Emily’s piggy bank contains twice as many dimes as nickels. It contains two more quarters than nickels. Emily calculates that if she had as many quarters as she has nickels, as many dimes as she has quarters, and as many nickels as she has dimes, then she would have $1.50 less than she actually has. How many nickels, how many dimes, and how many quarters are actually in Emily’s piggy bank? Emilyhas nickels, dimes, and quarters in her piggy bank. Clear all
The Correct Answer and Explanation is:
Let’s define the number of coins using a variable and then solve step-by-step:
Step 1: Define the variables
Let:
- nn = number of nickels
- d=2nd = 2n = number of dimes (twice as many dimes as nickels)
- q=n+2q = n + 2 = number of quarters (two more quarters than nickels)
Step 2: Calculate the actual value of the coins
- Value of nickels: 0.05n0.05n
- Value of dimes: 0.10⋅2n=0.20n0.10 \cdot 2n = 0.20n
- Value of quarters: 0.25(n+2)0.25(n + 2)
So the actual value is: 0.05n+0.20n+0.25(n+2)0.05n + 0.20n + 0.25(n + 2)
Simplify: 0.05n+0.20n+0.25n+0.50=0.50n+0.500.05n + 0.20n + 0.25n + 0.50 = 0.50n + 0.50
Step 3: Value if coins were swapped
According to the problem:
- Quarters become nickels → value: 0.05(n+2)0.05(n + 2)
- Dimes become quarters → value: 0.25⋅2n=0.50n0.25 \cdot 2n = 0.50n
- Nickels become dimes → value: 0.10n0.10n
Total new value: 0.05(n+2)+0.50n+0.10n0.05(n + 2) + 0.50n + 0.10n
Simplify: 0.05n+0.10+0.50n+0.10n=0.65n+0.100.05n + 0.10 + 0.50n + 0.10n = 0.65n + 0.10
Step 4: Use the condition from the problem
Emily says this new total is $1.50 less than the actual amount. 0.50n+0.50=0.65n+0.10+1.500.50n + 0.50 = 0.65n + 0.10 + 1.50
Simplify the right-hand side: 0.50n+0.50=0.65n+1.600.50n + 0.50 = 0.65n + 1.60
Step 5: Solve for nn
Subtract 0.50n0.50n from both sides: 0.50=0.15n+1.600.50 = 0.15n + 1.60
Subtract 1.60 from both sides: −1.10=0.15n-1.10 = 0.15n
Solve for nn: n=−1.100.15=−7.33n = \frac{-1.10}{0.15} = -7.33
This gives a negative value, which is not possible.
Let’s check the equation setup.
Error Check:
Emily says: if she had as many quarters as nickels, as many dimes as quarters, and as many nickels as dimes.
So she swaps:
- quarters → nickels: qq nickels
- dimes → quarters: dd quarters
- nickels → dimes: nn dimes
New value:
- Nickels (was quarters): 0.05q0.05q
- Quarters (was dimes): 0.25d0.25d
- Dimes (was nickels): 0.10n0.10n
So: New value=0.05q+0.25d+0.10n\text{New value} = 0.05q + 0.25d + 0.10n
Original value: 0.05n+0.10d+0.25q0.05n + 0.10d + 0.25q
Now set up: 0.05n+0.10d+0.25q=0.05q+0.25d+0.10n+1.500.05n + 0.10d + 0.25q = 0.05q + 0.25d + 0.10n + 1.50
Substitute d=2nd = 2n, q=n+2q = n + 2
Left side: 0.05n+0.10(2n)+0.25(n+2)=0.05n+0.20n+0.25n+0.50=0.50n+0.500.05n + 0.10(2n) + 0.25(n + 2) = 0.05n + 0.20n + 0.25n + 0.50 = 0.50n + 0.50
Right side: 0.05(n+2)+0.25(2n)+0.10n+1.50=0.05n+0.10+0.50n+0.10n+1.50=0.65n+1.600.05(n + 2) + 0.25(2n) + 0.10n + 1.50 = 0.05n + 0.10 + 0.50n + 0.10n + 1.50 = 0.65n + 1.60
Now set: 0.50n+0.50=0.65n+1.600.50n + 0.50 = 0.65n + 1.60
Same result: solve −1.10=0.15n⇒n=−7.33-1.10 = 0.15n \Rightarrow n = -7.33
Still wrong — let’s try reinterpreting the swap:
If she had:
- as many quarters as nickels → quarters = nn
- as many dimes as quarters → dimes = qq
- as many nickels as dimes → nickels = dd
Now, we recalculate the new value:
New:
- Nickels (now = d): 0.05d0.05d
- Dimes (now = q): 0.10q0.10q
- Quarters (now = n): 0.25n0.25n
Total new value: 0.05d+0.10q+0.25n0.05d + 0.10q + 0.25n
Original: 0.05n+0.10d+0.25q0.05n + 0.10d + 0.25q
Now use: 0.05n+0.10d+0.25q=0.05d+0.10q+0.25n+1.500.05n + 0.10d + 0.25q = 0.05d + 0.10q + 0.25n + 1.50
Use d=2nd = 2n, q=n+2q = n + 2:
Left: 0.05n+0.10(2n)+0.25(n+2)=0.05n+0.20n+0.25n+0.50=0.50n+0.500.05n + 0.10(2n) + 0.25(n + 2) = 0.05n + 0.20n + 0.25n + 0.50 = 0.50n + 0.50
Right: 0.05(2n)+0.10(n+2)+0.25n+1.50=0.10n+0.10n+0.20+0.25n+1.50=0.45n+1.700.05(2n) + 0.10(n + 2) + 0.25n + 1.50 = 0.10n + 0.10n + 0.20 + 0.25n + 1.50 = 0.45n + 1.70
Now solve: 0.50n+0.50=0.45n+1.700.50n + 0.50 = 0.45n + 1.70
Subtract: 0.05n=1.20⇒n=1.200.05=240.05n = 1.20 \Rightarrow n = \frac{1.20}{0.05} = 24
Final Answer:
- Nickels n=24n = 24
- Dimes d=2n=48d = 2n = 48
- Quarters q=n+2=26q = n + 2 = 26
✅ Correct and verified.