Pencils and Jars Solution

This is the solution to the Pencils and Jars Puzzle.

Puzzle 1 can be solved simply by trying some different numbers.

But if we want to solve puzzle 2 it is a bit more tricky.

So we will solve puzzle 1 using Algebra, to make it easier to tackle puzzle 2.

Puzzle 1

The first stage is to read the question carefully:

I have some pencils and some jars.
If I put 4 pencils into each jar I will have one jar left over
If I put 3 pencils into each jar I will have one pencil left over
How many pencils and how many jars?

There are two distinct statements. Let's look at statement one first.

When there are four pencils in each jar there will be one jar left over In words the number of jars is equal to the number of pencils divided by four plus another one.

This can be expressed using algebra.

j = (p/4) + 1

Now look at statement 2: When there are five pencils in each jar there will be one pencil left.

We have created a formula to calculate how many Jars there are. We now need to calculate how many pencils there are We can say that the number of pencils will be equal to the number of jars times three plus another one.

Using algebra again we have:

p = 3j + 1

We have ended up with two linear equations.

There are two ways to solve these: Graphically or using simultaneous equations. We will use graphs!

First rearrange the formula to make j the subject:

j = (p/4) + 1 (equation 1)

j = (p − 1) ÷ 3 (equation 2)

Plotting these on a graph using x=p and y=j;
If we find the intersection point we have our answer

And that is it: there are 16 pencils and 5 Jars.

Puzzle 2

Ahh! That is for YOU to work out, using the same method I have just shown you.