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The given formula is Young's Rule, which is used to calculate the dosage for a child based on the adult dosage. The formula is:

\[
C = \frac{DA}{A + 12}
\]

Where:
- \( A \) = child's age (in years),
- \( D \) = adult dosage,
- \( C \) = proper child's dosage.

The problem asks to find the difference in the dosage between a 13-year-old child and a 4-year-old child, using the formula.

### Step-by-Step Calculation:

1. **For a 13-year-old child**:
   Using \( A = 13 \), the dosage for the 13-year-old is:
   \[
   C_{13} = \frac{D \cdot 13}{13 + 12} = \frac{13D}{25}
   \]

2. **For a 4-year-old child**:
   Using \( A = 4 \), the dosage for the 4-year-old is:
   \[
   C_{4} = \frac{D \cdot 4}{4 + 12} = \frac{4D}{16} = \frac{D}{4}
   \]

3. **Difference in dosage**:
   Now, subtract the dosage for the 4-year-old from the dosage for the 13-year-old:
   \[
   \text{Difference} = C_{13} - C_{4} = \frac{13D}{25} - \frac{D}{4}
   \]

   To subtract these, we need a common denominator. The least common denominator between 25 and 4 is 100. Rewrite both fractions with denominator 100:
   \[
   \frac{13D}{25} = \frac{52D}{100}, \quad \frac{D}{4} = \frac{25D}{100}
   \]

   Now, subtract:
   \[
   \frac{52D}{100} - \frac{25D}{100} = \frac{27D}{100}
   \]

Thus, the difference in dosage between a 13-year-old and a 4-year-old is:

\[
\boxed{\frac{27D}{100}}
\]

### Explanation in terms of the model:
This result means that the dosage for the 13-year-old is \( \frac{27}{100} \) (or 27%) of the adult dosage higher than that for the 4-year-old.

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Would you like more details or have any questions?

### Related questions:
1. How would the dosage difference change if the adult dosage \( D \) were given as a specific value?
2. What is the dosage difference for children aged 10 and 5 using Young’s Rule?
3. How does the formula change if the child is older than 13? 
4. Can Young’s Rule be used for infants under 2 years old? Why or why not?
5. How would you derive a general formula for the difference in dosage between two children of different ages?

**Tip:** When dealing with fractions, always find the least common denominator to simplify addition or subtraction effectively!

A formula that approximates the dosage of a drug prescribed for children is shown below. In the formula, A = the child's age in years, D = an adult dosage, and C = the proper child's dosage. The formula applies for ages 2 through 13, inclusive. Use Young's rule to find the difference in a child's dosage for a 13-year-old and a 4-year-old child. Express the answer as a single rational expression in terms of D. Then describe what your answer means in terms of the variables in the model.

Explore how to calculate the difference in medication dosage between a 13-year-old and a 4-year-old using Young's Rule. This tutorial covers the rational expression derivation and its implications in pediatric dosage calculations.

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