Quiz 6 — Numbers in Science

The Guide

A number by itself is not an answer

Ask how far it is to the next town and the answer "twelve" is useless. Twelve what? Every measurement in science is two things joined together: a number and a unit. Drop the unit and you have not simplified the answer, you have destroyed it.

This matters on the test in a small, practical way. Wrong answer choices are often the right number with the wrong unit attached, and they are easy to pick because the arithmetic feels finished.

The units worth knowing cold

Length	meter (m); centimeter, kilometer
Mass	gram (g); kilogram, milligram
Volume	liter (L); milliliter. 1 mL is the same as 1 cm³
Time	second (s); minute, hour
Temperature	degree Celsius (°C)
Force	newton (N)
Energy and work	joule (J)
Power	watt (W)

The prefixes are the whole trick. Kilo means a thousand of them. Centi means a hundredth. Milli means a thousandth. So a kilogram is 1,000 grams, a centimeter is a hundredth of a meter, and a milliliter is a thousandth of a liter. Converting between them only ever moves the decimal point.

Which way does the decimal go? Ask whether the new unit is bigger or smaller than the old one. Going to a bigger unit — grams to kilograms — means fewer of them, so the number gets smaller. Going to a smaller unit means more of them, so the number gets bigger. Do that check before you touch the calculator and you will catch nearly every conversion mistake you would have made.

Scientific notation. 3 × 10⁸ means 3 followed by eight zeros: 300,000,000. A negative exponent means the other direction: 8 × 10⁻⁶ means 0.000008. It exists because writing out the distance to the sun or the width of a cell in full is unreadable. To compare two numbers written this way, look at the exponents first — the bigger exponent wins, and you only compare the front numbers if the exponents are equal.

Formulas are recipes. The test gives you the formula. Your job is to put the right number in the right place and carry the units through. Density is mass divided by volume, so its unit is grams per cubic centimeter. Speed is distance divided by time, so its unit is kilometers per hour. If you know any two parts of a formula, you can find the third.

Does the answer make sense? Last step, every time. A person does not walk at 900 kilometers an hour. A rock is not 4,000 degrees. A calculator will happily hand you an absurd number, and the only defense is a moment of ordinary judgment about the size of things.

Worked Examples

Two questions, worked through

Example 1. A cyclist rides 45 kilometers in 1.5 hours. What is her average speed?

Write the formula first: speed = distance ÷ time. Then substitute, keeping the units attached: 45 km ÷ 1.5 h. That gives 30, and the units divide the same way the numbers do, so the answer is 30 km/h.

Answer: 30 km/h. Notice that the unit came out of the formula on its own. You did not have to remember that speed is measured in kilometers per hour — dividing kilometers by hours produced it.

Example 2. A tablet contains 2,500 milligrams of a substance. How many grams is that?

Grams are bigger than milligrams, so the number must get smaller — that is the check, and it takes two seconds. A milligram is a thousandth of a gram, so divide by 1,000: 2,500 ÷ 1,000 = 2.5.

Answer: 2.5 g. If you had come out with 2,500,000 g, the size check would have caught it: a tablet does not weigh two and a half tons.

The Quiz

Ten questions

Answer all ten, then press the button at the bottom. A calculator is fine — the real test gives you one.

Data A — questions 1 to 4

A student measured a rock sample in the laboratory and recorded the following.

Mass	240 g
Volume	30 cm³
Longest side	6.2 cm
Temperature of the water bath	22 °C

Formula density = mass ÷ volume

1.What is the density of the rock?

A0.125 g/cm³
B8 g/cm³
C210 g/cm³
D7,200 g/cm³

2.What is the rock's mass expressed in kilograms?

A0.24 kg
B2.4 kg
C24 kg
D240,000 kg

3.The student writes the rock's density in her notebook as simply "8." Why is that not a complete answer?

ABecause the number should have been rounded to one decimal place.
BBecause density must always be written as a fraction.
CBecause a measurement needs its unit: 8 grams per cubic centimeter is not the same quantity as 8 kilograms per cubic meter.
DBecause she should have weighed the rock a second time.

4.A second rock is made of the same material and has a volume of 60 cm³. What is its mass?

A120 g
B480 g
C240 g
D960 g

Diagram B — question 5

Read the level at the bottom of the curved water surface.

5.What is the volume of the stone?

A20 mL
B50 mL
C70 mL
D120 mL

Table C — questions 6 and 7

Quantity	Measurement
Distance from Earth to the Sun	1.5 × 10¹¹ m
Width of a red blood cell	8 × 10⁻⁶ m
Speed of light	3 × 10⁸ m/s
Speed of a radio wave in glass	9 × 10⁷ m/s

6.Written out in full, the width of a red blood cell is

A8,000,000 m
B0.00008 m
C0.8 m
D0.000008 m

7.Which is the greater speed, and why?

A9 × 10⁷ m/s, because 9 is larger than 3.
B3 × 10⁸ m/s, because when both are written this way the larger exponent means the larger number.
CThey are equal.
DThere is no way to compare numbers written in this form.

Formula sheet — questions 8 to 10

Formulas speed = distance ÷ time work = force × distance density = mass ÷ volume

8.A car travels 240 kilometers in 3 hours. What is its average speed?

A720 km/h
B243 km/h
C80 km/h
D0.0125 km/h

9.A worker pushes a crate along the floor with a steady force of 50 newtons for a distance of 4 meters. How much work has she done?

A12.5 J
B54 J
C200 J
D200 N

10.A student times a friend walking the length of a hallway and calculates a speed of 900 km/h. What should she do next?

ACheck the calculation, because a walking person moving at 900 km/h is far outside any reasonable value.
BReport the result as it stands, since the arithmetic is what matters.
CChange to different units so the number looks smaller.
DConclude that her friend walks unusually fast.

Send this line to your teacher

The line records which questions you missed and which answer you chose. That is more useful to your teacher than the score, because it shows where a question went wrong. If a question felt unclear even though you got it right, add its number with a question mark — for example 5? — before you send it.

Score ______ / 10 Missed — write the question number and the letter you chose:
______________________________________________________________

The Key

Answers, and the trap in each one

1. B — 8 g/cm³. 240 g ÷ 30 cm³ = 8, and grams divided by cubic centimeters gives grams per cubic centimeter. A divides the wrong way round, which happens when you reach for the calculator before writing the formula down. C subtracts and D multiplies — both are what happens when the numbers get used without the recipe.

2. A — 0.24 kg. A kilogram is bigger than a gram, so the number has to get smaller. 240 ÷ 1,000 = 0.24. D multiplies by a thousand instead of dividing, which gives a rock heavier than a locomotive. The size check catches that one instantly, and it costs nothing to run.

3. C — the unit is missing. "8" does not say what quantity was measured or on what scale. Written properly, 8 g/cm³ is the density of a fairly heavy metal. The same figure in kilograms per cubic meter would describe something a thousand times lighter. A and D are reasonable-sounding laboratory advice about entirely different problems. The question asked why the recorded value is incomplete, and there is only one thing missing from it.

4. B — 480 g. Same material means same density, 8 g/cm³. Rearranging the recipe: mass = density × volume = 8 × 60 = 480 g. Twice the volume of the same stuff weighs twice as much, which you could also have reasoned without any formula at all. C is the first rock's mass, sitting in the table waiting to be copied. When a question introduces a second object, check which one it is asking about.

5. A — 20 mL. The water alone reads 50 mL. With the stone in it the level reads 70 mL. The stone pushed 20 mL of water out of its way, and that displaced water is exactly the stone's volume. C is the second reading taken straight off the cylinder without subtracting. The question asked for the stone, not for the water plus the stone — and 1 mL is the same as 1 cm³, so this is also a way of measuring an object too lumpy to measure with a ruler.

6. D — 0.000008 m. A negative exponent of six means the decimal point moves six places to the left of the 8. Count them: 0.000008. B moves it five places instead of six, which is the mistake almost everyone makes at least once. Count the zeros deliberately rather than by eye.

7. B — 3 × 10⁸ m/s. Written out, they are 300,000,000 and 90,000,000. The exponent tells you how many places the number stretches, and a difference of one exponent outweighs any difference in the front number. A compares only the visible digits, which is what the eye wants to do. Compare exponents first, every time, and look at the front numbers only when the exponents match.

8. C — 80 km/h. 240 km ÷ 3 h = 80 km/h. And it passes the sense check: motorway speed, roughly. A multiplies instead of dividing, giving a car faster than a passenger jet. B adds. Both survive the arithmetic and fail the last question you should always ask.

9. C — 200 J. Work = force × distance = 50 N × 4 m = 200 J. Newtons multiplied by meters give joules. D is the correct number with the wrong unit, and it is the most instructive wrong answer on this page. Newtons measure force. Joules measure the work that force does over a distance. Getting the arithmetic right and the unit wrong is still getting it wrong.

10. A — check the calculation. Nine hundred kilometers an hour is roughly the cruising speed of an airliner. Something has gone wrong — most likely seconds were treated as hours, or meters as kilometers. B is the habit this quiz exists to break. A calculator will return whatever you type, without comment. Judgment about the size of ordinary things is a scientific skill, not a distraction from one.

Your Score

What the number means

8 to 10	Solid. Check topic 6 on your map and go on to Quiz 7, the last in Part I.
6 or 7	Close. Read the whole key, then take this again in a few days before you check the box.
5 or fewer	Worth real time, and it pays on the mathematics test as well. Work through the two examples again on paper, writing the formula before any number goes into the calculator.

Misses on 2 or 6 are decimal places, and the cure is the size check: bigger unit, smaller number. Misses on 1, 4, or 8 mean a formula got used before it was written down — write it first, substitute second. Misses on 3 or 9 are units, which is the theme of the whole quiz. And a miss on 10 is worth more attention than any of the others, because it is the habit that catches all the rest.

The plate for this quiz

The gallery image is the meter bar of 1799, a strip of platinum deposited in the French national archives as the standard length for everyone. Before it, measures were local and often owned: a foot could be the length of a particular lord's foot, and a bushel could differ from one market town to the next, which was convenient for whoever controlled the measure.

Two astronomers, Delambre and Méchain, spent seven years surveying the meridian between Dunkirk and Barcelona so that the new unit could be defined as one ten-millionth of the distance from the pole to the equator — a length taken from the Earth itself rather than from any person's body. Méchain found an inconsistency in his own readings, could not resolve it, and was tormented by it until his death. The motto attached to the project in those years was that the new measures should belong to all times and to all peoples. It is a reasonable thing to think about while converting grams to kilograms.