9.8 Units, Quantities, Precision, & Appropriate Accuracy
Key Takeaways
- Carrying units through every step of a multi-step problem is a checking tool: if the units of your answer are wrong, the arithmetic is wrong.
- The unit of a result tells you which operation was performed — dollars divided by hours yields dollars per hour, and dollars times hours is meaningless.
- Choosing the scale and origin of a graph is part of representing data honestly; a truncated axis exaggerates differences.
- The accuracy of a reported answer is limited by the least precise measurement used to produce it, so a length measured to the nearest foot cannot yield an area accurate to the square inch.
Units, Quantities, Precision, & Appropriate Accuracy
Two Level A standards are easy to overlook because they sound like advice rather than mathematics: "use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays" and "choose a level of accuracy appropriate to limitations on measurement when reporting quantities." Both generate real TABE items, and both are habits that catch errors elsewhere on the test.
Units as an Error-Checking Tool
Every quantity has a unit, and the units obey their own arithmetic. Track them and they will tell you when you have made a mistake.
| Operation | Unit result | Meaning |
|---|---|---|
| miles $\div$ hours | miles per hour | a speed |
| dollars $\div$ hours | dollars per hour | a wage |
| dollars $\div$ pounds | dollars per pound | a unit price |
| feet $\times$ feet | square feet | an area |
| feet $\times$ feet $\times$ feet | cubic feet | a volume |
| dollars per hour $\times$ hours | dollars | a total pay |
| dollars $\times$ hours | dollar-hours | meaningless — you multiplied when you should have divided |
Use this on test day. A problem gives you $378 and 21 hours and asks for an hourly rate. Dividing gives $18 per hour — correct units. Multiplying gives 7,938 "dollar-hours," which is not a thing. The unit check identifies the operation without any reasoning about the story.
Units in multi-step problems
A tank holds 4,500 liters. A pump moves 25 liters per minute. How many hours to empty it?
Liters cancel against liters-per-minute to leave minutes; minutes cancel against minutes-per-hour to leave hours. The units drive the solution, which is exactly what the standard says.
Consistent units inside a formula
A formula is only valid when every input shares a unit system.
A rectangular slab is 12 feet by 30 inches. What is its area in square feet?
You cannot multiply 12 by 30. Convert first: 30 inches is 2.5 feet, so the area is $12 \times 2.5 = \mathbf{30}$ square feet. Multiplying without converting gives 360, a number with no meaningful unit at all.
The same applies to $I = Prt$, where $t$ must be in years because $r$ is an annual rate. Eighteen months entered as 18 gives interest eighteen times too large.
Choosing the Scale and Origin of a Graph
The standard explicitly names scale and origin, and TABE asks about both.
| Design choice | Honest practice | Distortion to recognize |
|---|---|---|
| Origin | start the value axis at 0 | truncating the axis exaggerates small differences |
| Interval size | uniform, and matched to the data range | uneven intervals compress or stretch part of the data |
| Axis labels | quantity and unit on both axes | unlabeled axes make the graph unreadable |
| Aspect ratio | proportionate | stretching one axis manufactures a steeper trend |
Worked judgment. Sales are $50M and $52M. On an axis starting at 0, the bars look nearly identical — a truthful 4% difference. On an axis starting at $49M, one bar is three times the height of the other. The data did not change; the origin did.
Choosing an interval: if data run from 12 to 87, an axis from 0 to 90 with intervals of 10 is sensible. Intervals of 1 produce 90 unreadable gridlines; intervals of 50 hide every distinction that matters.
Appropriate Accuracy
The rule: a result cannot be more precise than the measurements that produced it.
A room is measured with a tape to the nearest foot: 12 ft by 15 ft. Area computes to 180 sq ft. Reporting "180.00 square feet" claims a precision the measuring tape never had. Report 180 square feet, and understand that the true area could be anywhere from about 11.5 × 14.5 = 167 to about 12.5 × 15.5 = 194 square feet.
| Situation | Report to | Why |
|---|---|---|
| Money | the nearest cent | currency has two decimal places |
| Bulk lumber or fencing | the next whole unit up | you cannot buy 0.3 of a board |
| A measurement made to the nearest $\frac{1}{8}$ in | eighths | the instrument's limit |
| An average of whole-number counts | one decimal is usually enough | 4.7 people is a valid average, not a valid count |
| A calculator display of 3.166666667 | the precision of the inputs | trailing digits are noise |
Rounding at the right moment
Round once, at the end. Rounding an intermediate value propagates error into every step that follows.
$$100 \div 3 = $33.333\ldots$ per person. Rounding to $33.33 immediately and multiplying by 3 gives $99.99 — one cent has vanished. Keep the full value through the computation and round only the final reported figure, then reconcile the remainder deliberately.
Significant context beats significant digits
TABE does not test formal significant-figure rules. It tests judgment:
- A dosage of 12.5 mg is meaningfully different from 12 mg → keep the decimal.
- A crowd estimate of 4,000 does not become better by writing 4,000.00.
- A car trip of "about 250 miles" should not produce a fuel-cost answer of $41.8367.
A contractor is paid $1,632 for a job that took 24 hours. Which computation produces the hourly rate, and how do the units confirm it?
A storage platform measures 9 feet by 42 inches. What is its area in square feet?
A technician measures a rectangular panel with a tape marked in whole inches and records 34 inches by 51 inches. Which is the most appropriate way to report the area?