Data Arrangement

Placing numbers given only in words into rows and columns.

The idea

A paragraph is a heap of numbers. Solving it straight from the words is like hunting for a shirt in an unsorted cupboard. A table is the cupboard in which every number has its own place. First a few words: a row (a line across), a column (a line down), a cell (where a row and a column meet; one number sits there) and a total (the sum of a row or a column). When every person is described by two features — say boy/girl and club member/not — there are four cells, and each person falls in exactly one cell, so nobody is counted twice.

Here $a,b,c,d$ are the numbers in the four cells and $N$ is the grand total. The check: the row totals add up to $N$ and so do the column totals — if not, there is a slip somewhere.

Steps

1. Read and underline: with every number note what it counts (“30 of the boys”, “240 in all”).
2. Choose the frame: put in the rows the groups that the sentence says “of the …” (these are the bases) and in the columns the conditions. Add one row and one column for totals.
3. Write the totals first, then the numbers that are given directly. Write “?” for what is unknown.
4. Look for a lone-blank line (a row or column with just one “?”) and fill it with $\text{total}-\text{rest}$; repeat.
5. Now read the answer from the table and check row totals against column totals.

Worked examples

A recruitment exam has 250 candidates. Of them, 50 are men and the rest are women. Among the men, 20 are successful, while among the women, 100 are successful. All the others are unsuccessful.

The finished table:

A library has 180 members, of whom 100 are men. In all, 120 of the members are users of the online portal, and 60 of these are men.

Try yourself

A library has 600 members. Of them, 300 are men and the rest are women. Among the men, 180 are users of the online portal, while among the women, 75 are users of the online portal. All the others are non-users of it.

A colony survey covers 200 families, of whom 100 are families with a car. In all, 150 of the families are users of LPG, and 75 of these are families with a car.

The dictionary of caselet language and the rule of the base.

The idea

The real difficulty of a caselet is not the maths but the language. The same figure changes its meaning when the base changes: in “A is 20% more than B” the base of 20% is B, but in “by what percent is B less than A” the base becomes A. So in every comparison sentence first ask: “compared with what?” — the thing after “than” is the base; treat it as 100 (or 1) and go on.

Here: $A,B$ are two quantities, $p,q$ are percents, $d$ is a plain difference, $T$ the total, $X,Y$ groups. The commonest slip: taking “more than” and “less than” as the same percent turned round — but when the base changes, the percent changes (see the example below).

Steps

1. In every comparison sentence underline the thing after “than” (the base).
2. Start from the quantity whose value is given and compute the next one from it.
3. The next sentence uses the value just found as its base — in a chain the base changes at every step.
4. Do not use a percent for a plain difference (₹ 40 more) and do not add or subtract plainly for a percent.
5. When the total is given and the base is unknown, call the smallest quantity $x$ and write an equation for the total.

Worked examples

Library F’s monthly issues are 240 books. Library D’s monthly issues are 25% more than Library F’s. Library C’s monthly issues are 25% less than Library D’s.

The total wheat yield of the three (Farm F, D, A) is 780 quintals. Farm D’s wheat yield is twice Farm F’s. Farm A’s wheat yield is 30 quintals more than Farm D’s.

Try yourself

Farm A’s wheat yield is 480 quintals. Farm F’s wheat yield is 50% more than Farm A’s. Farm B’s wheat yield is 50% less than Farm F’s.

The total monthly sales of the three (Shop A, C, D) are ₹ 420 thousand. Shop C’s monthly sales are twice Shop A’s. Shop D’s monthly sales are ₹ 20 thousand more than Shop C’s.

Going down from the total to a group and then to a part of the group.

When people can belong to two groups, make four regions.

The three-circle Venn diagram and inclusion–exclusion.

Numbers from a ratio and a total, matching two units, and chains of ratios.

Like sudoku: begin where a line has only one blank.

When a table mixes numbers and percents, or gives a ratio condition.

Which cell comes first, how many rounds it takes, and when the answer is not unique.

Giving five people five given values.

When the "people" are ordered days such as Monday to Saturday.

Arranging a month-wise or day-wise table in order to answer.

Finding revenue from a quantity table and a price table.

The table gives the totals, the chart the percents; together they give numbers.

Applying changes given in the text to the table’s figures.

“Definitely true”, “not true” and “how many are true”.

When the data is incomplete, which statement holds in every case.

Deciding whether a solution can be found, without solving.

Tricks that save the full calculation.

The slips that cost most marks and how to avoid each.

How to read a set, which question to do first, when to skip, and what to check at the end.

Each method: when to use it and the key move.

Questions in the style of SSC CGL / CHSL, RRB, bank and state exams.

Think of the answer first, then flip the card.