Problem 4 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Dental Health Milk Fights Cavities At a middle school, students can choose to drink milk or other beverages at lunch. Seventy-five students who chose milk were monitored for one year, as were 75 students who chose other beverages. At the end of the year, students in the "milk" group had \(25 \%\) fewer cavities than students in the other group.

Step-by-Step Solution

Answer

The solution to the problem involves understanding the concept of percentages, creating an equation based on the given information, and then solving it. If the number of cavities in the milk group were known, the number of cavities in the other group would be \(4/3\) times that of the milk group, and conversely, if the number of cavities in the other group were known, the number of cavities in the milk group would be \(3/4\) times that of the other group.

Step 1: Identify the percentage

From given information, it is gathered that the students in the 'milk' group had \(25 \%\) fewer cavities. This will have to be translated into actual number of cavity cases.

Step 2: Calculate the number of cavities in 'other' group

Let's denote the number of cavities in the other group as \(X\). We don't know the value of \(X\), but we know that the students in 'milk' group had \(25 \%\) fewer cavities than \(X\). So, this will be our equation: \(X - 0.25X\).

Step 3: Calculate the number of cavities in 'milk' group

From the previous step, it can be deduced that the number of cavities in the 'milk' group is \(75 \%\) of the cavities in the 'other' group, which equals to \(0.75 * X\).

Step 4: Relation between number of cavities in both groups

From the previous steps, we can conclude that the number of cavities in the other group is \(4/3\) times the number of cavities in the milk group. The next step would be to apply this ratio if the number of cavities in one of the groups is known.

Key Concepts

Understanding Dental Cavities

Cavities, also known as dental caries, are areas of tooth decay that result from bacterial action on food particles. These bacteria produce acids that can erode the tooth's enamel over time. Regular oral hygiene practices such as brushing and flossing, combined with a healthy diet, help prevent cavities. Furthermore, certain dietary choices can affect cavity formation, such as choosing milk over sugary drinks.
Milk is believed to contain nutrients like calcium and phosphates that contribute to strengthening tooth enamel, making teeth less susceptible to decay. In this exercise, milk consumption led to a noticeable reduction in cavities among participating students. Acknowledging the importance of diet alongside oral hygiene is crucial when addressing dental health.

The Concept of Ratio in Analyzing Data

Ratios are a way to compare two quantities, showing the relationship between them. When we talk about percentages like the 25% reduction in cavities, we're using a ratio to compare the occurrence of cavities between two groups.
This can be understood in terms of equations: if one group's cavity count is a fraction of another's, we're effectively working with ratios. For instance, if the milk group has 25% fewer cavities than the other group, we can express this as:

• Milk Group Ratio: 75% of the other group
• Other Group Ratio: - If X is the number of cavities in the other group, then the milk group's cavities would be represented by 0.75X

By understanding these ratios, we can draw meaningful conclusions about the effect of variable A (milk consumption) on variable B (cavity formation). It's a simple yet powerful tool for interpreting data differences in group studies.

Solving Using Equations

Equations are vital when transforming percentage data into actionable numbers. In our scenario, we use the equation to symbolize the relationship between cavity occurrences in two distinct groups.
Let's break down the key equation from the solution:

• Other group cavities: Represented by X
• Milk group cavities: Converting a 25% reduction means the equation becomes: - Cavities in Milk group = X - 0.25X = 0.75X

This equation illustrates how to calculate the actual number of cavities in one group based on the other's data. Understanding such equations allows us to not only calculate unknown values but also understand the relative difference effectively. Equally, it provides a method to predict future outcomes or compare different datasets within larger studies.