Problem Solving: The Basics

Many students struggling with math generally ask questions along the line of "How am I supposed to approach this problem?" and/or "What is this question asking of me?" Here are some basic tips to help you get started:

Problem: A steel rod 100 meters long is cut in half. If one piece is 28 meters longer than the other, what is the length, in meters, of the shorter piece?

1) Actively engage the question, breaking it down, step-by-step, into its constituent elements from a general to a specific framework.

With regards to the above problem, a shorter piece and a longer piece add up to the total length of the rod. This can be translated into the algebraic expression:

S + L = 100

Notice how I symbolized each part of the rod. The shorter piece is expressed as "S" for its shortness in comparison to the longer piece and the longer piece is expressed as "L" for its longness in comparison to the shorter piece. This helps you remember to represent your expressions with distinctive letters/symbols in order to maintain clarity throughout the question and avoid any confusion when you are answering the question.

Now, each letter can be expressed in relation to each other. As the problem states, the longer piece is 28m longer than the shorter piece. This can be expressed as follows:

L = S + 28

You can use hypothetical numbers to make sure you are on the right track. L must always be 28 m longer than S. So, if your shorter piece is longer than your longer piece (literally defying logic), then you have to go back and refurbish your equation. With that in mind, it's time to relate both pieces of the rod to its total length. Notice how the equation for the longer piece of rod (L = S + 28) can be substituted into the equation for the rod's total length:

S + L (S + 28) = 100

This can be simplified into:

S + S + 28 = 100

2S +28 = 100

2S = 100 -28

2S = 72

S = 72 / 2

S = 36

2) Verify your answer

You can never be too careful with math problems. But, the beauty of this system of equations is that, if done correctly, you can verify your answer by plugging it in to one of the other equations to see if it makes mathematical sense. Why don't we plug it in to the equation for the longer piece and see what we get?:

L = S +28

L = 36 + 28

L = 64

Is this right? Well, remember our equation for the total length of the rod:

S + L = 100

36 + 64 = 100

It checks out! Every single equation can be substituted into each other and can, thus, be verified.

3) Make sure you are answering what the question is asking!

You might be thinking, "duh, that's obvious" and it is. But, notice how easy it would be to get lost in our set of equations. Remember, the question asked for the length of the shorted piece of rod, not the length of the longer piece. So, the correct answer is 36 m (remember to always include your units of measurement in order to keep yourself engaged in the process), not 64 m. We can all make mistakes as silly as this if we are not careful. Take your time and don't rush through things!

This problem is meant to be simple because acquiring the basics is very important. The more complex the problems, the more complex the translations of information into algebraic equations. But, the process is nearly always the same. If you able to master these basic principles, no question will be too hard for a budding math whiz such as yourself!

  • Posted By Alejandro ("Al")
  • #math10-3 #math10c #math20-1 #math20-2 #math20-3 #math30-1 #math30-2 #math31 #mathgrades3-6 #mathgrades7-9