Estimation Strategies

A physics question asks for the kinetic energy of a 68 kg runner moving at 4.7 m/s. You could compute ($\frac{1}{2}$)(68)(4.7)² exactly, but that takes forever without a calculator. Instead: round 68 to 70, round 4.7 to 5, and $5^{2}$ = 25. KE = ($\frac{1}{2}$)(70)(25) = 875 J. The answer choices are 400 J, 750 J, 1100 J, and 2200 J. Only 750 J is close, and your rough estimate of 875 J (slightly high because you rounded both numbers up) confirms it. Twelve seconds, no calculator, correct answer.

This section ties together everything in this chapter into a unified estimation strategy. On the MCAT, speed and confidence matter more than precision.

Strategy 1: Round to One Significant Figure

This is your default move. Replace every number with its first non-zero digit.

• 6.02 x $10^{23}$ becomes 6 x $10^{23}$
• 4.7 becomes 5
• 0.082 becomes 0.08
• 9.8 becomes 10
• 3.14 becomes 3

Then do the arithmetic with these simpler numbers. You will typically be within 10-20% of the exact answer, which is more than enough given MCAT answer spacing.

Strategy 2: Use Known Approximations

Memorize these common values:

Strategy 3: Powers-of-10 Reasoning

When the question really just asks “how big is this?” you do not need the coefficient at all. Just track the powers of 10.

Example: Avogadro’s number is about $10^{23}$. The mass of a proton is about $10^{-27}$ kg. How many protons are in 1 kg?

Number = 1 / $10^{-27}$ = $10^{27}$. But each mole of protons is about 1 g = $10^{-3}$ kg and contains $10^{23}$ protons. So 1 kg contains about $10^{3}$ moles x $10^{23}$/mole = $10^{26}$ protons. The exact coefficient does not matter - you are on the right order of magnitude.

Strategy 4: Process of Elimination

Even a rough estimate can eliminate 2-3 answer choices. Then, if needed, refine your estimate to choose between the remaining options.

Example: A question asks for the wavelength of light with frequency 5 x $10^{14}$ Hz.

λ = c/f = (3 x $10^{8}$) / (5 x $10^{14}$) = 0.6 x $10^{-6}$ = 6 x $10^{-7}$ m = 600 nm

If the choices are: (A) 6 nm, (B) 60 nm, (C) 600 nm, (D) 6000 nm - you only needed the exponent to eliminate three choices.

Strategy 5: Bracket the Answer

If you are unsure about rounding direction, calculate twice: once rounding up and once rounding down. The correct answer must lie between those two estimates.

Example: What is 67 x 8.3?

Lower bound: 60 x 8 = 480
Upper bound: 70 x 9 = 630
Actual: 556

If the answer choices are 280, 420, 560, and 840 - only 560 falls in the bracket.

Strategy 6: Compensating Rounds

When you round one number up, try to round the other down to partially cancel the error.

Example: 4.2 x 7.8

If you round both up: 5 x 8 = 40 (too high)
If you compensate: round 4.2 up to 4.5 and 7.8 down to 7.5: 4.5 x 7.5. Or simpler: 4 x 8 = 32 (round one down, other up). Actual: 32.76. The compensating round was nearly exact.

Worked Practice Examples

Problem 1: What is the pH of a 0.003 M HCl solution?

HCl is a strong acid, so [H+] = 3 x $10^{-3}$ M. pH = -log(3 x $10^{-3}$) = -(log 3 + log $10^{-3}$) = -(0.5 - 3) = 2.5.

Problem 2: A sound increases from 40 dB to 70 dB. By what factor did the intensity increase?

Change = 30 dB. Every 10 dB = 10x intensity. 30 dB = 10 x 10 x 10 = 1000x increase.

Problem 3: An electron (m = 9.1 x $10^{-31}$ kg) moves at 2 x $10^{6}$ m/s. Estimate its kinetic energy.

KE = ($\frac{1}{2}$)m$v^2$ = (0.5)($10^{-30}$)(2 x $10^{6}$)² = (0.5)($10^{-30}$)(4 x $10^{12}$) = 2 x $10^{-18}$ J. Rounding 9.1 to 10 and doing quick exponent math gets you there in seconds.