Express the value (in cents) of the change in a purse that contains half as many dimes as pennies, four more nickels than dimes, and as many quarters as dimes and pennies combined, if the number of pennies in the purse is $p$ .
A) $46 p + 20$
B) $27 p - 13$
C) $100 p - 81$
D) $32 p - 14$
E) $107 p - 27$

Question

Quizplus · Accepted Answer

The value of the change can be calculated by considering the value of each type of coin and their quantities relative to the number of pennies, $p$. Let's denote the number of each type of coin as follows: pennies ($p$), dimes ($d$), nickels ($n$), and quarters ($q$). Given:- Dimes are half as many as pennies, so $d = \frac{1}{2}p$.- Nickels are four more than dimes, so $n = d + 4$.- Quarters are as many as dimes and pennies combined, so $q = d + p$.The total value in cents is:- Pennies: $p \times 1 = p$,- Dimes: $d \times 10 = \frac{1}{2}p \times 10 = 5p$,- Nickels: $n \times 5 = (d + 4) \times 5 = (\frac{1}{2}p + 4) \times 5 = 2.5p + 20$,- Quarters: $q \times 25 = (d + p) \times 25 = (\frac{1}{2}p + p) \times 25 = 1.5p \times 25 = 37.5p$.Adding these values together gives the total value in cents:$$p + 5p + 2.5p + 20 + 37.5p = 46p + 20$$

Express the Value (In Cents) of the Change in a Purse