In a bag, a child has 200 coins worth $$ \$ 11.75 $$. There are three types of coins: pennies, nickels, and dimes. If the bag contains the same number of nickels as dimes, how many of each type of coin is in the bag? There are: pennies nickels

Question

UpStudy AI · Accepted Answer

Let's define the variables for the problem:

- Let $$ p $$ be the number of pennies.
- Let $$ n $$ be the number of nickels.
- Let $$ d $$ be the number of dimes.

From the problem, we have the following conditions:

1. The total number of coins is 200:
   $$
   p + n + d = 200
   $$

2. The total value of the coins is $$ \$11.75 $$, which is equivalent to 1175 cents:
   $$
   p + 5n + 10d = 1175
   $$

3. The number of nickels is equal to the number of dimes:
   $$
   n = d
   $$

Now, we can substitute $$ n $$ with $$ d $$ in the equations.

Substituting $$ n = d $$ into the first equation:
$$
p + d + d = 200 \implies p + 2d = 200 \quad (1)
$$

Substituting $$ n = d $$ into the second equation:
$$
p + 5d + 10d = 1175 \implies p + 15d = 1175 \quad (2)
$$

Now we have a system of two equations:
1. $$ p + 2d = 200 $$
2. $$ p + 15d = 1175 $$

Next, we can solve this system of equations. Let's express $$ p $$ from equation (1) and substitute it into equation (2).

From equation (1):
$$
p = 200 - 2d
$$

Substituting into equation (2):
$$
200 - 2d + 15d = 1175
$$

Now, let's simplify and solve for $$ d $$:
Solve the equation by following steps:
- step0: Solve for $$d$$:
  $$200-2d+15d=1175$$
- step1: Add the terms:
  $$200+13d=1175$$
- step2: Move the constant to the right side:
  $$13d=1175-200$$
- step3: Subtract the numbers:
  $$13d=975$$
- step4: Divide both sides:
  $$\frac{13d}{13}=\frac{975}{13}$$
- step5: Divide the numbers:
  $$d=75$$
We found that $$ d = 75 $$.

Now, since $$ n = d $$, we also have:
$$
n = 75
$$

Next, we can substitute $$ d $$ back into equation (1) to find $$ p $$:
$$
p + 2(75) = 200
$$
$$
p + 150 = 200
$$
$$
p = 200 - 150 = 50
$$

Now we have the number of each type of coin:
- Pennies: $$ p = 50 $$
- Nickels: $$ n = 75 $$
- Dimes: $$ d = 75 $$

Thus, the bag contains:
- 50 pennies
- 75 nickels
- 75 dimes
There are 50 pennies, 75 nickels, and 75 dimes in the bag.

In a bag, a child has 200 coins worth . There are three types of coins: pennies,
nickels, and dimes. If the bag contains the same number of nickels as dimes, how many of
each type of coin is in the bag?
There are:
pennies
nickels

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Algebra Questions

In a bag, a child has 200 coins worth . There are three types of coins: pennies, nickels, and dimes. If the bag contains the same number of nickels as dimes, how many of each type of coin is in the bag? There are: pennies nickels