Question

kaelyn has 14 coins that have a value of $1.20. she only has dimes and nickels. how many nickels does kaelyn have? 8 2 6 4

Explanation:

Step1: Define variables

Let \( n \) be the number of nickels and \( d \) be the number of dimes. We know two equations:

1. The total number of coins: \( n + d = 14 \) (so \( d = 14 - n \))
2. The total value of coins: A nickel is $0.05 and a dime is $0.10, so \( 0.05n + 0.10d = 1.20 \)

Step2: Substitute \( d \) into the value equation

Substitute \( d = 14 - n \) into \( 0.05n + 0.10d = 1.20 \):
\( 0.05n + 0.10(14 - n) = 1.20 \)

Step3: Simplify and solve for \( n \)

First, expand the equation:
\( 0.05n + 1.4 - 0.10n = 1.20 \)
Combine like terms:
\( -0.05n + 1.4 = 1.20 \)
Subtract 1.4 from both sides:
\( -0.05n = 1.20 - 1.4 \)
\( -0.05n = -0.2 \)
Divide both sides by -0.05:
\( n=\frac{-0.2}{-0.05}=4 \) Wait, no, wait, let's check again. Wait, maybe we made a mistake. Wait, let's try with the options. Let's test each option.

Option 1: 8 nickels. Then dimes = 14 - 8 = 6. Value: 80.05 + 60.10 = 0.4 + 0.6 = 1.00. Not 1.20.

Option 2: 2 nickels. Dimes = 12. Value: 20.05 + 120.10 = 0.1 + 1.2 = 1.30. Not 1.20.

Option 3: 6 nickels. Dimes = 8. Value: 60.05 + 80.10 = 0.3 + 0.8 = 1.10. Not 1.20.

Option 4: 4 nickels. Dimes = 10. Value: 40.05 + 100.10 = 0.2 + 1.0 = 1.20. Yes! Wait, but earlier when we solved, we got n=4? Wait, but the problem says "how many nickels". Wait, but let's check the calculation again. Wait, 0.05n + 0.10(14 - n) = 1.20. 0.05n +1.4 -0.10n =1.20. -0.05n = 1.20 -1.4 = -0.2. Then n = (-0.2)/(-0.05) =4. So the number of nickels is 4? But wait, the options have 4 as an option. Wait, but let's check again. Wait, 4 nickels (45 cents=20 cents) and 10 dimes (1010 cents=100 cents). 20+100=120 cents=1.20 dollars. Yes. So the answer is 4. Wait, but the options are 8,2,6,4. So the correct option is 4. Wait, but earlier when I thought maybe I made a mistake, but no, the calculation is correct.

Wait, maybe I messed up the variable definition. Let me re-express:

Let n = number of nickels, d = number of dimes.

n + d =14 --> d=14 -n.

Value: 5n +10d =120 (in cents, to avoid decimals).

Substitute d=14 -n:

5n +10(14 -n)=120

5n +140 -10n=120

-5n=120 -140

-5n= -20

n=4. Yes, so n=4. So the number of nickels is 4. So the correct option is 4.

Answer:

4 (the option with 4)