How To Solve This Tricky Algebra Problem (XIII) - Whiteboard-style graphics asking the following question: 98*99*100*101*102 = ?? Below this question, the following text is written in a box: "Use Algebra"

Welcome to the thirteenth entry in the tricky algebra problem series. This series celebrates the joy of algebra, where the featured problems range from beginner to advanced levels.

Before we proceed with this entry’s problem, let me address the elephant in the room. This problem is not your typical algebra problem; it can be solved using a calculator in a snap.

The catch is that you are not allowed to use a calculator and have to use algebra to solve it. In short, this is not an algebra problem because it is, but because it wants to be one.

So, give it some math love by giving it a shot. How would you proceed to solve this problem using algebra?

This essay is supported by Generatebg

Spoiler Alert

If you wish to solve this problem on your own, I recommend you pause reading this essay at this point. I will be explicitly discussing the solution to this problem beyond this section.

Once you are done with your attempt, you may continue reading and comparing approaches.

Setting Up the Tricky Algebra Problem

This problem becomes vastly simple if you think in algebra terms. Look at it a little closer, and we could see straight away that we could represent all the numbers involved in terms of 100 as follows:

How To Solve This Tricky Algebra Problem (XIII) — Whiteboard-style graphics showing the following mathematical operations: 98*99*100*101*102 = (100 − 2)*(100 − 1)*(100)*(100 + 1)*(100 + 2)
Thinking in algebra-terms — Math illustrated by the author

When we slightly rearrange the terms, we get the following outcome:

How To Solve This Tricky Algebra Problem (XIII) — Whiteboard-style graphics showing the following mathematical operations: (100 − 2)*(100 − 1)*(100)*(100 + 1)*(100 + 2) = (100)*(100 + 1)*(100 − 1)*(100 + 2)*(100 − 2)
The rearranged expression — Math illustrated by the author

Do you notice any familiar algebra patterns here? If not, don’t worry. In the next few steps, things will get much clearer. For ease of solving and in the spirit of algebra, let us replace 100 with the variable ‘x’:

How To Solve This Tricky Algebra Problem (XIII) — Whiteboard-style graphics showing the following mathematical operations: (100)*(100 + 1)*(100 − 1)*(100 + 2)*(100 − 2). Let x = 100. Then, the expression becomes: x*(x+1)*(x − 1)*(x+2)*(x − 2)
Replacing 100 with ‘x’ — Math illustrated by the author

Do you see the pattern now? We have two occurrences of the following algebraic identity:

(a + b) * (a − b) = a² − b²

By applying this algebraic identity, we get the following outcome:

How To Solve This Tricky Algebra Problem (XIII) — Whiteboard-style graphics showing the following mathematical operations: x*(x+1)*(x − 1)*(x+2)*(x − 2). Applying the identity (a+b)*(a−b) = a² − b², we get the following result: x*(x² − 1²)*(x² − 2²) =x* (x² − 1)*(x² − 4)
Applying the algebraic identity — Math illustrated by the author

With this step, we are now very close to solving our problem.

The Solution to the Tricky Algebra Problem

Let us start by substituting 100 in place of ‘x’ back in the above expression:

How To Solve This Tricky Algebra Problem (XIII) — Whiteboard-style graphics showing the following mathematical operations: x* (x² − 1)*(x² − 4); Substituting x=100 in this expression, we get: 100*(100²−1)*(100²−4)
Substituting x = 100 — Math illustrated by the author

Next, we could proceed using simple cross-multiplication as follows:

How To Solve This Tricky Algebra Problem (XIII) — Whiteboard-style graphics showing the following mathematical operations: 100*(10⁰²−1)*(10⁰²−4) = 100*(100⁴ − 4*(100²) − 1*(100²) + 4) = 100*(100000000 − 50000 + 4) = 100*(99950000+4) = 9995000400
The solution — Math illustrated by the author

There we go. We have solved this problem using algebra!

Final Thoughts

The astute reader might have observed that we did not need to introduce the variable ‘x’ to solve this problem. However, I did that to make the pattern clearer for the reader who did not see it straight away.

Furthermore, this problem serves as a good example to see the application of algebra to a problem relatable to real-life situations.

People are simply more likely to come across multiplication problems involving large numbers without a calculator than, let’s say, number theory problems.

I hope you enjoyed this engaging and simple algebra problem. If such problems are your thing, keep an eye on this space for more in the future. Thanks for reading!

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