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Sylvia has all dimes and nickels, totaling up to 20 coins. The total value of the coins is $1.35. Let's assume n = nickles, and d = dimes. What equation can be used to solve the value?

5 answers

To solve the problem, we can set up two equations based on the information given about the coins.

  1. Total number of coins: Since Sylvia has a total of 20 coins made up of dimes and nickels, \[ n + d = 20 \] where \( n \) is the number of nickels and \( d \) is the number of dimes.

  2. Total value of the coins: The total value of the coins is $1.35. The value of a nickel is $0.05 and the value of a dime is $0.10. Thus, the total value can be expressed as: \[ 0.05n + 0.10d = 1.35 \]

Now, you can use these two equations to find the values of \( n \) and \( d \).

You could also multiply the second equation by 100 to eliminate the decimals: \[ 5n + 10d = 135 \]

Now, you have the following system of equations to solve:

  1. \( n + d = 20 \)
  2. \( 5n + 10d = 135 \)

You can now solve this system of equations for the values of \( n \) and \( d \).

My bad, I meant to say: what equation(s) can we use to find the number of each coin?

No problem! To find the number of each coin (nickels and dimes), we can set up the following equations based on the given conditions:

Let:

  • \( n \) = number of nickels
  • \( d \) = number of dimes

  1. Equation for the total number of coins: \[ n + d = 20 \] This equation states that the total number of coins (nickels plus dimes) is 20.

  2. Equation for the total value of the coins: Since nickels are worth $0.05 each and dimes are worth $0.10 each, the total value of the coins can be represented as: \[ 0.05n + 0.10d = 1.35 \] To make calculations simpler, you can multiply this equation by 100 to get rid of the decimals: \[ 5n + 10d = 135 \]

Now you have a system of equations to work with:

  1. \( n + d = 20 \)
  2. \( 5n + 10d = 135 \)

Solving the equations

You can solve these equations using substitution or elimination.

Substitution Method:

  1. From the first equation, solve for \( d \): \[ d = 20 - n \]

  2. Substitute \( d \) into the second equation: \[ 5n + 10(20 - n) = 135 \] Simplify: \[ 5n + 200 - 10n = 135 \] Combine like terms: \[ -5n + 200 = 135 \] Subtract 200 from both sides: \[ -5n = -65 \] Divide by -5: \[ n = 13 \]

  3. Now substitute \( n = 13 \) back into the first equation to find \( d \): \[ d = 20 - 13 = 7 \]

Conclusion

Thus, Sylvia has 13 nickels and 7 dimes.

Thank you for the answer, but we don't need it right now. I simply need the equations only.

Of course! Here are the equations you need:

  1. Total number of coins: \[ n + d = 20 \]

  2. Total value of the coins: \[ 0.05n + 0.10d = 1.35 \]

Alternatively, you can multiply the second equation by 100 to eliminate decimals: \[ 5n + 10d = 135 \]

These are the equations you can use to find the number of each coin.