Minimum and Maximum Questions in DILR: Bound with Extremes

DILR

Some DILR questions ask what the highest possible value could be, or the least possible. They do not ask what the value is. The data is incomplete on purpose, and the answer depends on how far the unknown parts can be pushed.

The method is to bound with the extremes: push everything else as far as the rules allow in the opposite direction, then see what is left for your target. This guide shows the method in three steps with small worked cases, and lists the checks that stop you from claiming a bound that cannot be reached.

What These Questions Look Like

In a table set, one cell or row may be unknown, but the totals and some constraints are given. A question then asks: what is the maximum possible number of units sold by the third branch, or the minimum possible score of the top student? The data does not pin the value, so the answer is a range endpoint.

In a selection or ranking set, the same logic appears as the best or worst position an item can take. In an arrangement set, it can appear as the largest possible gap between two people. The common thread is a set of rules that leave freedom, and a question that probes the edge of that freedom.

Students often try to find the actual value and get stuck, because there is none. The shift in thinking is to treat the rules as walls and ask how close to a wall the target can get.

The Three Step Bounding Method

The method works because extremes interact. If the total is fixed, anything you take away from the others goes to the target. So to make one value as large as possible, make the rest as small as the rules permit.

The check step is where marks are won or lost. An extreme that violates a rule is not valid, and you must back off until all rules hold. The confirm step, writing the full case, catches arithmetic slips and shows that the bound is reachable.

Example One: The Largest Possible Value

Three departments sold 90 units in total, and each sold at least 1 unit. What is the largest number of units the first department could have sold? To maximise the first, push the other two down to their minimum, which is 1 each. That leaves 90 minus 2, or 88 units for the first.

Now add a rule: the three departments sold distinct numbers, with the first selling the most, then the second, then the third. The minimum for the third is 1, and for the second it must be larger, so 2. The first then sells 90 minus 3, which is 87 units. The case is 87, 2 and 1, and it satisfies every rule.

Notice how the extra rule changed the answer from 88 to 87. This is typical. Each added condition tightens the extreme. If you ignore a condition, the bound looks easier than it is, and the answer will be wrong.

Example Two: The Smallest Possible Maximum

Five students scored distinct whole number marks, and the five scores add to 100. What is the smallest possible score of the top student? To make the highest as small as possible, spread the total as evenly as the rules allow. The scores must be distinct, so the closest set is five consecutive numbers.

Five consecutive numbers with sum 100 are 18, 19, 20, 21 and 22, since their average is 20. The top score is 22. Could it be 21? Then the best case is 21, 20, 19, 18 and 17, which sums to 95, less than 100, so it is not possible. The least top score is 22.

This pattern, the smallest possible maximum, uses the idea of balance. It is the mirror image of pushing others to extremes. When you minimise the largest item, you do not push others down. You lift them up, as close to the top as the rules allow.

Goal	Move the others	Result
Largest possible value of one item	As low as allowed	Remainder goes to the target
Smallest possible value of one item	As high as allowed	Remainder is as small as possible
Smallest possible maximum	Spread evenly	Top value sits near the average, with gaps added if values must be distinct
Largest possible minimum	Spread evenly	Bottom value sits near the average, with gaps added if values must be distinct

Example Three: The Largest Possible Minimum

Use the same five students and the same total of 100 with distinct whole number scores. What is the largest possible score for the lowest student? Now you want the bottom as high as possible, so lift everyone toward the average and keep the gaps tight.

The same set, 18, 19, 20, 21 and 22, gives a lowest score of 18. Test 19: five distinct values starting at 19 would be at least 19, 20, 21, 22 and 23, which add to 105, more than 100. So 19 is impossible, and the answer is 18.

The pair of answers, 22 for the smallest top and 18 for the largest bottom, comes from one balanced case. That is a useful check. If your two answers for the same set come from different cases, one of them probably breaks a rule.

Keep this example as a template. Whenever a set gives a fixed total with distinct whole values, the balanced consecutive case is where you start.

Integers, Ties and Order

Three types of restriction change the answer, and each deserves a check. The first is whole numbers. If the average is 20.4 and values must be whole, the smallest possible maximum is 21, not 20.4. Round in the direction that keeps the rule true.

The second is ties. If two items may be equal, the extreme case is simpler, since equal values fit the balance idea. If they must be distinct, as in a ranking, you need a gap between each pair, and the bounds shift by the size of that gap.

The third is order. A rule such as the first is greater than the second forces a gap, and the bound moves accordingly. If several items are tied by rule, treat them together. A short list of the rules at the top of your rough work, written as symbols, keeps all of this visible while you push values.

Bounds in Larger Sets

In a real set, the bound may involve several rows and columns. The method scales if you stay organised. Draw the table, mark the known cells, and mark the unknowns with a symbol. Then write each constraint as an inequality on the unknowns.

To bound one unknown, find which constraints involve it and which push it up or down. Often a row total and a column total give two different limits, and the real limit is the tighter one. Check both before you claim a value.

Our guide on tables with missing data and how to fill the blanks covers the first stage, which is recovering everything that can be recovered. Only after that does bounding come in, for the cells that cannot be fixed.

A related skill is answering what if questions without rebuilding the set. See what if questions without rebuilding the set for a method that pairs naturally with bounding, since both ask how far a value can move.

Bounds in Ranking and Selection Sets

In ranking sets, a question may ask for the highest or lowest rank an item could have. Use the same idea: place everything that must be above it above, everything that must be below it below, and count the items left free. The free items can go on either side, so they define the range.

For example, if an item must be below two others and above one, with six items in total, then its best rank is third, since two items must be above it. Its worst rank is fifth, since one must be below. The range is third to fifth. Count the forced items on each side and the answer follows.

Our guide on ranking based sets and relative ordering covers the rules for building the chain, and selection practice gives you more cases where extremes matter.

A Practice Routine

Start with ten short bounding questions from tables, one rule at a time. Add one rule per round: whole numbers, then distinct values, then order. This builds the habit of listing the rules before computing.

Then move to full sets. For each minimum or maximum question, write the extreme case in full and tick the rules. After a session, note which rule you missed. Over a few sessions, you will see the one you forget most often, such as distinctness, and you can build a specific check for it.

A CAT study planner can slot these short sessions into the weeks you have left, and you can talk to a CAT mentor if you want help with sets that keep costing you time. DILR is more unpredictable than Quant, so focus on the method rather than on any chapter weight.