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Solution Pencils and Jars

Straight to the answer
Of course problem 1 can be solved simply by trying some different numbers However, if we want to solve problem 2 it is a bit more tricky. What we will do here is show how to solve the first problem using some algebra. Once you have looked at this bit of maths try and apply the same principles to problem 2.

The first stage is to read the question carefully.

Problem 1

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 an 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 and using simultaneous equations. The approach taken here will be using a graph.

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.