Bill and Rosa have only nickels and dimes. Rosa has 75 cents and Bill has 90 cents. Each has the same number of coins and Bill has the same number of dimes as Rosa has nickels.How many nickels does Rosa have?
Rosa:
number of dimes --- x
number of nickels ---- y
Bill:
number of dimes --- y
number of nickels --- x, must be x because they both have x+y coins
Rosa's equation: 10x + 5y = 75 , or
2x + y = 15
Bill's equation: 5x + 10y = 90 , or
x + 2y = 18 ====> x = 18-2y
plug Bill's into Rosa's
2(18-2y) + y = 15
36 - 4y + y = 15
y = 7
in my head, x = 4
state your conclusion
Let's solve this step-by-step:
1. Let's assume the number of nickels that Rosa has is "x". Since Bill has the same number of dimes as Rosa has nickels, Bill also has "x" dimes.
2. The value of each nickel is 5 cents, so the total value of the nickels that Rosa has is 5x cents.
3. The value of each dime is 10 cents, so the total value of the dimes that Bill has is 10x cents.
4. According to the given information, Rosa has 75 cents, so the equation can be written as:
5x = 75
5. Now we can solve this equation to find the value of "x" (the number of nickels that Rosa has):
5x = 75
x = 75 / 5
x = 15
Therefore, Rosa has 15 nickels.
To solve this problem, let's break it down into steps:
Step 1: Convert the given amounts into cents.
Rosa has 75 cents and Bill has 90 cents.
Step 2: Setup the equations.
Let's assume that Rosa has a certain number of nickels (N) and Bill has the same number of dimes as Rosa has nickels. Therefore, Bill also has N dimes.
Step 3: Calculate the value of nickels and dimes.
A nickel is worth 5 cents, so the total value of nickels is 5N cents.
A dime is worth 10 cents, so the total value of dimes is 10N cents.
Step 4: Write the equation based on the given information.
Based on the given information, we can write the equation:
5N + 10N = 75 (since Rosa has 75 cents)
Combine the like terms:
15N = 75
Step 5: Solve the equation for N.
Divide both sides of the equation by 15 to isolate N:
N = 75 / 15
N = 5
Step 6: Determine the number of nickels.
Rosa has 5 nickels, as N represents the number of nickels.
Therefore, Rosa has 5 nickels.