is in its simplest form.
- State the final ratio as \(\boxed{32:65}\).
Explanation
- Identifying Data from the Table
To find the part-to-part ratio of students who earned an average in the '100-90' range to those who earned an average in the '89-80' range, we first extract the relevant data from the provided table. The table indicates that there are 32 students in the '100-90' range and 65 students in the '89-80' range. We denote the number of students in the first range as \(N_1 = 32\) and in the second range as \(N_2 = 65\).
- Constructing and Simplifying the Ratio
A part-to-part ratio compares two quantities and is expressed in the form \(N_1:N_2\). Substituting our values, we get the ratio \(32:65\). To determine if this ratio can be simplified, we check for common factors between 32 and 65. The prime factorization of 32 is \(2^5\), and the prime factorization of 65 is \(5 \times 13\). Since they share no common prime factors, the greatest common divisor is 1, meaning the ratio is already in its simplest form.
- Final Ratio Calculation
The ratio of students in the '100-90' average range to students in the '89-80' average range is \(32:65\). This means that for every 32 students in the top range, there are 65 students in the second range. Thus, the final ratio is \(\boxed{32:65}\).
Examples
Ratios are frequently used in everyday situations, such as cooking, where you might follow a recipe that requires a specific ratio of ingredients (e.g., a 2:1 ratio of water to rice). In this problem, the ratio helps us compare the size of two student groups, which is a fundamental skill in statistics for understanding the distribution of data points across different categories. This same technique is used by teachers to understand grade distributions or by companies to analyze sales performance across different regions.