BPSC 71st CCE Prelims General Studies · Question 9 of 150

Correct — B, 35. Read the grid row by row and test division first, because in every row the outer pair is a multiple and its own factor: 28 and 7, 45 and 9, 84 and 12. In the first row 28 ÷ 7 = 4 and the middle entry is 20, which is 4 × 5. In the third row 45 ÷ 9 = 5 and the middle entry is 25, which is 5 × 5. The rule that fits both complete rows is therefore middle = 5 × (first ÷ last), and applying it once to the incomplete row gives 84 ÷ 12 = 7 and 7 × 5 = 35. There is a cleaner form of the same rule that avoids fractions altogether and is worth carrying into the exam hall: cross-multiplied, it says middle × last = 5 × first. Row one gives 20 × 7 = 140 = 5 × 28, row three gives 25 × 9 = 225 = 5 × 45, and row two therefore needs ? × 12 = 5 × 84 = 420, so ? = 35. Read as a proportion it is tidier still — first : last is always equal to middle : 5, which is 4 : 1 in row one, 5 : 1 in row three and 7 : 1 in row two. Two disciplines carry the mark here. The first is proving the rule on both visible rows before touching the gap, because a rule that fits one row proves nothing: middle = first − last − 1 reproduces row one exactly, since 28 − 7 − 1 = 20, and then collapses on row three, where 45 − 9 − 1 = 35 rather than 25. The second is checking the columns before committing, and here they carry nothing — 28, 84, 45 down the first column and 7, 12, 9 down the third form no pattern at all, which confirms the relation is horizontal.

• (a)30 — 30 is 5 × 6, so it would require 84 ÷ 12 to come out as 6 — a slip of one in a quotient that is genuinely easy to fumble at speed, because 72 ÷ 12 is 6 and 84 ÷ 12 is 7. It also passes the two cosmetic tests a hurried candidate applies: it is a multiple of 5, like every option here, and it is larger than both visible middle entries, as the rule requires. Only the exact quotient separates it from the answer, and the exact quotient is 7.
• (c)20 — This simply repeats the first row's middle entry, one of two distractors the setter has lifted straight out of the grid so that the number looks familiar rather than derived. It belongs to the quotient 28 ÷ 7 = 4 and to no other row. It also fails an ordering check that takes a second: 84 ÷ 12 = 7 is the largest quotient in the grid, so the missing entry must be larger than both 20 and 25, not equal to either.
• (d)25 — The second number copied out of the grid, this time the third row's middle entry, which follows from 45 ÷ 9 = 5. The second row's quotient is 7, so its middle entry cannot be the same. Tested against the fraction-free form of the rule, 25 × 12 = 300 while 5 × 84 = 420, so the row does not balance — and like option (c) it violates the ordering check, since the missing value must exceed every middle entry already shown.

A number grid of this kind hides one arithmetic relation that every row obeys. The relation may use addition, subtraction, multiplication, division, squares, or a mix of them, and it will normally connect the two outer entries to the middle one. Finding it is not guesswork but a short, ordered search, and the order is dictated by what the numbers look like. Outer pairs that are a multiple and its factor point at division; outer pairs of similar size point at sums and differences; entries such as 16, 25, 36 and 49 point at squares. Whatever rule you propose, the grid itself supplies the proof set: with two complete rows on show, the rule must reproduce both of them exactly before you are entitled to apply it to the row with the gap. That test matters more than the search, because plausible rules that fit a single row are abundant — in this grid, middle = first − last − 1 fits the first row perfectly and is wrong. Two habits round the method out. Always restate the rule in a fraction-free, cross-multiplied form, since middle × last = 5 × first is easier to check under time pressure than middle = 5 × (first ÷ last) and survives rows where the division is not exact. And always glance down the columns as well as across the rows, because a minority of grids hide the relation vertically or diagonally, and thirty seconds spent confirming which direction carries the pattern is never wasted.

Work in four moves. First, look at the shape of the numbers rather than at their values: 28 and 7, 45 and 9, 84 and 12 are each a multiple and its factor, which points straight at division before anything else is tried. Second, compute the quotients across the visible rows — 4 and 5 — and set them beside the middle entries, 20 and 25; the constant multiplier 5 falls out immediately. Third, prove the rule on both complete rows, not one. Fourth, apply it once. The single discriminating computation is the quotient 84 ÷ 12 = 7, because the multiplier 5 was already established from the rows you can see; get the quotient right and the answer is forced. Notice how carefully the option set has been built to defeat shortcuts. Every one of the four choices is a multiple of 5, so the obvious screen — does it look like five times something — eliminates nothing. Two of the three wrong options, 20 and 25, are numbers already printed in the grid, which makes them feel verified when they have merely been copied. That leaves one genuinely useful screen: since 84 ÷ 12 = 7 is a larger quotient than either 4 or 5, the missing entry must be larger than both 20 and 25, which kills options (c) and (d) at a glance and reduces the question to a choice between 30 and 35 — that is, to whether 84 ÷ 12 is 6 or 7.

• Every row divides cleanly and the outer pair is always a multiple and its factor: 28 ÷ 7 = 4, 45 ÷ 9 = 5 and 84 ÷ 12 = 7.
• The middle entry is five times that quotient — 4 × 5 = 20 in row one and 5 × 5 = 25 in row three — so row two gives 7 × 5 = 35, option (b).
• Fraction-free form of the same rule, safer to check under time pressure: middle × last = 5 × first, which reads 20 × 7 = 140 = 5 × 28, 25 × 9 = 225 = 5 × 45, and ? × 12 = 420, so ? = 35.
• As a proportion the rule is first : last = middle : 5 — 4 : 1 in row one, 5 : 1 in row three and 7 : 1 in row two, which is why the missing entry must exceed both 20 and 25.
• A rule that fits one row proves nothing: middle = first − last − 1 reproduces row one exactly, since 28 − 7 − 1 = 20, but fails row three, where 45 − 9 − 1 = 35 instead of 25.
• The columns carry no pattern here — 28, 84, 45 in the first and 7, 12, 9 in the third — which is what confirms that the relation runs horizontally across each row.

• Fixing on a rule that explains one row and never testing it on the second — here first − last − 1 fits row one exactly and is still wrong
• Looking only for sums and differences when the outer numbers are obvious multiples and factors, which is the signal to try division first
• Applying the rule in the wrong direction, multiplying where the row demands division or vice versa
• Choosing a number simply because it already appears in the grid — 20 and 25 are both printed middle entries and both wrong
• Slipping the quotient: 84 ÷ 12 is 7, not the 6 that would produce option (a)

Missing-number grids appear in BPSC's opening reasoning block in almost exactly this shape — three rows of three, one blank, options that are all near neighbours of the answer — and the examiner always leaves at least two complete rows, which is the candidate's proof set. BPSC also keeps the arithmetic deliberately clean, so the mark is decided by finding the rule, not by computing with it. UPSC dropped this format from the General Studies paper when mental ability moved to Prelims Paper-II in 2011, and that paper has been merely qualifying at 33 per cent since 2015; in the older General Studies papers the same skill was set as a series rather than a grid — a missing term inside a run of numbers, or a missing group inside a run of letter triplets — where the rule has to be inferred from consecutive terms rather than from parallel rows, but the discipline of validating the rule on every visible term before using it is identical.

• practice — not a real PYQ

  Find the missing number in the grid: row 1 is 6, 12, 2; row 2 is 8, 24, 3; row 3 is 9, ?, 4.

  • (a)30
  • (b)32
  • (c)36
  • (d)45

  Answer(c) 36 — in each row the middle number is the product of the outer two: 6 × 2 = 12 and 8 × 3 = 24, so 9 × 4 = 36. Both complete rows confirm the rule before it is used.

• practice — not a real PYQ

  Find the missing number in the grid: row 1 is 4, 9, 5; row 2 is 7, 15, 8; row 3 is 6, ?, 3.

  • (a)12
  • (b)9
  • (c)18
  • (d)10

  Answer(b) 9 — the middle number is the sum of the outer two: 4 + 5 = 9 and 7 + 8 = 15, so 6 + 3 = 9. Here the outer numbers are of similar size, which is the signal to try sums before quotients.