NRICH: Which Is Bigger?

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

You may wish to explore the problem Which Is Cheaper? before working on this task.

Which is bigger, n+10, or 2n+3?
How did you decide?

Here’s how I decided:


“I wonder what happens when n=4.”
4+10=14 but 2×4+3 is only 11.”
“So it looks like n+10 is bigger.”


But then my friend said:

“I wonder what happens when n=10.”
10+10=20 but 2×10+3 is 23.”
“So it looks like 2n+3 is bigger.”


Can you explain why we have come to different conclusions?
Is there a diagram you could draw that would help?

For the following pairs of expressions, can you work out when each expression is bigger?

2n+7 and 4n+11
2(3n+4) and 3(2n+4)
2(3n+3) and 3(2n+2)
Here are some challenges to try:
  • Find two expressions so that one is bigger whenever n<5 and the other is bigger whenever n>5.
  • Find three expressions so that the first is biggest whenever n<0, the second is biggest whenever n is between 0 and 4, and the third is biggest whenever n>4.
  • Find three expressions so that the first is biggest whenever n<3, the second is biggest when n>3, and the third is never the biggest.
  • Find three expressions so that one of them is the biggest regardless of the value of n.

Age 14 to 16

Algebra & Pre-Algebra, Equations
High School, Educator

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