algebra help needed

Question

a jar contains 145 coins, all nickels and dimes. if the total value of the coins is $12, how many of each type of coin is in the jar

Answer 1

Hi Bob,

When tackling a math problem, the first step is to gather and organize the information given. I like to jot these down under the heading "Given" – it's a simple way to keep track of what we already know.

Given:

The total number of coins (nickels and dimes) is 145.

The total value of these coins is $12.

Next, it's often helpful to assign symbols or letters to represent the unknowns in our problem. This might seem a bit abstract, but it's really just like using x and y in equations. For clarity, I usually use initials to represent these variables.

Let's denote: (denote means using a symbol to represent something or labeling)

- The number of nickels as N.

- The number of dimes as D.

Once we've outlined our given information, we need to pinpoint exactly what the problem is asking us to find. I label this as "Find" to help focus on the end goal.

Find:

The specific numbers of nickels N and dimes D among the 145 coins that together add up to $12.

With this structure in place, we can now start piecing together the solution and exploring the concepts at play. This is where the fun in math lies – in experimenting and learning through problem-solving.

Concept & Solution:

From the problem, we have two equations:

(1) N + D = 145 (Total number of coins)

(2) 0.05N + 0.10D = 12 (Total value of coins)

The second equation can be multiplied by 100 to make it easier to work with, getting rid of decimals as I tend to like working with round numbers:

5N + 10D = 1200

Now, we can use the substitution method, which involves expressing one variable in terms of the other from one equation and substituting it into the other equation. From equation (1), we can express N as

N = 145 - D (we do this by subtracting D from both sides).

Substituting this expression for in the modified second equation:

5(145 - D) + 10D = 1200

Expanding and simplifying:

725 - 5D + 10D = 1200

5D = 1200 - 725

5D = 475

Dividing both sides by 5:

D = 475/5 = 95

Now, substitute D = 95 back into N = 145 - D :

N = 145 - 95 = 50

Therefore, there are 50 nickels (N) and 95 dimes (D) in the jar.

I hope this breakdown makes the problem clearer and shows how breaking it into smaller, manageable parts can simplify the process. Remember, staying organized and methodical is key to solving these kinds of challenges.

Regards,

John

Answer 2

a jar contains 145 coins, all nickels and dimes. if the total value of the coins is $12, how many of each type of coin is in the jar ?

Let N = number of nickels and D = number of dimes

The total value of coins in the jar is V = V(N) + V(D) = 0.05 * N + 0.10 * D = 12

In terms of number of coins, N + D = 145 so .D = 145 - N

Substitute the 2nd equation into the 1st gives 0.05*N + 0.10*(145-N) = 12

0.05*N + 14.5 - 0.10*N = 12

0.05*N - 0.10*N = 12 - 14.5

-0.05N = -2.5

N = -2.5/-0.05 = 50

Now using the 1st equation we have 0.05*50 + 0.10*D = 12

0.10*D = 12 - 0.05*50 = 12 - 2.5 = 9.5

D = 9.5/0.10 = 95

Note that the same result would be obtained if we substituted (145-N) for D in equation 1.

Try it yourself ............. :-)

Answer 3

Hey Bob! I have an answer for ya:

To set this one up, lets call the amount of nickels "x" and the number of dimes "y". When we do this, we can describe the situation as a system of equations:

X+Y=145 (1)

0.05x+0.1y=12 (2)

The first equation describes the total amount of coins you have in the jar (number of nickels and dimes), and the second equation describes the total value of the coins inside the jar.

To solve, we can subtract X from both sides of the first equation to obtain

Y=145-X. (3)

Then we can use substitution and place the right side of this equation in place of Y in (2):

0.05X+0.1(145-X)=12. This produces a linear equation in the Variable X, which may be solved in the following fashion:

0.05X+14.5-0.1X=12

-0.05X=-2.5

X=50.

We can the use this value for X and substitute in into equation (3) to find Y:

Y=145-50

Y=95

Therefore, the jar will contain 50 nickels and 95 dimes. Hope this helps!