Math first: units, measurement, and matter

What you know: 1 g, and 1 μg = 10⁻⁶ g. What you want: micrograms.

1. Write the input and put g underneath so it cancels: 1 g × (1 μg / 10⁻⁶ g).
2. g cancels, leaving μg. Divide: 1 ÷ 10⁻⁶ = 10⁶, because dividing by 10⁻⁶ is the same as multiplying by 10⁶.
3. The answer is 10⁶ μg (1,000,000 μg). This is a definition, so it is exact.

Check: A microgram is far smaller than a gram, so it takes many of them. Reverse it: 10⁶ μg × 10⁻⁶ g/μg = 10⁰ g = 1 g.

Watch out: 1000 μg is only 1 mg, not 1 g. Micro is 1000 times smaller than milli.

Your route: Write the link, 1 μg = 10⁻⁶ g → Put g underneath → Cancel and calculate.

That’s right: 10⁶ μg (1,000,000 μg)

What you know: 3.5 MW, and 1 MW = 10⁶ W. What you want: watts.

1. Put MW underneath so it cancels: 3.5 MW × (10⁶ W / 1 MW).
2. MW cancels, leaving W. Multiply: 3.5 × 10⁶ W, which is 3.5 million watts.
3. 3.5 has two significant figures, and 1 MW = 10⁶ W is exact. The answer is 3.5 × 10⁶ W.

Check: A watt is much smaller than a megawatt, so the number grows. Reverse it: 3.5 × 10⁶ W ÷ 10⁶ W/MW = 3.5 MW.

Watch out: 3.5 mW, with a lowercase m, would be 3.5 milliwatts, a billion times less power.

Your route: Write the input → Put MW underneath → Multiply by 10⁶ → Keep two significant figures.

That’s right: 3.5 × 10⁶ W

What you know: 650 nm, and 1 nm = 10⁻⁹ m. What you want: meters.

1. Put nm underneath so it cancels: 650 nm × (10⁻⁹ m / 1 nm).
2. nm cancels, leaving m. Multiply: 650 × 10⁻⁹ m.
3. Fix the coefficient: 650 = 6.5 × 10², so the result is 6.5 × 10² × 10⁻⁹ m. Add the exponents: 2 + (−9) = −7.
4. The trailing zero in 650 has no decimal point after it, so it is ambiguous (see 0.3); here treat it as not significant, giving two significant figures. The answer is 6.5 × 10⁻⁷ m.

Check: A meter is far bigger than a nanometer, so the number gets very small: 6.5 × 10⁻⁷ m = 0.00000065 m. Reverse it: 6.5 × 10⁻⁷ m × 10⁹ nm/m = 6.5 × 10² nm = 650 nm.

Watch out: Moving the decimal point two places left makes the coefficient smaller, so the exponent goes up by 2, from −9 to −7. Writing 6.5 × 10⁻¹¹ m moves it the wrong way.

Your route: Put nm underneath → Multiply by 10⁻⁹ → Fix the coefficient → Round.

That’s right: 6.5 × 10⁻⁷ m

What you know: 2.5 km, and 1 km = 1000 m. What you want: meters.

1. Put km underneath so it cancels: 2.5 km × (1000 m / 1 km).
2. km cancels, leaving m. Multiply: 2.5 × 1000 = 2500. The answer is 2500 m.

Check: A meter is smaller than a kilometer, so you need more of them. Reverse it: 2500 ÷ 1000 = 2.5 km. At 10 to 12 minutes per kilometer, this is about a 25 to 30 minute walk.

Watch out: 2.5 has two significant figures. Writing 2.5 × 10³ m shows that more clearly than 2500 m.

Your route: Write the input → Put km underneath → Multiply by 1000.

That’s right: 2500 m

What you know: 4.2 × 10⁶ μL, and 1 μL = 10⁻⁶ L. What you want: liters.

1. Put μL underneath so it cancels: 4.2 × 10⁶ μL × (10⁻⁶ L / 1 μL).
2. μL cancels, leaving L. Add the exponents: 6 + (−6) = 0, so the result is 4.2 × 10⁰ L.
3. 10⁰ = 1, and 4.2 has two significant figures. The answer is 4.2 L.

Check: A microliter is tiny, so it takes millions of them to fill a few liters. Reverse it: 4.2 L × 10⁶ μL/L = 4.2 × 10⁶ μL.

Watch out: Multiplying by 10⁶ instead gives 4.2 × 10¹², which means the factor was upside down and μL would not cancel.

Your route: Write the input → Put μL underneath → Add the exponents.

That’s right: 4.2 L

What you know: 1 mm = 10⁻³ m and 1 μm = 10⁻⁶ m, because milli means 10⁻³ and micro means 10⁻⁶. What you want: which is larger, and by how many times.

1. Compare the exponents: −3 is less negative than −6, so 10⁻³ m is the larger length.
2. Divide the larger by the smaller: 10⁻³ m ÷ 10⁻⁶ m = 10^{−3 − (−6)} = 10³ = 1000. The meters cancel, so the ratio has no unit.
3. The answer is 1 mm is larger, by 1000 times.

Check: In plain numbers, 0.001 m ÷ 0.000001 m = 1000. That fits the picture: you can see a splinter, but you need a microscope for a bacterium.

Watch out: 6 looks bigger than 3, but 10⁻⁶ is the smaller number. A more negative exponent means a smaller amount.

Your route: Write both in meters → Compare the exponents → Divide.

That’s right: 1 mm is larger, by 1000 times

What you know: mass = 12.0 g, volume = 4.0 mL. What you want: density in g/mL.

1. The unit g/mL tells you mass goes on top: d = mass ÷ volume.
2. Divide: 12.0 g ÷ 4.0 mL = 3.0 g/mL.
3. Round: this is division. 12.0 has three significant figures and 4.0 has two. Keep the fewest, two. The answer is 3.0 g/mL.

Check: Reverse it: 3.0 g/mL × 4.0 mL = 12.0 g.

Watch out: Writing 3.00 claims more precision than 4.0 has.

That’s right: 3.0 g/mL

What you know: mass = 24.0 g, volume = 8.0 mL. What you want: density in g/mL.

1. The unit g/mL tells you mass goes on top: d = mass ÷ volume.
2. Divide: 24.0 g ÷ 8.0 mL = 3.0 g/mL.
3. Round: this is division. 24.0 has three significant figures and 8.0 has two. Keep the fewest, two. The answer is 3.0 g/mL.

Check: Reverse it: 3.0 g/mL × 8.0 mL = 24.0 g.

Watch out: Writing 3.00 claims more precision than 8.0 has.

That’s right: 3.0 g/mL

What you know: mass = 36.0 g, volume = 12.0 mL. What you want: density in g/mL.

1. The unit g/mL tells you mass goes on top: d = mass ÷ volume.
2. Divide: 36.0 g ÷ 12.0 mL = 3.00 g/mL.
3. Round: this is division. 36.0 has three significant figures and 12.0 has three. Keep the fewest, three. The answer is 3.00 g/mL.

Check: Reverse it: 3.00 g/mL × 12.0 mL = 36.0 g.

Watch out: Both inputs have three significant figures, so write 3.00, not just 3.

That’s right: 3.00 g/mL

What you know: mass = 48.0 g, volume = 16.0 mL. What you want: density in g/mL.

1. The unit g/mL tells you mass goes on top: d = mass ÷ volume.
2. Divide: 48.0 g ÷ 16.0 mL = 3.00 g/mL.
3. Round: this is division. 48.0 has three significant figures and 16.0 has three. Keep the fewest, three. The answer is 3.00 g/mL.

Check: Reverse it: 3.00 g/mL × 16.0 mL = 48.0 g.

Watch out: Both inputs have three significant figures, so write 3.00, not just 3.

That’s right: 3.00 g/mL

What you know: mass = 60.0 g, volume = 20.0 mL. What you want: density in g/mL.

1. The unit g/mL tells you mass goes on top: d = mass ÷ volume.
2. Divide: 60.0 g ÷ 20.0 mL = 3.00 g/mL.
3. Round: this is division. 60.0 has three significant figures and 20.0 has three. Keep the fewest, three. The answer is 3.00 g/mL.

Check: Reverse it: 3.00 g/mL × 20.0 mL = 60.0 g.

Watch out: Both inputs have three significant figures, so write 3.00, not just 3.

That’s right: 3.00 g/mL

What you know: Two numbers, 0.00450 and 0.0045. What you want: the one with three significant figures.

1. Leading zeros are not significant. They only place the decimal point. So skip the "0.00" in both.
2. 0.0045 leaves 4 and 5: two significant figures.
3. 0.00450 leaves 4, 5 and a trailing zero after the decimal point. Trailing zeros after a decimal point count. That makes three. The answer is 0.00450.

Check: In scientific notation, 0.00450 = 4.50 × 10⁻³ (three digits) and 0.0045 = 4.5 × 10⁻³ (two digits).

Watch out: The final zero is not decoration. It tells you the measurement was precise to that place.

That’s right: 0.00450

What you know: 2.31 + 1.2. What you want: the sum, rounded by the addition rule.

1. Add first, keeping all digits: 2.31 + 1.2 = 3.51.
2. For addition, look at decimal places, not significant figures. 2.31 goes to the hundredths place. 1.2 goes only to the tenths place.
3. The least precise place is the tenths. Round 3.51 to the tenths: 3.5.

Check: 1.2 is unknown past the tenths, so the sum cannot be known past the tenths either.

Watch out: Do not count significant figures here. That rule is for multiplication and division.

That’s right: 3.5

What you know: Readings cluster at 9.0; the accepted value is 10.0. What you want: precise, accurate, both, or neither.

1. Precision means repeated results are close to one another. The readings cluster tightly, so they are precise.
2. Accuracy means closeness to the accepted value. 9.0 is a full unit away from 10.0, so they are not accurate.
3. The answer is precise but inaccurate.

Why it matters: A tight cluster in the wrong place often means a mistake built into the method, like a scale that reads low every time.

Watch out: Precise and accurate are not the same word. Agreeing with each other is not the same as being right.

That’s right: Precise but inaccurate

What you know: 1000 mL = 1 L. What you want: whether this factor limits significant figures.

1. Ask: is this number measured or defined? Nobody measured it. The prefix "milli" means one thousandth by definition.
2. The rule says exact counts and defined conversions do not limit precision. They have unlimited significant figures.
3. So the answer is No. Only your measured inputs decide how many figures to keep.

Why it matters: If you treated 1 L as one significant figure, every conversion would be rounded to one digit, which throws away real information.

That’s right: No

What you know: density = 2.0 g/mL, volume = 5.0 mL. What you want: mass in g.

1. Rearrange d = mass ÷ volume to get mass alone: mass = density × volume.
2. Multiply: 2.0 g/mL × 5.0 mL = 10 g. mL cancels, leaving g.
3. Round: both inputs have two significant figures, so the answer has two. The answer is 10 g (two significant figures). Writing it as 1.0 × 10¹ g makes the two figures clear.

Check: Reverse it: 10 g ÷ 5.0 mL = 2.0 g/mL.

Watch out: A whole number ending in zero, like 10, hides whether the zero is significant. Scientific notation removes the doubt.

That’s right: 10 g (two significant figures)

What you know: (5.0 × 10⁴) × (2.0 × 10⁻⁶). What you want: the product in scientific notation.

1. Multiply the coefficients: 5.0 × 2.0 = 10.
2. Add the exponents: 4 + (−6) = −2. So far: 10 × 10⁻².
3. Fix the coefficient. 10 is not less than 10, so move the decimal one place left and raise the exponent by 1: 1.0 × 10⁻¹. Keep two significant figures to match the inputs. The answer is 1.0 × 10⁻¹.

Check: 50000 × 0.000002 = 0.1, and 1.0 × 10⁻¹ = 0.10.

Watch out: When you make the coefficient smaller, the exponent must go up, not down.

That’s right: 1.0 × 10⁻¹

What you know: (9.0 × 10⁸) ÷ (3.0 × 10³). What you want: the quotient in scientific notation.

1. Divide the coefficients: 9.0 ÷ 3.0 = 3.0.
2. Subtract the exponents (top minus bottom): 8 − 3 = 5.
3. 3.0 is already between 1 and 10, so no fixing is needed. Both inputs have two significant figures. The answer is 3.0 × 10⁵.

Check: Multiply back: (3.0 × 10⁵) × (3.0 × 10³) = 9.0 × 10⁸.

Watch out: Dividing exponents (8 ÷ 3) is wrong. For division, subtract them.

That’s right: 3.0 × 10⁵

What you know: 0.000720. What you want: scientific notation, with a coefficient from 1 up to less than 10.

1. Move the decimal point until one nonzero digit is in front of it: 0.000720 → 7.20. That is 4 places to the right.
2. The original number is less than 1, so the exponent is negative: 10⁻⁴.
3. Keep the significant figures. The leading zeros drop away, but the trailing zero after the decimal point counts. So keep 7.20. The answer is 7.20 × 10⁻⁴.

Check: 7.20 × 10⁻⁴ means move the decimal 4 places left: 0.000720.

Watch out: Writing 7.2 × 10⁻⁴ loses a significant figure.

That’s right: 7.20 × 10⁻⁴

What you know: Table salt has the formula NaCl. What you want: element, compound, or mixture.

1. A single chemical formula means a pure substance, not a mixture.
2. Count the different element symbols: Na (sodium) and Cl (chlorine). That is two.
3. Two or more different elements combined in a fixed ratio (here 1:1) make a compound.

Why it matters: Sodium and chlorine are combined chemically, so salt has its own properties, very different from either element alone.

Watch out: Two elements do not make a mixture when they share one formula. A mixture has no single formula.

That’s right: A compound

What you know: Air is mostly nitrogen (N₂) and oxygen (O₂). What you want: pure substance or mixture, and which kind.

1. Ask: one substance or several? Air is several gases physically mixed. It has no single formula, and its makeup can vary, for example in humidity. So it is a mixture.
2. Ask: can you see separate parts? No. Air looks the same throughout.
3. A mixture that looks the same throughout is a homogeneous mixture.

Why it matters: Homogeneous mixtures are also called solutions. Air is a gas solution, just as salt water is a liquid one.

Watch out: N₂ and O₂ are each elements, but air contains both mixed together, so air itself is not an element.

That’s right: A homogeneous mixture

What you know: The state has a fixed volume but no fixed shape. What you want: which state.

1. Picture pouring water into a different glass. The amount stays the same, but the shape changes to fit the glass.
2. At the particle level, liquid particles stay close together, so the volume is fixed. They slide past one another, so they take the container's shape.
3. That describes a liquid.

Why it matters: A solid keeps both its shape and volume. A gas keeps neither; it spreads to fill its container.

That’s right: Liquid

What you know: A gas can be squeezed; a solid cannot. What you want: the particle-level reason.

1. Compare the space between particles. In a gas, particles are far apart with large empty gaps.
2. Pushing on a gas closes those gaps, so the volume shrinks.
3. In a solid, particles are already packed together and touching. There is almost no empty space to remove. So: gas particles are far apart; solid particles are already touching.

Why it matters: This is why gas tanks can hold a large amount of gas in a small space.

Watch out: Squeezing a gas does not make the particles smaller. It only shrinks the space between them.

That’s right: Gas particles are far apart; solid particles are already touching.

What you know: Boiling point is a property. What you want: physical or chemical.

1. Ask: does measuring it turn the substance into something new?
2. When water boils, liquid water becomes water vapor. It is still H₂O, only in a different state.
3. A property you can measure without making a new substance is physical.

Why it matters: A chemical property, like flammability, can only be seen when the substance changes into something new.

Watch out: Bubbles in boiling water are water vapor, not a new gas. A change of state is a physical change.

That’s right: Physical

What you know: Wood is burning. What you want: physical or chemical change.

1. Look for the clues of a chemical change: a new gas, a color change, light or heat given off, a new odor.
2. Burning wood gives off heat and light, smoke and a new smell. It leaves ash, which is not wood.
3. New substances form (carbon dioxide, water vapor and ash). That makes it a chemical change.

Why it matters: You cannot turn the ash and gases back into wood. Physical changes, like melting ice, are usually easy to reverse.

That’s right: Chemical

What you know: Hot cocoa cools on a table. What you want: what happens to its particles' motion.

1. Heat flows from hotter to cooler. The cocoa is hotter than the room, so heat flows out of the cocoa.
2. Temperature measures the average kinetic energy, the energy of motion, of the particles.
3. A falling temperature means lower average kinetic energy. So the particles slow down on average.

Why it matters: Particles never stop moving. Cooling only lowers their average speed.

Watch out: "On average" matters. Some particles still move fast; the average speed drops.

That’s right: They slow down on average.

What you know: The statement "If I add salt, the water will boil at a higher temperature." What you want: observation, hypothesis, or law.

1. Ask: has it been tested yet? No. It says what will happen.
2. An observation records what you saw. A law describes a pattern confirmed many times. A hypothesis is a testable explanation or prediction.
3. This is a testable prediction, so it is a hypothesis.

Why it matters: The next step is an experiment. If the data do not support it, revise the hypothesis and test again.

Watch out: Wording like "If I do this, that will happen" predicts an untested result. That is a strong clue for a hypothesis.

That’s right: A hypothesis

What you know: Two kinds of scientific statement, laws and theories. What you want: the one that explains why.

1. A scientific law describes what happens, often as a pattern or equation.
2. A theory explains why it happens, and it is supported by many tests.
3. So the one that explains is a theory.

Why it matters: In science, "theory" does not mean a guess. It is a well-tested explanation.

Watch out: A theory does not "grow up" into a law. They do different jobs: one explains, the other describes.

That’s right: A theory