Estimation and checking reasonable answers

Decide what the estimate needs to tell you

An estimate can help you plan a purchase, predict a calculation’s scale or notice a misplaced decimal point. State its purpose before choosing how coarse it may be. Knowing that a rectangular model covers about fifty square metres can help check an area calculation; it may not settle whether a particular number of whole flooring packs will cover it.

Separate the given information, the estimate and the exact result available within the stated model. The models in this lesson stipulate their numbers exactly. Replacing those numbers with convenient ones creates an approximation for checking; it does not introduce a new measurement uncertainty or supply a guarantee about a real building, journey or package. Real applications still need appropriate measurements and assumptions.

Choose numbers that make the operation easy

Convenient numbers fit the operation. To estimate 396 ÷ 8, use 400 ÷ 8 = 50; the original exact quotient is 49.5. For 38 × 21, the nearby product 40 × 20 = 800 is easy to compare with the exact product 798. Use ≈ when relating the original calculation to its estimate, and explain which inputs you changed.

Rounding each input to the same number of places is not always the most useful mental strategy. A nearby multiple, a simple fraction of an hour or numbers whose product is easy may serve the purpose better. Increasing one factor while decreasing another does not automatically produce an upper or lower bound. The closeness of 800 to 798 in this example must be checked; mixed directions alone do not justify it in every product.

Exact compensation preserves the result: 198 + 37 = (200 + 37) − 2 = 235, because the added two is subtracted again. Without that compensation, 198 + 37 ≈ 200 + 37 = 237 is a rough estimate. Convenient numbers can therefore support either an exact calculation or an approximation; show which relationship you used.

Give a reason when you claim a bound

For positive rectangle dimensions, increasing either side increases the area. The stipulated sides 9.8 m and 4.9 m satisfy 9 m ≤ 9.8 m ≤ 10 m and 4 m ≤ 4.9 m ≤ 5 m. Therefore 36 m² ≤ 9.8 m × 4.9 m ≤ 50 m². Here 10 × 5 = 50 supplies both a convenient estimate and a justified upper bound; 9 × 4 = 36 supplies a lower bound.

These bounds enclose the exact model area, 48.02 m². They follow from the stated numbers, positive dimensions and rectangle formula. They are not measured uncertainty intervals or probability claims. A rough estimate made without a directional argument is a heuristic, not automatically a bound. An enclosure can rule out some answers while allowing several unequal answers, including 49 m².

Check the kind of quantity as well as its number

Multiplying two lengths gives an area unit: m × m = m². Adding lengths gives a length, so a rectangle’s perimeter is reported in m. For sides 6 m and 4 m, area is 24 m² and perimeter is 2 × (6 + 4) m = 20 m. A number can be familiar while attached to the wrong quantity or unit. Check what the question asks before comparing sizes.

A rate calculation also needs compatible units. At a stipulated constant 72 km/h for 25 min, rewrite 25 min = 25/60 h = 5/12 h before multiplying. The hour units cancel and the distance is 30 km. Multiplying the bare numbers 72 and 25 to obtain 1800 overlooks the different time units. A proposed answer in km/h would still be a speed, while this question requests distance. Correct dimensions are necessary, but they alone cannot verify a numerical answer.

Compare numerical scale in a fixed unit

A coarse magnitude check asks whether the result is in ones, tens, hundreds or a very different scale. Using the earlier lesson’s exponent-bin convention in a fixed unit, a nonzero numerical magnitude belongs to 10^n ≤ |x| < 10^(n+1). The area 48.02 m² lies in the numerical bin [10, 100) when expressed in m². This is a useful coarse check, not a replacement for the area calculation.

The same physical area is 480200 cm² because 1 m² = 10000 cm². Its numerical bin in cm² is [100000, 1000000). The quantity has not changed, but the unit and numerical bin have. Compare like units before judging magnitude. A tenfold error such as 480.2 m² is plainly inconsistent with the rectangle’s 36–50 m² enclosure; a nearby wrong answer such as 49 m² needs a finer check.

State the reference for a difference or percentage

An absolute difference keeps the quantity’s unit. Comparing the area estimate 50 m² with exact model area 48.02 m² gives |50 − 48.02| m² = 1.98 m². A relative difference divides that magnitude by the magnitude of a named nonzero reference. Taking 48.02 m² as the reference gives 1.98/48.02 × 100% ≈ 4.12%. The units cancel in the ratio. Using 50 as the denominator would answer a different comparison question.

A declared checking policy might allow an estimate within 5% of that reference. The estimate would pass that policy, yet it would still differ from the exact model result. A tolerance is a chosen criterion for a purpose, not a truth guarantee or a measurement uncertainty unless separately defined as such. If the reference is zero, this relative formula cannot be used; use an appropriate absolute or contextual comparison instead.

Reverse the operation and return to the given relationship

An inverse operation can check arithmetic. Since 38 × 21 = 798, dividing 798 by 21 returns 38. For the rectangle, 48.02 m² ÷ 4.9 m = 9.8 m returns the other given side. Keep units in the reverse calculation. A candidate 49 m² instead gives 10 m, which does not match the stipulated 9.8 m side.

You can also substitute a candidate into the original situation. Five boxes with capacity eight give 5 × 8 = 40 available places; four give 32. A candidate distance 32 km would require 32/72 h = 4/9 h, or 26⅔ min, at the stated constant speed, rather than the given 25 min. Reversing a calculation helps check its relation to the inputs. It cannot establish that an assumed rectangle, rate or capacity model describes a real situation correctly.

Keep the question’s whole and constraints visible

A percentage needs its reference whole. Twenty is 25% of eighty because 20/80 = 1/4; the same twenty is 10% of two hundred. Both comparisons are valid, but only the whole named in the question answers it. Similarly, distinguish a total budget from the amount remaining, distance travelled from distance still to go, and available capacity from the number of containers actually required.

Check the allowed answers in context. A count of whole boxes must be a nonnegative integer. A part cannot exceed its stated whole, and a proposed minimum must both work and have every smaller possibility ruled out. These constraints can reject impossible answers. They still leave room for multiple plausible answers unless the required calculation and conditions are checked.

Use a rate estimate only under its stated model

For constant speed 72 km/h throughout 25 min, a half-hour estimate gives 72 × 1/2 = 36 km. It is an upper estimate because the actual model time is less than half an hour and the speed is positive. Bracketing the time between 20 and 30 min gives 24 km ≤ distance ≤ 36 km. Exact calculation gives 30 km, and the estimate’s absolute difference is 6 km, or 20% relative to 30 km.

The enclosure also admits 32 km, so passing it does not verify the exact answer. Constant speed throughout is an explicit condition here. Stops, changing speed or a different interpretation of the time require a suitable model before the same calculation can be used. A plausible travel distance does not validate assumptions about traffic, a route or a real vehicle.

Finish a capacity calculation with a whole-number check

Suppose 37 indivisible objects must be placed in boxes holding at most eight objects each, and any grouping is allowed. The ratio 37/8 = 4.625 compares the count with one box’s capacity; it is not a permitted number of whole boxes. Four boxes hold at most 32 objects, while five hold at most 40. Thus 4 × 8 < 37 ≤ 5 × 8 proves that the minimum is five boxes in this model.

Five boxes are feasible: fill four with eight objects each and put the remaining five in the fifth. Together they leave three unused places. Six boxes can also hold the objects but do not answer the request for the minimum. The conclusion relies on the stated count capacity and free grouping; extra restrictions about object sizes, weights or permitted combinations would need to be included separately.

Compare a candidate with estimates, bounds and exact checks

Start with the explorer’s rectangle case and correct candidate 48.02 m². Its exact sides are 9.8 m and 4.9 m, its easy upper estimate is 50 m² and its model enclosure is [36, 50] m². Compare the nearby candidate 49 m²: it fits the enclosure, but dividing by the given 4.9 m side gives 10 m rather than 9.8 m. The tenfold candidate and the candidate with a length unit reveal different faults.

The constant-speed case uses exactly 72 km/h throughout 25 min, giving 30 km. Its 36 km half-hour estimate and [24, 36] km enclosure are justified by the time comparison. Candidate 32 km fits the enclosure but fails the exact time check; 1800 km exposes the missing minute-to-hour conversion, and a speed unit does not answer a distance question. The packing case compares five boxes, 37 boxes, six boxes and the capacity ratio 4.625. Check both feasibility and the requested minimum, as well as the whole-number requirement.

Three cases and four candidate positions give 3 × 4 = 12 curated states. The cases keep their exact model inputs fixed; choosing a different candidate does not change those inputs. Use the display to identify what each check can settle. It does not assign measurement uncertainties or turn a rough estimate, an enclosure or a plausible candidate into proof of exact correctness.

Use several checks and keep their limits explicit

Restate the question, identify the inputs and unit, and make an estimate before relying on a detailed calculation. Check the dimensions and numerical scale, then any justified bounds. Compare against a named reference when a difference or percentage helps. Reverse the operation or substitute the candidate, and check the whole, count restrictions and model conditions.

Passing a rough check means the answer survived that check. It does not prove that every digit, unit, interpretation or assumption is correct. An exact calculation can establish the answer within a stated model; applying it to the world still needs a suitable model and input information. The next lesson, Variables, expressions and substitution, introduces names for quantities and ways to evaluate the relationships they describe.

Worked examples

Try it yourself

1. Estimate 38 × 21 using 40 × 20, then calculate the exact product. Does increasing one factor and decreasing the other automatically prove that the estimate is an upper bound?

   Show the answer and reasoning

   The estimate is 800 and the exact product is 798, so this estimate is two above the exact result. Mixed rounding directions alone do not prove an upper bound for every product.

   Convenient numbers make the operation easy. Check how changing the inputs affects the requested quantity before claiming a direction. Here exact calculation establishes the comparison, and 798 ÷ 21 = 38 supplies an inverse arithmetic check.

2. Stipulated prices are €17.80, €9.60 and €12.40. Compare the easy sum 18 + 10 + 12 with the exact total. Then justify an upper total using 18 + 10 + 13. Do these replacements state measurement uncertainties?

   Show the answer and reasoning

   The exact total is €39.80. The first estimate is €40, but its mixed rounding alone does not justify an upper bound. Each term in the second sum is at least the given price, so €41 is a justified upper total. These are arithmetic estimates and bounds, not measurement uncertainties.

   For addition, replacing every term by a greater or equal value gives an upper sum. In the first estimate the third price was decreased, so its direction needs a separate check. Both estimates should be identified as replacements of the exact stipulated prices.

3. For a rectangle with exact sides 6 m and 4 m, find area and perimeter. Which claim has the wrong kind of unit: area 24 m or perimeter 20 m? Convert the correct area to cm² and explain why its larger numeral does not describe a larger physical area.

   Show the answer and reasoning

   Area is 24 m² and perimeter is 20 m. The claim area 24 m has a length unit. Since 1 m² = 10000 cm², 24 m² = 240000 cm²; the area is unchanged.

   Multiplying lengths gives square units, while adding boundary lengths gives a length unit. A unit conversion changes numerical representation. Compare magnitudes only after expressing quantities in compatible units.

4. An exact model reference distance is 30 km and an estimate is 36 km. Find their absolute difference and the relative difference with 30 km as reference. A checking policy accepts differences up to 25% of this reference: does passing it prove that 36 km is exact?

   Show the answer and reasoning

   The absolute difference is 6 km and the relative difference is 6/30 × 100% = 20%. The estimate passes the declared 25% policy, but it is still unequal to the exact model reference 30 km.

   Name the nonzero reference used in the denominator. A checking tolerance is a chosen criterion with a stated purpose; passing it does not prove exact equality, establish a physical truth or supply measurement uncertainty.

5. At exact constant speed 72 km/h for 25 min, candidates are 30 km, 32 km, 1800 km and 30 km/h. Use units, time conversion and a reverse check to explain which answers the question. Is fitting [24, 36] km sufficient?

   Show the answer and reasoning

   30 km answers the stipulated model. 32 km fits the enclosure but takes 26⅔ min at this speed; 1800 km comes from ignoring the minute-to-hour conversion; 30 km/h has the wrong quantity kind. Fitting the enclosure alone is insufficient.

   Convert 25 min to 5/12 h before multiplying, then check 30/72 h = 5/12 h. Coarse bounds remove some wrong answers but can admit other wrong candidates. The constant-speed and elapsed-time assumptions must remain explicit.

6. Twenty tiles are blue. One question compares them with a set of eighty tiles; another with a set of two hundred. Calculate both percentages. Why does one percentage not automatically answer the other question?

   Show the answer and reasoning

   Twenty of eighty is 25%; twenty of two hundred is 10%. The whole named in each question differs, so each percentage answers its own comparison.

   A percentage is a ratio to a stated whole. Preserve the whole and the counted part rather than carrying a familiar percentage into a different context. The same part can give different valid percentages against different totals.

7. Exactly 37 indivisible objects may be grouped freely into whole boxes with capacity at most eight each. Compare candidates four, five, six and 4.625 boxes. Which is feasible, and which answers the request for the minimum?

   Show the answer and reasoning

   Four boxes are insufficient because their capacity is 32. Five and six can hold the objects, but five is the minimum because 32 < 37 ≤ 40. The ratio 4.625 is not a permitted whole-box count.

   A minimum needs two checks: the proposed count works, and every smaller allowed count fails. Four full boxes plus five objects in a fifth demonstrate feasibility. Extra packing restrictions could change the model and must be stated rather than assumed away.

What to carry forward

• Choose an estimate for its purpose and explain the convenient numbers used; a rough heuristic needs a separate argument before it can be called a bound.
• Check quantity kind, compatible units and numerical magnitude before trusting a familiar-looking number.
• Keep the unit in an absolute difference; for a relative comparison, name the nonzero reference. A declared tolerance does not establish exact correctness.
• Reverse or substitute into the given relationship, then check the chosen whole, whole-number constraints, feasibility and any requested minimum.
• Reasonableness checks can reject errors or admit plausible candidates; exact model calculations and real-world model suitability require their own checks.