Updated October 6, 2026·Reference — Math Refresher·By Kevin L., CRSP & CRST

CIH Calculator & Math Basics

Every CIH equation eventually comes down to the same handful of math moves — logs, exponents, roots, ratios — done on a scientific calculator. If it's been a few years since you've touched any of this, or you've just never thought of yourself as a "math person," this page is the refresher: what each skill actually means, a worked example, and the exact calculator keys to press.

Why this page exists

You don't need to be good at math to pass the CIH exam — you need to recognize a small, repeating set of calculator moves and be able to execute them without hesitating under time pressure. This page covers every one of those moves in plain language, with a real practice example for each, and a working calculator below so you can try them yourself before you ever open a real CIH equation.

The CIH exam calculator: what you get on exam day

The Board for Global EHS Credentialing (BGC) decides what you can calculate with, and its rules changed in ways that matter for practice. This is what its applicant handbook says about calculating on exam day:

No personal calculator. Personal calculators are not allowed in the testing session. A calculator is built into the exam, and the pre-test instructions show where its icon is.
No phone calculator either. Calculators built into mobile phones are not permitted in the test center.
A whiteboard, not paper. Each candidate gets a whiteboard and a dry marker for equation-based questions. Scratch paper is also built into the screen, in the upper left of each question next to the equation sheets and the calculator icon.
Equation sheets on screen. The BGC Equation Sheets open from an Equation Sheet button on the exam screen, so the job is choosing the right equation and keying it quickly, not memorizing it.
An official practice link. BGC lists pearsonvue.com/athena for practicing with the exam calculator.
The clock. The exam has 180 questions (150 scored and 30 pilot questions) in two 2.5-hour sections. That is 300 minutes, or about 100 seconds per question on average, so a calculation cannot be allowed to take four minutes.

Test-prep sources describe the on-screen calculator as an emulation of the TI-30XS MultiView, and the practice calculator on this page uses that key layout. The handbook itself does not name a model, so treat the layout as a close match and use the Pearson link above as your final check.

Source: BGC Applicant CIH Handbook, “What to Expect on Exam Day” and “Examination Purpose and Format” (checked October 6, 2026). Rules can change, so confirm against the handbook before your test window.

Five habits before you press enter

Check the calculator's mode every time you open it. Float or scientific display changes how a result looks, and degree or radian matters for any trig. Some candidates report the on-screen calculator going back to its default mode between questions. The handbook does not mention this, so make the check a habit rather than a hope.
Write the setup on the whiteboard first. The formula, each value with its unit, and which value is which. Then type once.
Type the whole expression with brackets and round only at the end. Chaining rounded intermediate answers is the quiet way to drift into the wrong multiple-choice option.
Close every bracket you open. log, ln, square root and 10^x each open a bracket for you.
Ask whether the answer is plausible. Two equal noise sources add 3 dB, not double. Poor room mixing makes a purge take longer, not shorter. A decayed source reads lower than it did.

What's covered

Ten skills, each showing up over and over across the actual CIH formulas:

Rearranging a formula
Keeping units consistent
Exponents & powers
Square roots & nth roots
Logarithms & antilogs
Scientific notation
Ratios & proportions
Squares, sums of squares & reciprocals
Weighted averages & signed numbers
Basic areas

Open the practice calculator now and keep it open while you work through the examples below — it's the same simulated calculator used throughout the CIH Equation Master app.

The Math, One Skill at a Time

1. Rearranging a Formula

Most exam questions don't hand you the exact formula you need — they give you the answer and ask for one of the inputs. Whatever you do to one side of an equation, you must do to the other: multiply to move a divisor, divide to move a multiplier.

Example

From $a = bc$: to solve for $c$, divide both sides by $b$, giving $c = a/b$.

Harder example

From $y = \frac{x}{2} + 5$: subtract 5 from both sides first, then multiply both sides by 2 — giving $x = 2(y-5)$. Check it: if $y=9$, then $x=2(9-5)=8$, and $8/2+5=9$. ✓

Do the rearrangement on paper first, then type once: 2 × ( 9 - 5 ) enter

2. Keeping Units Consistent

A formula only works if every value going into it is in the same unit system — mixing inches with feet, or seconds with minutes, silently wrecks the answer without ever throwing an error.

Example

An 8-inch measurement must become $8/12 = 0.667$ ft before it can combine with a value already in feet.

Harder example

A rate quoted as "90 seconds per cycle" needs converting to minutes before it can combine with a flow rate in cfm (cubic feet per minute): $90 \div 60 = 1.5$ minutes.

90 ÷ 60 enter

3. Exponents & Powers

$x^n$ means $x$ multiplied by itself $n$ times. Small numbers raised to a power can grow — or shrink — much faster than intuition suggests, especially once the exponent gets past 2 or 3.

Example

Repeated halving: $0.5^4 = 0.0625$ — four halvings leaves just 6.25% of the original amount.

.5 ^ 4 enter

Harder example

A small percentage change raised to a higher power creates a much bigger swing: $1.1^5 \approx 1.61$ — a 10% increase becomes a 61% increase once raised to the 5th power. Also try $2^{10}=1{,}024$ — ten doublings takes you from 1 to over a thousand.

1.1 ^ 5 enter | 2 ^ 10 enter

4. Square Roots & nth Roots

A square root undoes a square: $\sqrt{x^2}=x$. An nth root is the same idea generalized — it's really just a fractional power: $\sqrt[n]{x}=x^{1/n}$.

Example

$\sqrt{0.04+25}=\sqrt{25.04}\approx5.004$. When a formula adds two squared quantities before rooting, both must already be in the same unit.

2nd x² types sqrt(

Harder example

$\sqrt[5]{3125} = 3125^{1/5} = 5$. And roots aren't always positive-only: $\sqrt[3]{-27}=-3$, since $(-3)^3=-27$ — odd-numbered roots can take a negative input, unlike square roots.

3125 2nd ^ 5 ) enter

5. Logarithms & Antilogs

Read $\log(x)$ as "what power of 10 gives me $x$?" — it converts a ratio into an order of magnitude. Antilog reverses it: $10^{\log(x)}=x$. Natural log ($\ln$) and $e^x$ are the same relationship using the natural base $e\approx2.71828$ instead of 10.

Example

$\log(1000)=3$, because $10^3=1000$. And $e^{-1.5525}\approx0.2117$ — a negative exponent always means decay toward zero.

log 1000 ) enter | 2nd ln types e^(

Harder example

Log-scale quantities never add directly. Two values that are each $10^9$ don't combine to "$10^{18}$" — add the actual numbers first, then take the log: $10^9+10^9=2\times10^9$, and $\log(2\times10^9)\approx9.3$, not 18.

log 2 × 2nd log 9 ) enter — this calculator has no "implied multiplication," so open with log first and use × to combine the 2 with the 10⁹, rather than typing the 2 before 2nd log (which would be read as the single number "210", not "2 times 10").

6. Scientific Notation

Numbers that span an enormous range are written as a number times a power of ten, so you're never stuck counting zeros by hand.

Example

1 Curie $= 3.7 \times 10^{10}$ Becquerels — 37 billion, written compactly.

3.7 ×10ⁿ 10

Harder example

Small numbers work the same way, with a negative exponent: $0.000045 = 4.5\times10^{-5}$. Count how many places the decimal point moves — 5 places right to reach 4.5, so the exponent is $-5$.

4.5 ×10ⁿ (−) 5 enter — press the single (−) key (bottom row, next to enter) before the 5, not after. It's one "make the next number negative" key — don't build it out of the separate (, −, ) keys elsewhere on the pad, which type literal parentheses and a subtraction sign instead and will error.

7. Ratios & Proportions

When a formula is built from a ratio, the actual units cancel out — only consistency between the two sides matters. Squaring a ratio means small changes get amplified, not just doubled.

Example

$\left(\frac{2}{4}\right)^2=0.25$ — a ratio of one-half becomes one-quarter once squared.

( 2 ÷ 4 ) x²

Harder example

If two quantities are inversely proportional (their product stays constant) and one doubles, the other must halve to compensate: with $a \times b = 12$, if $a$ goes from 3 to 6, $b$ must go from 4 down to 2 — $6\times2=12$, same as $3\times4=12$.

12 ÷ 6 enter

8. Squares, Sums of Squares & Reciprocals

Squaring makes every deviation positive so they can't cancel each other out. Summing reciprocals (then inverting the sum) is how you combine rates or parallel contributions — never just average them.

Example

Deviations of $+2$ and $-2$ sum to zero, but their squares sum to 8 — squaring is what keeps variability formulas from cancelling to nothing.

x² after each value, then add

Harder example

Classic "two pipes" problem: one pipe alone fills a tank in 100 minutes, another alone in 50 minutes. Combined, they don't average to 75 minutes — sum the reciprocals (rates), then invert: $\frac{1}{\frac{1}{100}+\frac{1}{50}} = 33.3$ minutes.

( 1 ÷ 100 + 1 ÷ 50 ) x⁻¹ enter

9. Weighted Averages & Signed Numbers

A weighted average multiplies each value by its own weight (the weights always sum to 1.0) before adding — it is not the same as a plain average. Some formula terms can be positive or negative depending on the situation, so track signs deliberately rather than assuming everything adds.

Example

Three grades weighted 50% / 30% / 20%: $0.5(80)+0.3(90)+0.2(70) = 81$. A plain average of 80, 90, and 70 would give 80 — a meaningfully different number.

.5 × 80 + .3 × 90 + .2 × 70 enter

Harder example

A running total with a gain and two losses: net $=500-200-150=150$. When a term's sign can flip depending on conditions (a gain in one scenario, a loss in another), use the calculator's (−) key to enter it as negative rather than trying to track the sign in your head.

500 - 200 - 150 enter

10. Basic Areas

A rectangular area is width × height; a circular area is $\pi r^2$. Keep every length in the same unit before multiplying — area formulas amplify a unit mistake instead of just carrying it through.

Example

A 2 ft × 3 ft rectangle is 6 ft².

2 × 3 enter

Harder example

A circular opening with radius 1.5 ft: $\pi(1.5)^2 \approx 7.07$ ft².

π × 1.5 x² enter

Six exam-style calculations, with every keystroke

The skills above are easier to trust once you have used them on a whole problem. These six follow the multi-step pattern of the calculation questions: a short scenario, two or three equations chained together, and answer choices that match the common slips. Try each on paper first, then open the answer. Every number here was worked out and checked, and the keystrokes follow the practice calculator on this page.

Noise · distance, then combine

Question 1

Two identical air compressors each produce 97 dB at 1 m in an open yard (treat each as a point source in a free field). A technician stands 4 m from both. What is the combined sound pressure level at the technician?

85.0 dB
88.0 dB
94.0 dB
100.0 dB

Show the answer and every keystroke

Answer: B — 88.0 dB

Step 1 — move each source out to 4 m

Sound pressure falls with distance as $SPL_2 = SPL_1 + 20\log(d_1/d_2)$, so each compressor is $97 + 20\log(1/4) = 84.96$ dB at the technician.

97 + 20 × log 1 ÷ 4 ) enter → 84.96

Step 2 — combine the two equal sources

Decibels do not add directly. For $n$ identical sources the total is $L + 10\log n$, which is what $L_{PT}=10\log\sum 10^{L_i/10}$ collapses to: $84.96 + 10\log 2 = 87.97 \approx 88.0$ dB.

84.96 + 10 × log 2 ) enter → 87.97

The traps: Skipping the distance step gives 100.0 dB. Using $10\log$ instead of $20\log$ for the distance gives 94.0 dB. Skipping the combine step gives 85.0 dB.

Equations used: Distance Change (Point Source) and Total Level (Combining Sources)

Noise · dose across three periods, then TWA

Question 2

A press operator's day is 2 h at 90 dBA, 3 h at 87 dBA and 3 h at 82 dBA. The program uses an 85 dBA criterion level, an 8-hour criterion duration and a 3 dB exchange rate, and all three periods count toward the dose. What is the 8-hour TWA?

85.9 dBA
86.4 dBA
87.0 dBA
93.3 dBA

Show the answer and every keystroke

Answer: C — 87.0 dBA

Step 1 — permissible time at each level

$T_p = 8 / 2^{(L-85)/3}$ gives 2.52 h at 90 dBA, 5.04 h at 87 dBA and 16.0 h at 82 dBA.

8 ÷ 2 ^ ( ( 90 - 85 ) ÷ 3 ) enter → 2.52 (repeat with 87 and 82)

Step 2 — add the dose fractions

$\%D = 100\,(2/2.52 + 3/5.04 + 3/16.0) = 157.6\%$.

100 × ( 2 ÷ 2.52 + 3 ÷ 5.04 + 3 ÷ 16 ) enter → 157.6

Step 3 — convert the dose to a TWA

$TWA = 10\log(\%D/100) + 85 = 10\log(1.576)+85 \approx 87.0$ dBA.

10 × log 1.576 ) + 85 enter → 86.98

The traps: Leaving out the 82 dBA period gives a dose of 138.9% and 86.4 dBA. Averaging the three levels by time gives 85.9 dBA, which ignores that decibels are logarithmic. Using the 5-dB exchange-rate TWA formula gives 93.3 dBA.

Equations used: Permissible Time, Dose Percentage and TWA (3-dB Exchange Rate)

Ventilation · purge time with a mixing factor

Question 3

After a solvent spill stops, a 12,000 ft³ storage room holds 400 ppm of vapor. A fan exhausts 1,200 cfm and the room's mixing factor is 1.5. Assuming nothing is still evaporating, how many minutes until the room reaches 25 ppm?

18.1 min
18.5 min
27.7 min
41.6 min

Show the answer and every keystroke

Answer: D — 41.6 minutes

Step 1 — effective flow rate

Poor mixing penalizes the fan: $Q' = Q/m_i = 1{,}200/1.5 = 800$ cfm.

1200 ÷ 1.5 enter → 800

Step 2 — purge time

$t = -\dfrac{V_r}{Q'}\ln\!\left(\dfrac{C_{g2}}{C_{g1}}\right) = \dfrac{12{,}000}{800}\ln\!\left(\dfrac{400}{25}\right) = 15 \times 2.7726 = 41.6$ min. Flipping the fraction to $400/25$ removes the minus sign, so you never need the (−) key.

12000 ÷ 800 × ln 400 ÷ 25 ) enter → 41.59

The traps: Ignoring the mixing factor gives 27.7 min. Multiplying by 1.5 instead of dividing gives 18.5 min. Using log (base 10) in place of ln gives 18.1 min.

Equations used: Effective Flow Rate and Purge Time

Ventilation · hood airflow from two pressure readings

Question 4

At a hood's duct, the velocity pressure is 1.10 in. wg and the hood static pressure is −1.90 in. wg. The duct cross-sectional area is 3.0 ft² and the air is standard density. What is the exhaust flow?

9,590 cfm
12,600 cfm
16,560 cfm
17,370 cfm

Show the answer and every keystroke

Answer: B — about 12,600 cfm

Step 1 — coefficient of entry

$C_e = \sqrt{VP/|SP_h|} = \sqrt{1.10/1.90} = 0.761$.

2nd x² 1.1 ÷ 1.9 ) enter → 0.761 (2nd x² types sqrt()

Step 2 — airflow

$Q = 4005\,C_e\,A\sqrt{|SP_h|} = 4005 \times 0.761 \times 3.0 \times \sqrt{1.90} \approx 12{,}600$ cfm.

4005 × 0.761 × 3 × 2nd x² 1.9 ) enter → 12,603

Check — a shortcut that must agree

Because $C_e\sqrt{|SP_h|} = \sqrt{VP}$, you can also write $Q = 4005\,A\sqrt{VP}$.

4005 × 3 × 2nd x² 1.1 ) enter → 12,601

The traps: Putting $VP$ under the root in the hood equation (instead of $SP_h$) gives 9,590 cfm. Dropping $C_e$ gives 16,560 cfm. Using $SP_h$ without its square root gives 17,370 cfm.

Equations used: Coefficient of Entry and Hood Airflow (Standard Air)

Radiation · decay, then shielding

Question 5

A sealed iridium-192 source (half-life 73.8 days) reads 64 mR/h at a work position today. The job happens in 120 days, and the target is 2 mR/h or less at that position. The shield material has a half-value layer of 0.40 in. What thickness is needed?

1.35 in
2.00 in
2.65 in
3.37 in

Show the answer and every keystroke

Answer: A — 1.35 in

Step 1 — decay to day 120

$A = A_i(0.5)^{t/T_{1/2}}$, so the rate is $64 \times 0.5^{120/73.8} = 64 \times 0.324 = 20.7$ mR/h.

64 × .5 ^ ( 120 ÷ 73.8 ) enter → 20.73

Step 2 — thickness for 2 mR/h

$x = 3.32\log(I_1/I_2)\,(HVL) = 3.32\log(20.73/2)\times 0.40 = 3.37 \times 0.40 = 1.35$ in.

3.32 × log 20.73 ÷ 2 ) × .4 enter → 1.35

The traps: Ignoring the decay gives 2.00 in. Running the decay the wrong way (the source growing) gives 2.65 in. Stopping at 3.37 reports the number of half-value layers, not inches.

Equations used: Radioactive Decay and Shielding Thickness Required

Science & statistics · a mixture limit from reciprocals

Question 6

A paint-line blend has vapor fractions of 50% solvent A, 30% solvent B and 20% solvent C, all acting on the same target organ with additive effects. Their TLVs are 100 ppm (A), 50 ppm (B) and 200 ppm (C). What is the TLV of the mixture?

50 ppm
83.3 ppm
105 ppm
116.7 ppm

Show the answer and every keystroke

Answer: B — 83.3 ppm

One step — reciprocal of the weighted reciprocals

$TLV_{mix} = 1\big/\left(\frac{F_1}{TLV_1}+\frac{F_2}{TLV_2}+\frac{F_3}{TLV_3}\right) = 1/(0.005+0.006+0.001) = 83.3$ ppm.

( .5 ÷ 100 + .3 ÷ 50 + .2 ÷ 200 ) x⁻¹ enter → 83.33

The traps: Weighting the TLVs themselves gives 105 ppm. A plain average of the three TLVs gives 116.7 ppm. Picking the lowest TLV gives 50 ppm, which is safe but is not the mixture formula.

Equations used: Mixture TLV

Which equations lean on which calculator moves?

Equation family	The moves you will repeat	Reference
Noise	Logs, antilogs (10^x), powers of 2, ratios	Noise equations
Ventilation	Square roots, natural logs, e^x, reciprocals, unit conversion	Ventilation equations
Hood airflow	Square roots of pressures, area from diameter	Hood airflow equations
Heat stress	Weighted averages	Heat stress equations
Radiation	Fractional exponents, logs, inverse-square ratios	Radiation equations
Science & statistics	Logs, sums of squares, reciprocals, geometric means	General sciences & statistics

Ready for the Real Equations?

Everything above is the math skill on its own — the CIH Equation Master puts it to work on all 98 real CIH equations across Radiation, Ventilation, Noise, Science & Statistics, Hood Airflow, and Heat Stress, with worked practice problems and this exact calculator built in.

Try the CIH Equation Master →

Radiation equations are free to try — no sign-up required

Frequently asked questions

Do I need to be good at math to pass the CIH exam?

No. The CIH exam's calculation questions reuse a small, repeating set of math moves — rearranging a formula, converting units, exponents, roots, logarithms, scientific notation, ratios, and weighted averages. Recognizing those moves and executing them confidently on a calculator matters far more than raw math ability.

What calculator is allowed on the CIH exam, and does it matter which one I practice with?

On exam day the test center provides an on-screen calculator that emulates the TI-30XS MultiView, and you cannot bring your own. Practicing on the same key layout beforehand, including the 2nd-key shifted functions, removes the fumbling that costs time.

Why do I need logarithms for an industrial hygiene exam?

Several CIH calculations deal with quantities that span many orders of magnitude — sound intensity, radioactive decay, and similar exponential relationships. Logarithms convert those enormous ranges into manageable numbers, so several exam formulas are built directly on log or ln.

What's the difference between a square root and an nth root?

A square root undoes a square: √(x²) = x. An nth root generalizes that idea to any power and is mathematically the same as raising a number to a fractional exponent, 1/n — for example, a 5th root equals raising a number to the power 1/5.

Why does scientific notation matter for the CIH exam?

Several CIH calculations involve values that are either extremely large or extremely small (radioactive activity, particle counts). Scientific notation — a number times a power of ten — lets those values be entered on a calculator accurately without miscounting zeros, which is a common source of exam errors.

What's a common mistake candidates make with the calculator's negative-entry key?

The (−) key is a prefix operator you press before typing a number, not a sign-flip you apply afterward — pressing it in the wrong order produces a calculator error or, worse, a silently wrong answer instead of the intended negative value.

Do I need a separate study resource just for calculator practice, or is this covered elsewhere?

This math-basics refresher and its built-in practice calculator are free and cover the underlying skills in isolation. The CIH Equation Master then applies those same skills to all 98 real CIH equations with worked practice problems, using this exact calculator.

Can I bring my own calculator to the CIH exam?

No. According to BGC's applicant handbook, personal calculators are not allowed in the testing session. A calculator is built into the exam, and phone calculators are not permitted in the test center.

Do I get scratch paper on the CIH exam?

Not paper and pencil. BGC's handbook says each candidate receives a whiteboard and dry marker for equation-based questions, and scratch paper is also built into the exam screen, in the upper left of each question next to the equation sheets and the calculator icon.

Where can I practice with the real CIH exam calculator?

BGC's handbook lists www.pearsonvue.com/athena for practicing with the exam calculator. The practice calculator on this page uses the TI-30XS MultiView key layout that test-prep sources describe, so you can drill the same keystrokes here.

How much time do I have per question on the CIH exam?

The exam has 180 questions, 150 scored and 30 pilot, in two 2.5-hour sections. That is 300 minutes in total, or about 100 seconds per question on average, which is why calculation steps need to be fast and clean.

Do I need to memorize the CIH equations?

BGC's handbook says the BGC Equation Sheets open from an Equation Sheet button on the exam screen, so memorizing is not the main challenge. Choosing the right equation, entering it with the right brackets and checking the answer is. Our equation reference and the six worked calculations above are built for that practice.