Annna wants to buy her grandmother a gift. Anna decides to buy a piece of jewelry. At the store she sees that 1/2 of the jewelry is necklaces. 1/4 of the jewelry is pins. The rest of the jewelry is a total of 16 bracelets and rings. There are 3 times as many bracelets as rings. How many of each piece of jewelry does the store have for Anna to decide what to buy for her grandmother? If necklaces average $25, pins average $12, bracelsts average $8 and rings average $30,about how much inventory is in this jewelry store?
please explain steps along with answers
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If you tell me what you've figured out on your own, I'll be happy to walk you through the rest of it!
To solve this problem, we will use algebraic equations to represent the information given in the problem statement.
Let's start by assigning variables to each unknown quantity:
Let N be the number of necklaces.
Let P be the number of pins.
Let B be the number of bracelets.
Let R be the number of rings.
Now, let's set up the equations using the information given:
1) "1/2 of the jewelry is necklaces"
This can be written as: N = (1/2)(N + P + B + R)
2) "1/4 of the jewelry is pins"
This can be written as: P = (1/4)(N + P + B + R)
3) "The rest of the jewelry is a total of 16 bracelets and rings."
This can be written as: B + R = 16
4) "There are 3 times as many bracelets as rings"
This can be written as: B = 3R
Now, let's solve these equations step by step.
From equations (1) and (2), we can eliminate the variables N and P to solve for B and R:
N = (1/2)(N + P + B + R)
P = (1/4)(N + P + B + R)
Rearranging equation (1):
(N + P + B + R) = 2N
P + B + R = 2N
Now substitute P + B + R = 2N into equation (2):
(1/4)(P + B + R) = (1/4)(2N)
P = (1/8)(2N)
Substitute P in equation (3):
B = 3R
Now substitute P = (1/8)(2N) and B = 3R into equation (3):
(1/8)(2N) + 3R + R = 16
N/4 + 4R = 16
Multiply both sides of the equation by 8 to eliminate fractions:
2N + 32R = 128
Now we have a system of two equations:
N/4 + 4R = 16
2N + 32R = 128
Solve equation 1 for N:
N = 16 - 4R
Substitute N in equation 2:
2(16 - 4R) + 32R = 128
32 - 8R + 32R = 128
24R = 96
R = 4
Now substitute R = 4 into equation B = 3R:
B = 3(4) = 12
Substitute R = 4 into equation N = 16 - 4R:
N = 16 - 4(4) = 16 - 16 = 0
Finally, substitute R = 4 into equation N = (1/8)(2N):
0 = (1/8)(2N)
0 = N
Since N = 0, there are no necklaces in the store.
To calculate the inventory value, multiply the quantity of each type of jewelry by its average price and sum them up:
Value of pins = P x average price of pins = (1/4)(N + P + B + R) x $12 = (1/4)(0 + P + 12 + 4) x $12 = (1/4)(P + 16) x $12
Value of bracelets = B x average price of bracelets = 12 x $8 = $96
Value of rings = R x average price of rings = 4 x $30 = $120
Total inventory value = Value of pins + Value of bracelets + Value of rings
= (1/4)(P + 16) x $12 + $96 + $120
Plug in the value of P (P = 1/8)(2N)
=(1/4)(((1/8)(2N)) + 16) x $12 + $96 + $120
=(1/4)(((1/8)(2 x 0)) + 16) x $12 + $96 + $120
=(1/4)((0) + 16) x $12 + $96 + $120
=(1/4)(16) x $12 + $96 + $120
=4 x $12 + $96 + $120
=$192 + $96 + $120
=$408
Therefore, the inventory value in this jewelry store is $408.