Nathalie has some quarters, dimes and nickels. The ratio of the number of quarters to the number of dimes to the number of nickels that she has is $9: 3: 1$. The total value of these coins is $\$ 18.20$. How many coins does Nathalie have?
(A) 130
(B) 117
(C) 98
(D) 91
(E) 140
Solution 1

If Nathalie had exactly 9 quarters, 3 dimes and 1 nickel, she would have $9 \times 25+3 \times 10+5=260$ cents or $\$ 2.60$ in total.

Since the coins she has are in this same ratio $9: 3: 1$, then we can split her coins up into sets of 9 quarters, 3 dimes and 1 nickel, with each set having a value of $\$ 2.60$.

Since the total value of her coins is $\$ 18.20$ and $\frac{\$ 18.20}{\$ 2.60}=7$, then she has 7 of these sets of coins. Since each of these sets contains 13 coins, then she has $7 \times 13=91$ coins in total.

## Solution 2

Suppose Nathalie had $n$ nickels. Since the ratio of the number of quarters to the number of dimes to the number of nickels that she has is $9: 3: 1$, then she must have $3 n$ dimes and $9 n$ quarters.

This tells us that the total value of her coins in cents is $(9 n \times 25)+(3 n \times 10)+(n \times 5)=260 n$.

Since we know that the total value of her coins is 1820 cents, then $260 n=1820$ or $n=7$.

Therefore, she has 7 nickels, 21 dimes and 63 quarters, or 91 coins in total.

ANSWER: (D)