Without using a calculator or paper can you think which of these two expressions has the largest value?
23 x 25
24^{2}
Put your left hand up if you think 23 x 25 is the largest.
Put your right hand up if you think 24^{2} is the largest.
Topics: Starter  Mental Methods  Mixed
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24^{2} is the largest.
Here is a method for working out the answer to the first pair of values you see when this page is loaded; Which is the largest: 23 × 25 or 24^{2}?
You could imagine a 23 by 25 rectangle overlapping a 24 by 24 square with one common corner.
Let the area of the common, overlapping region be A then the area of the rectangle is A + 23 and the area of the square is A + 24. It is then quite clear that the area of the square is greater so the answer is 24^{2}.
Another strategy would be to realise that the results of the two calculations are quite close so think only of the final digit.
A third strategy (for those with more algebra) might be to consider the general case when the first expression is (n − 1)(n + 1) and the second expression is n^{2}
The first expression can be simplified to n^{2} − 1, clearly smaller than the second expression.
There is a printable worksheet to go with this activity.
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