Estimate the square root of 150

Question

No calculator. Solve in your head. Estimate \sqrt{150} to two decimal places. Aim: under 15 seconds.

Accepted Answer

150 sits just above 144 = 12^2. Using \sqrt{a^2 + d} \approx a + \dfrac{d}{2a} with a = 12, d = 6: \sqrt{150} \approx 12 + \frac{6}{24} = 12 + 0.25 = 12.25. Check: 12.25^2 = 150.06, accurate to two decimals. Takeaway. The correction 	frac{d}{2a} is remarkably good when d is small next to a^2. Here 	frac{6}{24} = 0.25 landed the answer almost exactly, because 12.25^2 overshoots by only the tiny \left(	frac{d}{2a}ight)^2 term.

Estimate the square root of 150

Your answer

More Mental Math questions