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Ladder and Cube


Doesn't Add Up


From All Corners


Which Is Bigger?

Age 14 to 16
Challenge Level


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 $2 \times 4 + 3$ 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 $2 \times 10 +3$ 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 $n.$

You may also be interested in the other problems in our Many ways to see Feature.