Which Is Bigger?

'Which Is Bigger?' printed from https://nrich.maths.org/

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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.