7.2 Estimation & Reasonableness Checks

7.2 Estimation & Reasonableness Checks

Quick Answer: Estimate first, then refine. Round to friendly numbers, judge the order of magnitude (tens vs hundreds vs thousands), and eliminate options that are impossible for the story. A correct exact answer should still feel plausible in size and unit.

Mathematics on civil-service entrance papers rewards candidates who stay accurate under time pressure. Estimation and reasonableness are not “lazy math.” They are professional habits: engineers, shopkeepers, and public servants all check whether a figure is in the right ballpark before they trust it.

For the SCD Mathematics subject area, practice estimating with civilian Trinidad and Tobago situations—grocery bills, bus fares, school counts, rainfall millimetres, and market weights. Do not rely on firefighting hydraulics or technical fireground formulas.

What estimation is (and is not)

When the question says exact or options are very close, finish the precise arithmetic. Use estimation to catch errors and discard impossible choices.

Rounding for friendly numbers

Common rounding targets:

• To the nearest 10: 47 → 50; 142 → 140.
• To the nearest 100: 380 → 400; 1,240 → 1,200.
• To 1 significant figure for rough size: 67 ≈ 70; 0.048 ≈ 0.05.
• To simple fractions/decimals: 0.49 ≈ 0.5; 19 ≈ 20; 3.1 ≈ 3.

Worked example — market shopping estimate

A shopper in San Juan buys:

• 3 packs of rice at $28 each
• 2 bottles of oil at $42 each
• vegetables about $35

Exact path (for comparison):
Rice: 3 × 28 = 84. Oil: 2 × 42 = 84. Vegetables: 35. Total = 84 + 84 + 35 = $203.

Estimate path:
Rice: 3 × 30 = 90. Oil: 2 × 40 = 80. Vegetables: 35. Estimate ≈ 90 + 80 + 35 = $205.

The estimate is within a few dollars of the exact total. On an MCQ, options like $85, $2,030, or $20 are immediately suspicious.

Order of magnitude

Order of magnitude means the scale of a number: about 10, about 100, about 1,000, and so on.

Questions to ask yourself:

• Should the answer be less than 10, tens, hundreds, or thousands?
• Is the answer a count of people, a money total, a percentage (0–100), or a rate?
• Can the answer be larger than a stated total? (Usually no for parts of a whole.)

Worked example — attendance scale

A hall seats 200 people. If 45% attend, about how many people is that?

Estimate: 50% of 200 = 100, so 45% is a bit less → about 90.
Exact: 0.45 × 200 = 90.

Absurd options: 9 (missed a factor of 10), 900 (extra factor of 10), 200 (full hall), 45 (copied the percent as a count).

Worked example — rainfall magnitude

Four weekly rainfall readings (mm): 12, 8, 15, 10.
Weekly mean is roughly (12 + 8 + 15 + 10) ÷ 4. Sum ≈ 45; mean ≈ 11 mm.

An option of 110 mm as the weekly mean for these data is off by a factor of ten. An option of 45 mm is the total, not the mean—different error, still catchable.

Compatible units before estimating

Estimation fails if units are mixed.

• 1.5 km + 300 m → convert first: 300 m = 0.3 km → total 1.8 km, not 301.5 of anything mixed.
• 2 hours 30 minutes is 2.5 hours, not 2.30 hours in decimal math.
• Prices “in hundreds of dollars” on a scale: a mark at 4 may mean $400.

Always restate every quantity in one unit before rounding.

Benchmarks you should know cold

Memorise friendly benchmarks so estimation is automatic:

Percent of a bill

A restaurant bill is $186. Tip or service charge of 10%: about $19 (exact 18.60).
15% ≈ 10% + half of 10% ≈ 18.60 + 9.30 = 27.90 (estimate: 19 + 9.5 ≈ 28.5, still near).

If options are $1.86, $18.60, $186, and $1,860 for a 10% charge on $186, only $18.60 sits at the right magnitude (exact). $1.86 is 1%; $186 is 100%; $1,860 is nonsense scale.

Reasonableness of story problems

After you get a number, test it against the story:

1. Size: Can a maxi-taxi carry 500 passengers in one trip? Unlikely—recheck.
2. Direction: If a price is discounted, the final price must be lower than the original.
3. Bounds: A percentage share of a group cannot exceed 100% of that group.
4. Units: “Speed in km/h” should not be answered in minutes alone.
5. Integer sense: Counts of people are whole numbers unless the question asks for an average that can be decimal.

Worked example — change from a bill

Items total $67. Customer pays with $100. Change = 100 − 67 = $33.

Reasonableness: change must be positive and less than $100. Options $167 (added instead of subtracted) or −$33 (sign error) fail the check. Option $67 is the bill, not the change.

Worked example — multi-day mean

Five days’ sales: 120, 130, 110, 140, 150. Mean?

Rough estimate: values sit near 130. Exact total = 650; mean = 130.
If you mis-add to 750, mean 150 equals the maximum day—suspicious because most days are lower. That suspicion pushes you to re-add.

Eliminating absurd MCQ options

Use a filter pass before detailed work when options are far apart:

Worked example — proportion of attendees

Of 80 registered participants, 56 attended. What percent attended?

Exact: 56 ÷ 80 = 0.70 = 70%.
Estimate: 56/80 = 7/10 = 70%.

Absurd options: 56% (copied the numerator), 80% (copied the denominator), 136% (added then divided wrong), 7% (dropped a zero in thinking).

Estimation as error control on exact work

Even when you compute exactly:

1. Estimate the answer in 5–10 seconds.
2. Compute carefully.
3. Compare: if exact and estimate disagree wildly, find the bug (often place value or wrong operation).

Worked example — product check

Compute 48 × 21.

Estimate: 50 × 20 = 1,000.
Exact: 48 × 20 + 48 × 1 = 960 + 48 = 1,008. Close to 1,000—good.

If you accidentally computed 48 × 12 = 576, the estimate 1,000 would flag that 576 is far too small for “about 50 × 20.”

Front-end estimation vs compatible rounding

• Front-end: Use leading digits: 2,347 + 5,812 ≈ 2,000 + 6,000 = 8,000 (rough).
• Compatible rounding: Adjust both numbers so they cancel error: 48 × 21 → 50 × 20 works well; 19 × 21 → 20 × 20 = 400 (exact is 399).

Prefer compatible pairs when multiplying.

When not to rely on estimation alone

• Options are very close (e.g. 63.5 vs 63.75 vs 64).
• The stem demands an exact remainder, exact fraction, or exact money to the cent.
• Ranking merit on a competitive paper means careless rounding that picks the near-miss option still costs a mark.

In those cases, estimate only as a check, then finish exact arithmetic.

Practice drills (civilian)

1. Grocery basket: Round each price to the nearest dollar, sum, compare to exact sum.
2. Bus journey: Distance 47 km in about 1 hour 10 minutes—about what average speed? (≈ 47 ÷ 1.2 ≈ 40 km/h).
3. Class fraction: 18 of 25 students present—about what percent? (18/25 = 72%).
4. Rainfall: Three days 9 mm, 11 mm, 10 mm—mean about 10 mm.

Link to other math sections

Estimation complements averages, multi-step word problems, and data tables: after you read a table cell, ask “is this total of 2,500 people for a single maxi-taxi trip reasonable?” If not, you likely grabbed the wrong cell.

Bottom line for exam day

• Round → estimate magnitude → eliminate absurd options → compute exactly when needed → re-check against the story.
• Never invent official exam timing rules; use estimation to protect accuracy, not to skip understanding.
• Keep all practice in ordinary arithmetic and everyday T&T contexts—no technical firefighting math.