1.The ratio of nickels to dimes to quarters in 3 : 8 : 1. If all the coins were dimes, the amount of money would be the same. Show that there are infinitely many solutions to this problem.

2.A grocer wants to make a mixture of three dried fruits. He decides that the ratio of pounds of banana chips to apricots to dates should be 3:1:1. Banana chips cost $1.17/lb, apricots cost $3.00/lb, and dates cost $2.30/lb. What is the cost per pound of the mixture?

number of nickels = 3x

number of dimes = 8x
number of quarters = x

value of the money = 5(3x) + 10(8x) + 25x = 120x
number of coins = 12x

suppose we have 12x dimes as stated
value of all the those 12x dimes = 10(12x) = 120x

so 120x = 120x
0 = 0 !!!!!
the variables dropped out and the remaining statement is true.
Therefore, there are infinitely many solutions

btw, if the variables drop out, and the remaining statement is false, such as 5 = 9, there would be NO solutions to the problem

Thanks a lot!

You are welcome,

did you get the 2nd question?

Oh yes! I figured it out by myself.

good job

The ratio of 8 dime to 3 quarters is equivalent to?

1. To show that there are infinitely many solutions to the problem, we can consider the overall value of the coins. Let's assume the value of a nickel is $0.05, a dime is $0.10, and a quarter is $0.25.

Since the ratio of nickels to dimes to quarters is 3:8:1, let's assign variables to represent the number of nickels, dimes, and quarters. Let N represent the number of nickels, D represent the number of dimes, and Q represent the number of quarters.

We can set up the equation:
0.05N + 0.10D + 0.25Q = constant value

Now, if all the coins were dimes, the value of the coins would be:
0.10(3N) + 0.10(8D) + 0.10(Q) = constant value

Simplifying this equation:
0.30N + 0.80D + 0.10Q = constant value

Now we have two equations:
0.05N + 0.10D + 0.25Q = constant value
0.30N + 0.80D + 0.10Q = constant value

By subtracting the second equation from the first equation, we can eliminate Q:
(0.05N - 0.30N) + (0.10D - 0.80D) + (0.25Q - 0.10Q) = 0
-0.25N - 0.70D + 0.15Q = 0

These equations form a linear system. By solving the system, we can find that there will be infinitely many solutions. This means we can find different combinations of nickels, dimes, and quarters that would result in the same value of money.

2. To find the cost per pound of the mixture, we need to consider the ratio of each fruit and their respective costs per pound.

Let's assume the number of pounds of banana chips is 3x, the number of pounds of apricots is x, and the number of pounds of dates is x. Therefore, the total weight of the mixture is 5x pounds.

The cost of banana chips per pound is $1.17, so the cost of the banana chips in the mixture is 3x * $1.17 = $3.51x.

The cost of apricots per pound is $3.00, so the cost of the apricots in the mixture is x * $3.00 = $3.00x.

The cost of dates per pound is $2.30, so the cost of the dates in the mixture is x * $2.30 = $2.30x.

To find the total cost of the mixture, we sum up the costs of each fruit:
Total cost = $3.51x + $3.00x + $2.30x

The total weight of the mixture is 5x pounds, so the cost per pound of the mixture is:
Cost per pound = Total cost / Total weight
Cost per pound = ($3.51x + $3.00x + $2.30x) / (5x)

Now we can simplify the expression to find the cost per pound of the mixture.