Mary has $4.00 in nickels, dimes, and quarters. If she has twice as many dimes as quarters and forty-seven more nickels than dimes, how many coins of each type does she have?
number of quarters --- x
number of dimes ---- 2x
number of nickels ---- 2x+47
25x + (10)(2x) + 5(2x+47) = 400
solve for x, then sub into my definitions
i zunno
Let's use a step-by-step approach to solve this problem.
Step 1: Set up the equations based on the given information.
Let's assume Mary has x quarters. Since she has twice as many dimes as quarters, she must have 2x dimes. Additionally, she has 47 more nickels than dimes, so she has 2x + 47 nickels.
Step 2: Convert the values to cents for easier calculations.
1 quarter = 25 cents
1 dime = 10 cents
1 nickel = 5 cents
Step 3: Write the equation based on the total value of the coins.
The total value of her quarters (25 cents each), dimes (10 cents each), and nickels (5 cents each) is:
25x + 10(2x) + 5(2x+47) = 400 cents
Step 4: Simplify the equation.
25x + 20x + 10x + 235 = 400
55x + 235 = 400
Step 5: Solve for x.
55x = 165
x = 165 / 55
x = 3
Step 6: Calculate the number of each type of coin Mary has.
Mary has 3 quarters, 2x dimes, and 2x + 47 nickels.
Substituting the value of x:
Mary has 3 quarters, 2(3) = 6 dimes, and 2(3) + 47 = 53 nickels.
So Mary has 3 quarters, 6 dimes, and 53 nickels.
To solve this problem, let's start by assigning variables to represent the unknown quantities.
Let's say:
- The number of quarters is Q.
- The number of dimes is D.
- The number of nickels is N.
We are given the following information:
1. Mary has $4.00 in total: Since 1 dollar is equal to 100 cents, we know that the total value in cents must be 400.
So, we can write an equation:
Value of quarters (in cents) + Value of dimes (in cents) + Value of nickels (in cents) = 400.
2. Mary has twice as many dimes as quarters:
Since the number of quarters is Q, then the number of dimes is 2Q.
3. Mary has forty-seven more nickels than dimes:
Since the number of dimes is D, then the number of nickels is D + 47.
Now, let's convert the values of quarters, dimes, and nickels into cents:
- Value of quarters (in cents) = 25Q
- Value of dimes (in cents) = 10(2Q) = 20Q
- Value of nickels (in cents) = 5(D + 47)
Plugging these values into the equation from point 1, we get:
25Q + 20Q + 5(D + 47) = 400
Let's simplify this equation and solve for Q:
45Q + 5D + 235 = 400
45Q + 5D = 165
Now let's use the next given information: Mary has twice as many dimes as quarters.
Since the number of dimes is 2Q, we can substitute 2Q for D in the equation:
45Q + 5(2Q) = 165
45Q + 10Q = 165
55Q = 165
Q = 3
Now that we know the number of quarters, we can find the number of dimes and nickels:
D = 2Q = 2(3) = 6
N = D + 47 = 6 + 47 = 53
So Mary has 3 quarters, 6 dimes, and 53 nickels.