Hi I was wondering if I could get some help on this word problem...

Penny collected some change in preparation for a garage sale. She collected two more nickels that twice the number of dimes and eight fewer quarters that twice the number of nickels. If the value of the quarters was $1.60 greater than four times the value of the nickels and dimes together, what was the total value of the change that Penny collected?

I tried to get some equations of it, but I could only get one.
25q = 4 ( 5n ) + 10d +160
the first equation was too confusing.
Please help.

jason.

Your equation is close, but it should be...

25q = 4(5n + 10d) + 160
It says "four times the value of the nickels and dimes TOGETHER."

Since we have three variables, we must have three equations. Here are the other two. (They are a bit difficult to come up with.)

2d = n + 2
q - 8 = 2n

I believe those are the right equations, but check them out for yourself.

Now, solve the system of equations by substitution or elimination.

Hi Jason! I'd be happy to help you with your word problem. Let's break it down and see if we can solve it together.

Let's start with what we know:

1) Penny collected two more nickels than twice the number of dimes.
2) Penny collected eight fewer quarters than twice the number of nickels.
3) The value of the quarters was $1.60 greater than four times the value of the nickels and dimes together.

Let's translate these sentences into equations. Let's say the number of dimes is d, the number of nickels is n, and the number of quarters is q.

1) Penny collected two more nickels than twice the number of dimes: n = 2d + 2
2) Penny collected eight fewer quarters than twice the number of nickels: q = 2n - 8
3) The value of the quarters was $1.60 greater than four times the value of the nickels and dimes together: 25q = 4(5n + 10d) + 160

Now let's see if we can solve it.

Hmm, I see why you found the first equation confusing. It can get a bit tricky sometimes. Let's simplify it a bit:

25q = 4(5n + 10d) + 160
25q = 20n + 40d + 160

Now let's substitute n and q with the expressions we found in the first two equations:

25(2n - 8) = 20n + 40d + 160
50n - 200 = 20n + 40d + 160
30n - 200 = 40d + 160

We can bring the 40d to the left side:

30n - 40d - 200 = 160

Combining like terms:

30n - 40d = 360

That's as far as we can simplify it for now. We need another equation to solve this system. Do you see anything else we can work with here?

Sure, I can help you with that word problem.

Let's break down the problem step by step to find a solution:

Step 1: Define the variables
Let's assign variables to represent the unknown quantities in the problem. We'll use:
n = number of nickels
d = number of dimes
q = number of quarters

Step 2: Set up the equations
Based on the given information, we can set up the following equations:

Equation 1: "Penny collected two more nickels than twice the number of dimes."
n = 2d + 2

Equation 2: "Penny collected eight fewer quarters than twice the number of nickels."
q = 2n - 8

Equation 3: "The value of the quarters was $1.60 greater than four times the value of the nickels and dimes together."
25q = 4(5n + 10d) + 160

Step 3: Solve the equations
To solve the system of equations, we will substitute the values from one equation into the others to eliminate variables.

Substitute Equation 1 into Equation 2:
q = 2(2d + 2) - 8
q = 4d + 4 - 8
q = 4d - 4

Substitute Equation 1 and Equation 2 into Equation 3:
25(4d - 4) = 4(5(2d + 2) + 10d) + 160
100d - 100 = 4(10d + 10 + 10d) + 160
100d - 100 = 4(20d + 10) + 160
100d - 100 = 80d + 40 + 160
100d - 100 = 80d + 200
20d = 300
d = 15

Substitute d = 15 into Equation 1:
n = 2(15) + 2
n = 30 + 2
n = 32

Substitute n = 32 into Equation 2:
q = 4(15) - 4
q = 60 - 4
q = 56

Step 4: Calculate the total value of the change
To calculate the total value of the change, we need to determine the value of each coin and then add them together.

Value of nickels = 5n = 5(32) = 160 cents
Value of dimes = 10d = 10(15) = 150 cents
Value of quarters = 25q = 25(56) = 1400 cents

Total value of change = Value of nickels + Value of dimes + Value of quarters
Total value of change = 160 + 150 + 1400 = 1710 cents
Total value of change = $17.10

Therefore, the total value of the change that Penny collected is $17.10.

I hope this step-by-step explanation helps you understand how to solve the word problem. Let me know if you have any further questions!

Hi Jason! I'd be happy to help you with this word problem.

Let's break down the problem step by step and try to determine the equations we need to solve it.

Step 1: Define the variables.
We need to define the variables for the number of nickels, dimes, and quarters that Penny collected. Let's say:
n = number of nickels
d = number of dimes
q = number of quarters

Step 2: Translate the information given into equations.
Let's look at the information given and try to express it using equations.

1) Penny collected two more nickels than twice the number of dimes.
This can be translated into the equation: n = 2d + 2.

2) Penny collected eight fewer quarters than twice the number of nickels.
This can be translated into the equation: q = 2n - 8.

3) The value of the quarters was $1.60 greater than four times the value of the nickels and dimes together.
We can break this down into two parts. Let's find the value of the quarters and the value of the nickels and dimes together.

The value of the quarters can be calculated as 25q (since each quarter is worth 25 cents).

The value of the nickels and dimes together can be calculated as 5n + 10d (since each nickel is worth 5 cents, and each dime is worth 10 cents).

Therefore, the equation becomes: 25q = 4(5n + 10d) + 160.

Finally, we need to find the total value of the change Penny collected.

The total value can be calculated as the sum of the values of the nickels, dimes, and quarters:
Total value = 5n + 10d + 25q.

Step 3: Solve the equations.
Now we have a system of equations:
n = 2d + 2
q = 2n - 8
25q = 4(5n + 10d) + 160
Total value = 5n + 10d + 25q.

To solve this system of equations, you can either use algebraic methods such as substitution or elimination, or you can input the equations into a system of equation solver or graphing calculator to find the values of n, d, q, and the total value.

I hope this explanation helps you understand how to approach this word problem. Let me know if you have any further questions!