Find the missing number from the given alternatives:
28 | 20 | 7
84 | ? | 12
45 | 25 | 9

Question

UnlockIAS · Accepted Answer

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.

Why the other options are wrong — (a) 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) 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) 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.

Concept: 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.

State PCS 2025 Q9 — Science & Technology PYQ with Answer

State PCS 2025 Q9 — Science & Technology PYQ with Answer