Ruthie has 10 coins, all either nickels, dimes, or quarters. She has N nickels, D dimes, and Q
quarters, where N, D, and Q are all different, and are each at least 1. Amazingly, she would have
the same amount of money if she had Q nickels, N dimes, and D quarters. How many cents does
Ruthie have?
1.55
n=1 d =5 and q=4
To find out how many cents Ruthie has, we need to set up and solve a system of equations based on the given information. Let's break down the problem step by step.
1. We know that Ruthie has a total of 10 coins, which can be expressed as the equation N + D + Q = 10.
2. We also know that the value of her coins remains the same when she swaps the quantities of each coin, which can be expressed as the equation 5N + 10D + 25Q = Q + 10N + 10D.
Now, we can solve this system of equations:
From the first equation, we can isolate N: N = 10 - D - Q.
Substituting this into the second equation, we get:
5(10 - D - Q) + 10D + 25Q = Q + 10(10 - D - Q) + 10D.
Simplifying this equation will help us solve for Q:
50 - 5D - 5Q + 10D + 25Q = Q + 100 - 10D - 10Q + 10D.
Combine like terms:
50 + 5Q = Q + 100 - 10Q.
Simplify:
15Q = 50.
Divide both sides by 15:
Q = 50 / 15.
Since we know that Q is an integer, we can determine it by doing integer division.
50 divided by 15 equals 3 remainder 5.
This means that Q is equal to 3.
Now, we can substitute the value of Q back into the first equation to solve for N:
N + D + 3 = 10.
Rearranging the equation, we have:
N + D = 7.
Since N, D, and Q must all be different integers, we can test combinations of N and D that satisfy this equation:
- If N = 1 and D = 6, the sum is 7.
- If N = 2 and D = 5, the sum is 7.
- If N = 3 (which we already know Q = 3) and D = 4, the sum is also 7.
Since we are looking for distinct values, we can eliminate N = 3 and D = 4 as a viable solution because they are identical to Q = 3.
Now, we can calculate the total number of cents Ruthie has.
We can use the equation 5N + 10D + 25Q.
Substituting N = 1 and D = 6, we have:
5(1) + 10(6) + 25(3) = 65 cents.
Or, substituting N = 2 and D = 5, we have:
5(2) + 10(5) + 25(3) = 70 cents.
Therefore, Ruthie has either 65 cents or 70 cents.