BTT Mathematics / Primary Mathematics Learning Hub / Bounds and Guarantees
A bound tells us how small or large a quantity can be without claiming that we know its exact value. A guaranteed conclusion tells us what must be true for every arrangement satisfying the conditions, even when the precise arrangement is unknown.
If thirty-one counters are placed into five boxes, the exact box counts are not determined. But at least one box must contain at least seven counters. If every box had at most six, the five boxes could hold at most thirty. The thirty-first counter forces some box to reach seven.
This kind of reasoning is different from finding one possible arrangement. For example, 7,6,6,6,6 is possible, but so is 10,8,5,4,4. The guaranteed statement is the part shared by all valid arrangements.
This guide develops lower and upper bounds, rounded-number intervals, extremes under fixed totals, guaranteed occupancy, worst-case planning and the difference between possible, impossible and certain. It complements Rounded Data, Thresholds and Guaranteed Conclusions with a Primary worked-practice route focused on whole-number reasoning and transparent proofs.
Use the learner’s current school scope and the MOE Primary curriculum and syllabus page as reference points. The examples and 24 questions below are original educational material.
Lower and upper bounds · Rounded-number ranges · Minimum and maximum · Guaranteed occupancy · Worst-case reasoning · Possible, guaranteed or impossible? · 24 questions · Worked solutions
1. A bound is a limit, not necessarily an attained value
Suppose three positive whole numbers total twenty. Each number must be at least one. Therefore no single number can exceed eighteen, because the other two positive numbers need at least one each.
The bound eighteen is attainable: 18,1,1 is valid. So eighteen is not only an upper bound but the maximum possible value of one part.
Worked example A: Non-negative parts
Four non-negative whole numbers total fifty. One number can be as small as zero and as large as fifty because the other three are allowed to be zero.
If the numbers are required to be positive, the largest possible one becomes 50−1−1−1=47.
Worked example B: At least a stated minimum
Four boxes contain sixty objects altogether, and every box has at least eight. To maximise one box, put the minimum eight into each of the other three: 60−24=36.
To minimise one selected box under only the stated conditions, it can contain eight.
Worked example C: A fixed difference
Two positive whole numbers total forty. The larger is at most thirty-nine and the smaller at least one without more information. If the larger is exactly ten more than the smaller, those wide bounds collapse to the exact pair fifteen and twenty-five.
Bounds become tighter as conditions are added.
Open and closed limits
“Less than fifty” excludes fifty. “At most fifty” includes it. “Greater than twenty” excludes twenty. “At least twenty” includes it. These words change the endpoint of the candidate set.
When the quantities are whole numbers, “less than fifty” means forty-nine or below. For measured continuous quantities, there may be infinitely many values just below fifty.
2. Rounding replaces an exact value with an interval
A whole number rounded to the nearest ten is eighty. Under the usual halfway-up convention, the original could be any whole number from 75 to 84 inclusive.
Seventy-four rounds to seventy. Eighty-five rounds to ninety. These neighbouring failures define the lower and upper whole-number boundaries.
Worked example D: Nearest hundred
A whole number rounds to 1,200 to the nearest hundred. Possible originals are 1,150 through 1,249 inclusive.
Worked example E: Use a rounded value in a threshold question
A whole-number attendance is reported as 430 to the nearest ten. Could the actual attendance be at least 435? No. The largest possible original is 434.
Could it be at least 425? Yes; possible originals run from 425 through 434.
Worked example F: Compare two rounded values
Population A rounds to 600 to the nearest hundred, so its whole-number range is 550–649. Population B rounds to 700, giving 650–749.
Therefore B is definitely larger than A because B’s smallest possible value, 650, exceeds A’s largest possible value, 649.
Overlapping ranges prevent a guaranteed ordering
If one quantity rounds to 600 to the nearest hundred and another to 620 to the nearest ten, their possible ranges may overlap. A rounded display alone may not support a definite claim about which exact original is larger.
Always convert rounded reports back into their possible intervals before making a guaranteed conclusion.
3. To find an extreme, push the other quantities to their allowed limits
Worked example G: Maximum with minimum requirements
Five teams share one hundred points. Every team must receive at least twelve. To maximise Team A, give the other four teams twelve each. Team A can receive 100−48=52 points.
Worked example H: Minimum with maximum requirements
Four containers hold eighty litres altogether. No container can hold more than twenty-five litres. To minimise Container A, maximise the other three: 25+25+25=75. At least five litres must remain for A, so its minimum is 5 L.
Worked example I: Largest difference
Two positive whole numbers total thirty. The largest possible difference occurs when the smaller is as small as allowed: one and twenty-nine. The maximum difference is 28.
Worked example J: Smallest difference
Two whole numbers total thirty. To make their difference as small as possible, make them as equal as possible. Fifteen and fifteen give difference zero.
If the total were thirty-one, the closest whole pair would be fifteen and sixteen, giving minimum difference one.
Worked example K: Extreme product is a different problem
For two positive whole numbers with fixed sum ten, the maximum sum is already fixed, but the product changes. The products for 1×9,2×8,3×7,4×6,5×5 are 9,16,21,24,25. The largest product occurs at five and five.
Do not assume the arrangement maximising a difference also maximises a product. Each target quantity has its own extreme structure.
4. Guaranteed occupancy comes from testing the largest arrangement that still avoids the target
Thirty-one counters are placed into five boxes. To avoid any box having seven counters, each box could contain at most six. Five boxes at six each hold thirty. Since there is one more counter, avoiding seven is impossible.
Therefore at least one box must contain at least seven counters.
Worked example L: Students and groups
Twenty-five students are placed into four groups. If every group had at most six students, there would be at most twenty-four. The twenty-fifth student forces some group to have at least 7 students.
Worked example M: Birth months
Thirteen pupils have birthdays across twelve months. If every month contained at most one birthday, only twelve pupils could be accommodated. Therefore at least two pupils share a birth month.
This conclusion says nothing about which month and does not assume birthdays are equally likely.
Worked example N: Guarantee at least three in one category
Objects are placed into four categories. How many objects guarantee that one category contains at least three? To avoid three, each category may hold at most two. Four categories can then hold eight. The next object forces a third in some category.
The minimum guaranteed total is 9 objects.
Worked example O: Guarantee at least five
Six boxes receive objects. To guarantee at least five in one box, first ask how many objects can be distributed while keeping every box at four or fewer. Six times four equals twenty-four. Therefore 25 objects guarantee at least five in some box.
General whole-number rule as a reasoning sentence
With k boxes, to avoid r objects in any one box, place at most r−1 in each. That arrangement holds k(r−1) objects. One more object makes the target unavoidable.
Formal notation is optional. The key is the worst arrangement that still avoids the guaranteed event.
5. “Guarantee” questions use the most delaying valid arrangement
Worked example P: Coloured socks without seeing them
A drawer contains red, blue and green socks, with enough of each colour for the scenario. How many socks must be drawn blindly to guarantee two of the same colour?
In the most delaying arrangement, the first three socks could be one of each colour. The fourth sock must match one of those colours. Therefore 4 socks guarantee a matching-colour pair.
Worked example Q: Guarantee three of one colour
With three colours available, two of each colour could be drawn without yet obtaining three of one colour. That is six socks. The seventh guarantees a third sock of some colour.
The result assumes at least the needed stock exists and that only the three stated colours are possible. A hidden fourth colour would change the guarantee.
Worked example R: Guarantee a repeated digit category
Seven selected whole numbers are classified by their remainder when divided by three: remainder zero, one or two. There are three categories. To avoid three numbers in the same remainder class, each class could hold at most two, totalling six.
The seventh number guarantees at least three numbers with the same remainder on division by three.
Worst-case does not mean most likely
The most delaying arrangement may be very unlikely in a real random process. Guarantee reasoning does not use probability. It asks whether the target is forced even under the valid arrangement that postpones it longest.
Stock limits can make a guarantee happen earlier
If a drawer contains only one green sock but many red and blue socks, the worst case for delaying a colour pair changes. The composition of the actual collection matters. Use every known stock limit rather than assuming unlimited availability in each category.
6. Possible, guaranteed and impossible are different claims
“Possible” means at least one valid arrangement exists. “Guaranteed” means every valid arrangement has the property. “Impossible” means no valid arrangement has it.
Possible but not guaranteed
Ten counters in three boxes can be arranged 8,1,1, so it is possible for one box to contain eight. But eight is not guaranteed: 4,3,3 is also valid.
Guaranteed but not exact
Ten counters in three boxes guarantee at least four in some box, because three boxes capped at three hold only nine. The fullest box could have four, five or more. The guarantee gives a lower bound on the maximum occupancy, not its exact value.
Impossible
Ten positive whole-number parts cannot all exceed three, because four parts each at least four would already total sixteen. An impossibility proof compares minimum requirements with the available total.
A bound can be sharp
A guaranteed lower bound is sharp if there is an arrangement that meets it exactly while still satisfying the conditions. For ten counters in three boxes, 4,3,3 shows that “at least four” cannot be improved to “at least five.”
Check whether the question asks for an extreme or a guarantee
“What is the greatest possible number in one box?” and “What number must appear in at least one box?” are not the same. The first seeks a maximum allowed by some arrangement. The second seeks a lower bound that holds across all arrangements.
7. Practice: 24 questions
For each question, state whether you are finding an exact value, a possible range, an extreme or a guaranteed conclusion. Keep the worked solutions covered during the first attempt.
Questions 1–8: Bounds and rounding
1. Three positive whole numbers total 25. What is the largest possible value of one number?
2. Four non-negative whole numbers total 40. What is the largest possible value of one number?
3. Five boxes contain 90 objects. Each box has at least 10 objects. What is the largest possible number in one box?
4. Four containers hold 100 L altogether. No container holds more than 30 L. What is the smallest possible amount in one selected container?
5. A whole number rounds to 70 to the nearest ten. State the complete possible range.
6. A whole number rounds to 1,500 to the nearest hundred. State the complete possible range.
7. Attendance rounds to 320 to the nearest ten. Can the actual attendance be 325?
8. Quantity A rounds to 400 to the nearest hundred and B rounds to 500. Is B definitely greater than A?
Questions 9–16: Extremes and guaranteed occupancy
9. Two positive whole numbers total 36. Find the largest possible difference.
10. Two whole numbers total 37. Find the smallest possible difference.
11. Six teams share 150 points, with at least 15 points per team. What is the maximum possible score of one team?
12. Five containers hold 120 L, each holding at most 30 L. What is the minimum possible amount in one selected container?
13. Thirty-one counters are placed into five boxes. Show that some box contains at least seven.
14. Twenty-nine pupils are divided into four groups. What minimum group size is guaranteed in at least one group?
15. How many objects placed into six boxes guarantee at least four objects in one box?
16. How many objects placed into five categories guarantee at least six in one category?
Questions 17–24: Worst case and claim language
17. A drawer contains socks of three possible colours. How many blind draws guarantee two socks of the same colour?
18. With four possible colours, how many blind draws guarantee three socks of one colour, assuming enough stock of every colour?
19. Thirteen pupils have birthdays in twelve months. What is guaranteed?
20. Ten counters are placed into three boxes. Is it possible for one box to contain eight? Is eight guaranteed?
21. Ten counters are placed into three boxes. What minimum occupancy is guaranteed in at least one box, and show an arrangement proving the bound is sharp.
22. Seven whole numbers are classified by remainder on division by three. What must be true?
23. Five positive whole numbers total 18. Can every number be at least four? Explain.
24. A whole number is reported as 800 to the nearest hundred. Another is reported as 840 to the nearest ten. Is the second definitely larger? Justify from the possible ranges.
8. Worked solutions
Solutions 1–8
1. 23. Keep the other two positive numbers at their minimum one each: 25−2=23.
2. 40. The other three may be zero, so one number can take the whole total.
3. 50. Put the minimum ten into four other boxes, using forty. The remaining fifty can be in the selected box.
4. 10 L. Maximise the other three at thirty each, using ninety litres. At least ten litres remain for the selected container.
5. 65 through 74 inclusive. Sixty-four rounds to sixty; seventy-five rounds to eighty.
6. 1,450 through 1,549 inclusive. These are the whole numbers rounding to 1,500 to the nearest hundred under the usual halfway-up convention.
7. No. Whole numbers rounding to 320 to the nearest ten range from 315 through 324. Three hundred twenty-five rounds to 330.
8. Yes. A lies from 350 through 449; B lies from 450 through 549. B’s minimum exceeds A’s maximum.
Solutions 9–16
9. 34. Use one and thirty-five, whose difference is thirty-four.
10. 1. The closest whole-number pair is eighteen and nineteen.
11. 75 points. Put the minimum fifteen into five teams, using seventy-five. The remaining seventy-five can go to one team.
12. 0 L. Four other containers can hold thirty litres each, accounting for all one hundred twenty litres. No positive minimum was stated for the selected container.
13. At least 7. If every box had at most six, five boxes could contain at most thirty counters. The thirty-first forces seven in some box.
14. At least 8 pupils. Four groups of seven hold only twenty-eight. The twenty-ninth pupil forces some group to eight.
15. 19 objects. To avoid four in any box, keep all six boxes at three or fewer: maximum eighteen. The nineteenth forces a fourth object somewhere.
16. 26 objects. To avoid six in a category, each of five categories may contain at most five, totalling twenty-five. The twenty-sixth forces six in one category.
Solutions 17–24
17. 4 draws. The first three can be one of each colour. The fourth must repeat a colour.
18. 9 draws. Two of each of four colours gives eight without three matching. The ninth guarantees a third of some colour.
19. At least two pupils share a birth month. Twelve months could hold at most twelve pupils if all months had at most one.
20. Possible yes; guaranteed no. The arrangement 8,1,1 is possible. But 4,3,3 shows that no box need contain eight.
21. At least 4; sharp via 4,3,3. If every box had at most three, the total would be at most nine. Therefore some box has at least four. The arrangement 4,3,3 shows five cannot be guaranteed.
22. At least three numbers have the same remainder when divided by three. There are only three remainder categories. Two in each would account for six numbers; the seventh forces a third in one category.
23. No. Five numbers each at least four would total at least twenty, exceeding eighteen.
24. No; the second is not definitely larger. The first whole number may lie anywhere from 750 through 849. The second lies from 835 through 844. These ranges overlap: for example, the first could be 849 while the second is 835. Therefore the rounded reports do not guarantee the ordering.
9. Teaching and diagnosis
If a learner searches for one arrangement when the question asks what must happen, ask them to produce a different valid arrangement. Any claim that changes across the two arrangements is possible rather than guaranteed.
When maximum and guarantee are confused
Use ten counters in three boxes. One box can contain all ten, so ten is the maximum possible occupancy. But only four is guaranteed in at least one box. The two questions examine different directions of reasoning.
When rounding is treated as exact
Write the neighbouring halfway points before comparing rounded values. A displayed eight hundred is not one exact original; it represents an interval. Threshold conclusions must survive every value in that interval.
When a worst-case arrangement is missing
Ask how many objects can be placed while still avoiding the target. Fill every category to one below the threshold. Only then add one more. This makes the guarantee proof constructive and easy to inspect.
When a bound is not sharp
After proving “at least r,” try to build an arrangement whose largest category is exactly r. If such an arrangement exists, a stronger universal claim would be false.
Continue through the structural-reasoning collection
For equal-unit scaling, use Unitary Method and Equal-Unit Reasoning. For two sharing plans with excess or shortage, use Excess and Shortage Problems. For answers that must satisfy several conditions at once, use Simultaneous Conditions and Multiple Constraints.
Return to the BTT Primary Mathematics Learning Hub.
Original explanations and 24 original practice questions with separate worked solutions. Rounding examples use the usual school halfway-up convention for whole numbers. No examination prediction or performance guarantee is implied.