Okay, let's tackle this problem step by step. So, we have two equations here involving pencils and a jumbo eraser. The goal is to find the cost of a pencil in cents. Let's start by understanding what we're given.

First, the problem states that three pencils and one eraser cost $1.24. Since we need the answer in cents, I should convert that dollar amount to cents right away. $1 is 100 cents, so $1.24 is 124 cents. Got that. So, the first equation is 3 pencils plus 1 eraser equals 124 cents.

Similarly, the second part says five pencils and one eraser cost $1.82. Converting that to cents, $1.82 is 182 cents. So, the second equation is 5 pencils plus 1 eraser equals 182 cents.

Now, I need to find the cost of a single pencil. Let's denote the cost of a pencil as P (in cents) and the cost of the eraser as E (also in cents). So, translating the problem into equations:

1. 3P + E = 124
2. 5P + E = 182

Hmm, okay. Now, we have a system of two equations with two variables. The standard way to solve this is by elimination or substitution. Let me see which method would be easier here.

If I subtract the first equation from the second, the E terms will cancel out. Let's try that:

(5P + E) - (3P + E) = 182 - 124

Simplifying the left side: 5P - 3P + E - E = 2P

Right side: 182 - 124 = 58

So, 2P = 58. Then, dividing both sides by 2 gives P = 29.

Wait, that seems straightforward. So, the cost of a pencil is 29 cents. Let me check if this makes sense by plugging the value back into the original equations.

First equation: 3*29 + E = 124

3*29 is 87. So, 87 + E = 124. Subtracting 87 from both sides gives E = 124 - 87 = 37. So, the eraser would be 37 cents.

Second equation: 5*29 + E = 182

5*29 is 145. So, 145 + E = 182. Subtracting 145 gives E = 182 - 145 = 37. Yep, same result. So both equations are satisfied with P = 29 and E = 37. Therefore, the cost of a pencil is 29 cents.

But let me just make sure I didn't make any arithmetic errors. Let's redo the subtraction step. 182 - 124. 182 minus 100 is 82, minus another 24 is 58. Correct. Then 2P = 58, so P = 29. Then substituting back, 3*29 is indeed 87, 87 + 37 is 124. 5*29 is 145, 145 + 37 is 182. All checks out. So, the answer should be 29 cents.
Therefore, the cost of a pencil is $\boxed{29}$ cents.
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