The height of a triangle is increased by 40%. What can be the maximum percentage increase in length of the base so that the increase in area is restricted to a maximum of 60%?

Question

Mohit C · Accepted Answer

It&rsquo;s easier to treat this as a percentage compounding problem.
&bull; Original height: 100% &rarr; increased by 40% &rarr; 140% of original. &bull; Let base increase by x%. Then base = (100 + x)% of original. &bull; Area change = (140 &times; (100 + x)) / 10000 in relative terms.
For the area to grow by at most 60%, (140 &times; (100 + x)) / 10000 &le; 1.6 &rarr; 140 &times; (100 + x) &le; 16000 &rarr; 100 + x &le; 114.2857 &rarr; x &le; 14.29.
So the base can go up by a maximum of about 14.3%. Anything more pushes area growth over the 60% cap.

S Raghav · Answer

When height increases by 40%, it becomes 1.4 times the original. Since the area of a triangle is &frac12; &times; base &times; height, the new area factor is 1.4 &times; (new base &divide; old base).
If the maximum allowed increase in area is 60%, the new area can be at most 1.6 times the original. So: 1.4 &times; (base factor) &le; 1.6 &rarr; base factor &le; 1.6 / 1.4 &asymp; 1.1429.
That means the base can be increased by at most 14.29%. Any more, and the total area growth crosses the 60% limit. This kind of question is common in aptitude tests where proportional reasoning is faster than plugging into formulas repeatedly.