Can you please help on this Algebra problem?
Keana's piggy bank contains $4.30 in nickels and dimes only. If she has 59 coins in her ban, then what is the sum of the digits in the number of nickels in Keana's bank? Write a system of equations for this situation and find its solution.
I have so far:
n = number of nickels
d = number of dimes
n+d = 430
5n + 10d = 430
But I am not sure what step is next to figure out how to solve for nickels only. Thank you
first of all just divide the 2nd equation by 5 to get
n + 2d = 86, the other is
n + d = 59 <<---- you had 430 here.
subtract them:
d = 27
then in n+d = 59 , n+27 = 59
n = 32
She has 27 dimes and 32 nickels
check:
is 27+32 = 59 for the sum of coins? YES
for the value of coins, is
27(10) + 32(5) = 430 ?, YES
My answer is correct.
To solve the system of equations and find the number of nickels, you can use the method of substitution.
Starting with the first equation, n + d = 430, isolate n by subtracting d from both sides:
n = 430 - d
Substitute this expression for n in the second equation, 5n + 10d = 430:
5(430 - d) + 10d = 430
Distribute the 5:
2150 - 5d + 10d = 430
Combine like terms:
2150 + 5d = 430
Subtract 2150 from both sides:
5d = 430 - 2150
5d = -1720
Divide both sides by 5:
d = -1720 / 5
d = -344
However, since the number of coins cannot be negative, this value of d is not meaningful in the given context.
To find the solution, let's examine the problem constraints again. We know that there are 59 coins in the piggy bank. The total number of nickels and dimes must equal 59: n + d = 59.
Substituting the expression for n:
430 - d + d = 59
Combine like terms:
430 = 59
This equation is not true, so there must have been an error in setting up the equations. Let's correct it.
Since we know that the piggy bank contains 59 coins, we can write the equation as:
n + d = 59
The value of each nickel is 5 cents, so the value of n nickels is 5n cents. Likewise, the value of each dime is 10 cents, so the value of d dimes is 10d cents. The total value of the coins, $4.30, can be expressed as 430 cents. Therefore, we can form a second equation:
5n + 10d = 430
Now, we have the correct system of equations:
n + d = 59
5n + 10d = 430
To solve this system, we can use the method of substitution or elimination. Let's use substitution:
From the first equation, isolate n:
n = 59 - d
Substitute this expression for n in the second equation:
5(59 - d) + 10d = 430
Distribute the 5:
295 - 5d + 10d = 430
Combine like terms:
295 + 5d = 430
Subtract 295 from both sides:
5d = 430 - 295
5d = 135
Divide both sides by 5:
d = 135 / 5
d = 27
Now, substitute this value of d back into the first equation:
n + 27 = 59
Subtract 27 from both sides:
n = 59 - 27
n = 32
So, we have found that the number of nickels, n, is 32.
To find the sum of the digits in the number of nickels, we add the digits of 32:
3 + 2 = 5
Therefore, the sum of the digits in the number of nickels in Keana's bank is 5.