Use the three steps to solve the problem. Betty has 10 more dimes than quarters. If she has $3.45, how many coins does she have?
{
quarters,
dimes }
1. Represent the problem: Let's use the variables q for quarters and d for dimes.
Given that Betty has 10 more dimes than quarters, we can express this as: d = q + 10.
Additionally, we know that the total value of the coins is $3.45.
2. Translate the problem into an equation: The value of each quarter is $0.25, so the total value of the quarters is 0.25q. Similarly, the value of each dime is $0.10, so the total value of the dimes is 0.10d. The total value of all the coins can be represented by the equation: 0.25q + 0.10d = 3.45.
3. Solve the equation: Let's substitute the value of d from step 1 into the equation in step 2:
0.25q + 0.10(q + 10) = 3.45.
Simplifying the equation:
0.25q + 0.10q + 1 = 3.45.
Combining like terms:
0.35q + 1 = 3.45.
Subtracting 1 from both sides:
0.35q = 2.45.
Finally, dividing both sides by 0.35 gives:
q ≈ 7.
So, Betty has approximately 7 quarters.
Substituting this value back into the equation in step 2 gives:
0.25(7) + 0.10d = 3.45.
Simplifying:
1.75 + 0.10d = 3.45.
Subtracting 1.75 from both sides:
0.10d = 1.70.
Dividing both sides by 0.10 gives:
d ≈ 17.
Therefore, Betty has approximately 17 dimes.
In conclusion, Betty has 7 quarters and 17 dimes, totaling 7 + 17 = 24 coins.
Step 1: Set up a system of equations to represent the problem. Let's call the number of quarters "q" and the number of dimes "d".
According to the problem, Betty has 10 more dimes than quarters, so we can write the equation: d = q + 10.
We also know that the total value of her coins is $3.45. The value of a quarter is $0.25 and the value of a dime is $0.10. We can write the equation: 0.25q + 0.10d = 3.45.
Step 2: Solve the system of equations.
Substitute the expression for "d" from the first equation into the second equation: 0.25q + 0.10(q + 10) = 3.45.
Simplify and solve the equation: 0.25q + 0.10q + 1 = 3.45.
Combine like terms: 0.35q + 1 = 3.45.
Subtract 1 from both sides: 0.35q = 2.45.
Divide by 0.35: q = 7.
Step 3: Find the value of "d" using the first equation: d = q + 10.
Substitute the value of q into the equation: d = 7 + 10.
Simplify: d = 17.
So, Betty has 7 quarters and 17 dimes, for a total of 7 + 17 = 24 coins.
To solve this problem, we can follow these three steps:
Step 1: Define the variables:
Let's define our variables:
q = number of quarters
d = number of dimes
Step 2: Set up equations using the given information:
We are given two pieces of information:
- Betty has 10 more dimes than quarters: d = q + 10
- Betty has $3.45 in total, with each quarter being worth $0.25 and each dime being worth $0.10: 0.25q + 0.10d = 3.45
Step 3: Solve the equations:
Now we can substitute the value of d from the first equation into the second equation:
0.25q + 0.10(q + 10) = 3.45
Simplifying this equation:
0.25q + 0.10q + 1 = 3.45
0.35q + 1 = 3.45
0.35q = 2.45
q = 2.45 / 0.35
q ≈ 7
Now that we know q, we can substitute it back into the first equation to find d:
d = q + 10
d = 7 + 10
d = 17
Therefore, Betty has 7 quarters and 17 dimes.