4 Jack has 14 coins in his pocket, consisting entirely of dimes and quarters. If he has a total of $2.30 in coins, how many quarters are in his pocket?
The correct answer and explanation is:
To find how many quarters Jack has, we can solve the problem using a system of equations.
Step 1: Define the variables
Let:
- d = number of dimes
- q = number of quarters
From the problem:
- Jack has 14 coins total
→ Equation 1: d + q = 14 - The total value is $2.30, or 230 cents
- Dimes are worth 10 cents
- Quarters are worth 25 cents
→ Equation 2: 10d + 25q = 230
Step 2: Solve the system
Use Equation 1 to express d in terms of q:
d = 14 – q
Substitute this into Equation 2:
10(14 – q) + 25q = 230
140 – 10q + 25q = 230
140 + 15q = 230
15q = 90
q = 6
Step 3: Find the number of dimes
d = 14 – q = 14 – 6 = 8
✅ Final Answer: Jack has 6 quarters.
300-Word Explanation:
This problem involves solving a system of equations that represent a real-life scenario: counting coins of different denominations to meet a specific total. We are told that Jack has 14 coins in total, and they are only dimes (worth 10 cents) and quarters (worth 25 cents). Altogether, the coins add up to $2.30, which is 230 cents.
To begin solving, we define two variables: d for the number of dimes, and q for the number of quarters. From the total number of coins, we know that d + q = 14. From the total value, we get another equation: the value of the dimes plus the value of the quarters equals 230 cents. This gives us 10d + 25q = 230.
We now have a system of two equations:
- d + q = 14
- 10d + 25q = 230
To solve, we isolate d from the first equation, giving d = 14 – q, and substitute this into the second equation. This replacement eliminates one variable, allowing us to solve for q. Doing the arithmetic carefully, we find q = 6, meaning Jack has 6 quarters.
To confirm the solution, we find d = 14 – 6 = 8 dimes. Then, compute the total value:
6 quarters = 6 × 25 = 150 cents
8 dimes = 8 × 10 = 80 cents
Total = 150 + 80 = 230 cents = $2.30
Thus, the solution is correct: Jack has 6 quarters.