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.

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!