Tables With Missing Data in CAT DI: Fill the Blanks

DILR

A table arrives with five blank cells and a question about one of them. Your first thought is that something has gone wrong, because the number you need is not there. It is not meant to be. The table gives you the totals, and the blanks are a puzzle you solve with addition and subtraction.

Think of it as Sudoku with sums. Every row total and column total is a clue, and a few cells fall out in seconds once you know where to look. This guide shows a three part method for tables with missing data, including how to tell, before you start, whether the table can be solved at all.

Why Blank Cells Are a Clue, Not a Problem

DI sets in CAT use tables, charts and caselets, and set types and counts change from year to year. Tables with missing data are one recurring flavour. They look intimidating because the visible numbers do not answer the question, and the temptation is to hunt for a formula.

No formula is needed. A row total says the cells in that row add up to a known number. A column total says the same about a column. Every blank sits in one row and one column, so every blank is pinned by two sum statements. When one of those statements has only one unknown, the cell is solved.

The deeper point is that the table already contains more information than it shows. The totals are redundant by design: the row totals add up to the grand total, and so do the column totals. That redundancy is what lets you fill cells from two directions and cross check your work.

If you treat each blank as a mystery, you will search randomly. If you treat each as a variable in a few simple equations, you will solve in a sensible order. That change of view is most of the skill, and the method below puts it into steps.

The Sweep Chain Count Method: Three Moves for Any Blank Table

The method has three moves, in order. Most tables are finished after the first two.

Why this order and not another? Subtraction is the cheapest operation you can do, so it goes first. Chaining reuses the results of the sweep, so it comes second. Counting is mental overhead, so it waits until you are genuinely stuck. Spending effort on counting before you have swept every line is a common way to waste the first minute of a table set.

The method also scales. A table of four rows and three columns has twelve cells and seven totals, and the same three moves apply to a table twice that size. Larger tables just give the chain more links to follow.

The sweep is fast because it needs only subtraction. The chain is where the table opens up, since each cell you fill reduces the unknowns in two lines at once. Count is the safety move: it tells you whether you are stuck because you missed something or because the table genuinely lacks information.

Sweep and Chain: A Worked Table

Here is a table written for practice. Four regions sell three products. Totals are given and five cells are blank. North sells 30 of X, a blank of Y and 25 of Z, with a total of 75. South sells 40 of X, 15 of Y and a blank of Z, total 90. East sells a blank of X, 30 of Y and 20 of Z, total 60. West sells 20 of X, a blank of Y and a blank of Z, total 75.

The column totals are 100, 90 and 110, and the grand total is 300.

First, the consistency check. The row totals are 75, 90, 60 and 75, which add to 300. The column totals are 100, 90 and 110, which also add to 300. The table agrees with itself, so you can trust the totals.

Now sweep. North has one blank: 75 minus 30 minus 25 gives 20 for Y. South has one blank: 90 minus 40 minus 15 gives 35 for Z. East has one blank: 60 minus 30 minus 20 gives 10 for X. West still has two blanks, so skip it for now.

Chain. The Y column now has three known cells, 20, 15 and 30, so West Y is 90 minus 65, which is 25. The Z column has 25, 35 and 20 known, so West Z is 110 minus 80, which is 30. Check the West row: 20 plus 25 plus 30 is 75, which matches. The table is complete, and the last check confirms it.

Count: When the Table Stalls and Degrees of Freedom

Now change the table so that it stalls. Suppose South and West both have blanks in Y and Z. South sells 40 of X and the row total is 90, so its Y and Z add to 50. West sells 20 of X and the total is 75, so its Y and Z add to 55.

In column Y the known cells are North 20 and East 30, so the two blanks in Y add to 40. In column Z, North 25 and East 20 are known, so the two blanks in Z add to 65.

No row or column has a single blank, so the sweep finds nothing. Count the freedom. Call South Y a. Then South Z is 50 minus a, West Y is 40 minus a, and West Z is 55 minus 40 plus a, which is 15 plus a. Every cell depends on one unknown. The block has exactly one degree of freedom.

The rule is simple. A rectangle of blanks with r rows and c columns, surrounded by known totals, has (r minus 1) times (c minus 1) free choices. A two by two block has one, a two by three block has two, and so on. You need that many extra relations to fix the block.

That count tells you what to do next. If the set gives one extra relation, such as "South sells 20 more of Z than Y", the block is solved: a plus a plus 20 equals 50, so a is 15. If it gives none, then some questions about the block cannot be answered, and a question that asks for a value that depends on a is probably a trap or a leave it question.

Extra Relations: Percentages, Ratios and Rounding

Statements in the set text are often the missing equations. "East sells twice as much of X as of Z" gives a relation. "Product Y accounts for 30 percent of the grand total" gives a column total. Convert each statement into an equation as soon as you meet it, and add it to the pile before you declare the table stuck.

Percentages deserve care. If the table shows shares as rounded percentages, the totals may not add to exactly 100 and the figures may not reconcile exactly. Work with approximate values and choose the option that fits best, instead of searching for a perfect match that the rounding makes impossible.

Some sets ask whether a value can be determined at all. A table with a free block is the standard setting for that kind of question. If your count shows a degree of freedom that the question depends on, the answer is that it cannot be determined. That is a legitimate answer, and recognising it saves time.

Situation	What it means	What to do
Row or column with one blank	Solvable by subtraction	Solve and sweep again
Two blanks in a line, solved crosswise	Chain will resolve it	Fill the crossing line first
Rectangle of blanks	(r minus 1)(c minus 1) free choices	Find that many extra relations
Row and column totals disagree	A printed value is off or rounded	Use approximate values and compare options

One more habit pays off. Whenever you fill a cell, write the value in the table straight away, and do not hold it in your head. Tables are the easiest DI sets to solve on paper and the easiest to get wrong from memory. A filled cell on the page is available for the next sweep, and it is visible when you cross check at the end.

A Practice Routine for Blank Tables

Practise on tables of different shapes, so the sweep becomes automatic. For two weeks, solve three or four table sets a week. Before you solve, write down how many blanks there are and whether a single blank exists. That habit takes five seconds and points you to the first move.

After each set, ask which move cost you time. If it was the sweep, you were not scanning both directions. If it was the chain, you missed a new single blank. If it was the count, you guessed instead of checking freedom. For the bigger picture on picking sets, see can two DILR sets be enough, and for sets that look hopeless at first, read why some DILR sets feel unsolvable.

Past papers give you the real feel of how totals and blanks appear. Our topic wise CAT previous year questions include table based sets with solutions, and quant based CAT DI practice sets add the ratio and percentage twists described above.

Guessing in DILR has its own logic. Our note on whether guessing is good in CAT DILR helps you decide what to do with a block that your count has marked as undetermined.