sarah has dimes and nickels which total $1.65. the number of nickels she has is 3 less than the number of dimes. how many of each does she have?
Let n = # of nickels and d = # of dimes
d = n + 3
5n + 10d = 165
5n + 10(n+3) = 165
Solve for n then d.
12
To solve this problem, let's break it down step by step:
Step 1: Assign variables.
Let's assign variables to the unknown quantities.
Let:
- x = the number of dimes Sarah has
- y = the number of nickels Sarah has
Step 2: Form equations based on the given information.
From the problem, we know that:
- The value of dimes (in cents) = 10x
- The value of nickels (in cents) = 5y
- The total value (in cents) = 165
Since the value of the dimes and nickels equals $1.65 or 165 cents, we can write the equation:
10x + 5y = 165
Step 3: Translate the next sentence into an equation.
"The number of nickels she has is 3 less than the number of dimes."
This can be expressed as:
y = x - 3
Step 4: Solve the system of equations.
Now, we have a system of two equations with two variables. We can solve it using the substitution or elimination method.
Substitution Method:
Substitute the value of y (from equation 2) into equation 1:
10x + 5(x - 3) = 165
Simplify the equation:
10x + 5x - 15 = 165
15x - 15 = 165
15x = 180
x = 12
Substitute the value of x back into equation 2 to find y:
y = 12 - 3
y = 9
Step 5: Check the solution.
To verify our solution, we can substitute the values of x and y into the original equation:
10(12) + 5(9) = 120 + 45 = 165
The total value of the dimes and nickels is 165 cents, which matches the given amount.
Step 6: Answer the question.
Sarah has 12 dimes and 9 nickels.