a box of pencils cost the same as 11 erasers and 7 pencils. a box of erasers cost the same as 6 erasers and a pencil. a box of empty boxes and an erasers cost the same as a pencil. given that boxes cost a penny and each of the boxes contain an equal number of objects, how much does it costs to buy a box of erasers combined?
Let's break down the given information step by step to find the cost of buying a box of erasers.
Step 1: Let's assign variables to the unknown quantities:
Let "x" be the cost of a box of pencils.
Let "y" be the cost of a box of erasers.
Let "z" be the cost of a pencil.
Let "w" be the cost of an empty box.
Step 2: From the given information, we can form the following equations:
1. A box of pencils cost the same as 11 erasers and 7 pencils:
x = 11y + 7z
2. A box of erasers cost the same as 6 erasers and a pencil:
y = 6z + 1
3. A box of empty boxes and an eraser cost the same as a pencil:
w + y = z
Step 3: We can substitute equation 2 into equation 1 to eliminate "y":
x = 11(6z + 1) + 7z
x = 66z + 11 + 7z
x = 73z + 11
Step 4: We can substitute equation 3 into equation 1 to eliminate "y":
w + y = z
w = z - y
x = 11y + 7z
x = 11(z - w) + 7z
x = 11z - 11w + 7z
x = 18z - 11w
Step 5: Since the cost of boxes is given as a penny, we know that w = 0.01.
Substituting this value into the equations:
w = 0.01
x = 73z + 11
x = 18z - 11(0.01)
x = 18z - 0.11
Step 6: Now we can solve the system of equations:
x = 73z + 11
x = 18z - 0.11
By setting the two expressions for "x" equal to each other, we can solve for "z":
73z + 11 = 18z - 0.11
55z = -11.11
z ≈ -0.202 (rounded to three decimal places)
Step 7: Since the cost cannot be negative, we discard the negative value for "z".
Therefore, the cost of a pencil is approximately 0.202 cents.
Step 8: Now we can substitute the value of "z" into equation 2 to find the cost of a box of erasers:
y = 6z + 1
y = 6(0.202) + 1
y ≈ 2.212 (rounded to three decimal places)
Therefore, the cost of a box of erasers is approximately 2.212 cents.
To find out the cost of a box of erasers, we need to assign variables and set up equations based on the given information. Let's say the cost of a box of pencils is $x, the cost of a box of erasers is $y, and the cost of a pencil is $p.
From the first statement, "a box of pencils cost the same as 11 erasers and 7 pencils," we can write the equation:
x = 11y + 7p ----(1)
From the second statement, "a box of erasers cost the same as 6 erasers and a pencil," we can write another equation:
y = 6y + p ----(2)
Now, from the third statement, "a box of empty boxes and an eraser cost the same as a pencil," we know that an empty box costs a penny and the number of objects in each box is the same. Since an empty box costs a penny, we can represent this as:
y + 1 = p ----(3)
Now, we can solve these equations to find the value of y, which is the cost of a box of erasers.
First, we solve equation (3) for p:
p = y + 1
Substituting this value of p in equation (2), we get:
y = 6y + (y + 1) ----(4)
Simplifying equation (4):
y = 7y + 1
Bringing all the y terms to one side:
6y = 1
Dividing both sides by 6, we find:
y = 1/6
Therefore, the cost of a box of erasers is 1/6 of a penny (or approximately 0.1667 cents).
Note: The given information suggests that each box contains an equal number of objects, but it doesn't specify what that number is. Hence, we cannot determine the exact number of erasers in a box based on the given information, only the cost.