Question

Adam's piggy bank contains dimes and nickels, for a total of 236 coins. The contents are worth a total of $17.40. How many nickels does Adam's piggy bank contain?

Answer 1

To find the number of nickels in Adam's piggy bank, we can solve a system of equations based on the given information. By setting up two equations, one for the total number of coins and another for the total value, we can use the method of elimination or substitution to find the values of dimes and nickels. The solution shows that Adam's piggy bank contains 124 nickels.

Step-by-step explanation:

To solve this problem, we can set up a system of equations. Let's use the variables d and n to represent the number of dimes and nickels, respectively. From the problem, we have two pieces of information:

1) The total number of coins is 236. So, d + n = 236.

2) The total value of the coins is $17.40. Since a dime is worth 10 cents and a nickel is worth 5 cents, the total value can be expressed as 10d + 5n = 1740 (in cents).

We can solve this system of equations using substitution or elimination. Let's use elimination:

Multiply the first equation by 5 to match the coefficient of n in the second equation: 5(d + n) = 5(236) ==> 5d + 5n = 1180.

Now we have the system of equations:

5d + 5n = 1180

10d + 5n = 1740

Subtract the first equation from the second equation to eliminate n: (10d + 5n) - (5d + 5n) = 1740 - 1180 ==> 5d = 560.

Divide both sides of this equation by 5 to solve for d: 5d/5 = 560/5 ==> d = 112.

Now we can substitute the value of d back into the first equation to find n: 112 + n = 236 ==> n = 236 - 112 = 124.

Therefore, Adam's piggy bank contains 124 nickels.

Answer 2

Let D be the number of dimes and N be the number of nickels.

Since there is a total of 236 coins, then:

`D+N=236`

Since a dime is worth 0.1 and a nickel is worth 0.05, then for D dimes and N nickels, they will worth 0.1D + 0.05N. Then:

`0.1D+0.05N=17.40`

Isolate D from the first equation:

`D=236-N`

Replace D for the expression 236-N in the second equation and solve for N:

`\begin{gathered} 0.1(236-N)+0.05N=17.4 \\ \Rightarrow23.6-0.1N+0.05N=17.4 \\ \Rightarrow23.6-0.05N=17.4 \\ \Rightarrow-0.05N=17.4-23.6 \\ \Rightarrow-0.05N=-6.2 \\ \Rightarrow N=(-6.2)/(-0.05) \\ \therefore N=124 \end{gathered}`

Substitute N=124 into the expression for D to find the amount of dimes:

`\begin{gathered} D=236-124 \\ =112 \end{gathered}`

Therefore, there are 124 nickels in total.