Diego is collecting dimes and nickels in a jar. He has collected \( \$ 22.25 \) so \( * 25 \) points far. The relationship between the numbers of dimes and nickels, and the amount of money in dollars is represented by the equation . \( 10 \mathrm{~d}+.05 \mathrm{n}= \) 22.25 (d, n\( ) \) \( (0,445) \) \( (0.50,435) \) \( (1183,209) \)

To solve the problem, we need to analyze the equation given for the relationship between the number of dimes (d) and nickels (n) that Diego has collected. The equation is: \[ 10d + 0.05n = 22.25 \] This equation represents the total value of the dimes and nickels in dollars, where \(10d\) is the value of the dimes (each dime is worth $0.10) and \(0.05n\) is the value of the nickels (each nickel is worth $0.05). To find the values of \(d\) and \(n\) that satisfy this equation, we can rearrange it to express \(n\) in terms of \(d\): \[ 0.05n = 22.25 - 10d \] \[ n = \frac{22.25 - 10d}{0.05} \] \[ n = 445 - 200d \] Now, we can check the given points to see if they satisfy the equation: 1. **Point (0, 445)**: - \(d = 0\), \(n = 445\) - \(10(0) + 0.05(445) = 0 + 22.25 = 22.25\) (Valid) 2. **Point (0.50, 435)**: - \(d = 0.50\), \(n = 435\) - \(10(0.50) + 0.05(435) = 5 + 21.75 = 26.75\) (Not valid) 3. **Point (1183, 209)**: - \(d = 1183\), \(n = 209\) - \(10(1183) + 0.05(209) = 11830 + 10.45 = 11840.45\) (Not valid) From the calculations, only the point (0, 445) satisfies the equation \(10d + 0.05n = 22.25\). Thus, the correct answer is: \[ \text{(0, 445)} \]