A pencil, eraser and notebook cost a total of $1. If a pencil costs more than 2 erasers, 8 erasers cost more than 3 notebooks, and 5 notebooks cost more than 6 pencils, find the cost of a pencil. I don't know what to do. please help!!!!
n+p+4=100¢
p>2e
8e>3n
5n>6p
I sort of used systematic trial and error for this.
But I got:
n=45¢, p=37¢, and e=18¢
It works since:
37>2*18
8*18>3*45
5*45>6*37
Good luck (,from five years later)!
To find the cost of a pencil, let's assign variables to the unknowns. Let's say the cost of a pencil is "p" dollars, the cost of an eraser is "e" dollars, and the cost of a notebook is "n" dollars.
1. From the given information, we know that:
- A pencil, eraser, and notebook cost a total of $1, so we can write the equation: p + e + n = 1. (Equation 1)
- A pencil costs more than 2 erasers, so we can write the inequality: p > 2e. (Inequality 1)
- 8 erasers cost more than 3 notebooks, so we can write the inequality: 8e > 3n. (Inequality 2)
- 5 notebooks cost more than 6 pencils, so we can write the inequality: 5n > 6p. (Inequality 3)
2. Let's solve the inequalities to narrow down the range of possible values.
- Since p > 2e from Inequality 1, we can rewrite it as p = 2e + k1, where k1 is a positive constant.
- Similarly, we can rewrite Inequality 2 as e = (3/8)n + k2, where k2 is a positive constant.
- And we can rewrite Inequality 3 as n = (6/5)p + k3, where k3 is a positive constant.
3. Let's substitute these rewritten equations into Equation 1 to get rid of the variables.
p + e + n = 1 can be rewritten as:
(2e + k1) + ((3/8)n + k2) + ((6/5)p + k3) = 1
Simplifying, we get:
(6/5)p + (2/8)n + (1/2)e + k1 + k2 + k3 = 1
Multiplying both sides by 5, we get:
6p + (5/4)n + (5/2)e + 5k1 + 5k2 + 5k3 = 5
4. Let's simplify this equation further:
6p + (5/4)n + (5/2)e = 5 - 5k1 - 5k2 - 5k3
Because the right side is a constant, let's assume 5 - 5k1 - 5k2 - 5k3 = K, where K is a positive constant.
5. Our equation now becomes:
6p + (5/4)n + (5/2)e = K
6. Now, with K as a positive constant, we can solve for e in terms of p.
Rearranging the equation, we have:
(5/2)e = K - 6p - (5/4)n
e = (2/K)(K - 6p - (5/4)n)
7. Let's substitute the value of e from step 6 into Inequality 1 (p > 2e).
We replace e with (2/K)(K - 6p - (5/4)n), so we have:
p > 2(2/K)(K - 6p - (5/4)n)
Multiplying both sides by K/4, we get:
(K/2)p > (4/K)(K - 6p - (5/4)n)
Simplifying, we have:
p > (2/K)(K - 6p - (5/4)n)
8. We have now reduced the problem to finding the value of p such that p > (2/K)(K - 6p - (5/4)n).
Without specific values for K, n, and any known constraints, we cannot determine the exact cost of a pencil.
To solve the problem completely, you'll need additional information or specific values for K, n, or any constraints.
To solve this problem, let's assign variables to each item's cost. Let's say that the cost of a pencil is x dollars, the cost of an eraser is y dollars, and the cost of a notebook is z dollars.
We are given the following information:
1) A pencil, eraser, and notebook cost a total of $1, or:
x + y + z = 1
2) A pencil costs more than 2 erasers, or:
x > 2y
3) 8 erasers cost more than 3 notebooks, or:
8y > 3z
4) 5 notebooks cost more than 6 pencils, or:
5z > 6x
We need to find the cost of a pencil (x).
To solve this system of equations, we can use a technique called "substitution." Let's rearrange equation 1 to solve for z and substitute the values into equations 3 and 4:
From equation 1, we have:
z = 1 - x - y
Substituting z in equations 3 and 4, we get:
8y > 3(1 - x - y)
5(1 - x - y) > 6x
Simplifying equations 3 and 4, we have:
8y > 3 - 3x - 3y
5 - 5x - 5y > 6x
Combining like terms in both equations, we get:
8y + 3x + 3y > 3
5 + 6x + 5y > 6x
Further simplifying, we have:
3x + 11y > 3
5y > 0
To solve for y, we can solve equation 5 (5y > 0):
y > 0
Since y cannot be zero (as it represents the cost of an eraser), we know that y is a positive number.
Now, let's consider the inequality 3x + 11y > 3. Since we know y > 0, in order to obtain the minimum value for x, we need 3x to be as small as possible.
The smallest positive integer value for x that satisfies the inequality is x = 1. This means a pencil costs $1.
Therefore, the cost of a pencil is $1.