# Arithmetic: GATE CS question with solution

From the GATE Daily set of 27 October 2026, question 1 of 10 (https://gatedaily.in/q/2026-10-27-tea-coffee-ratio-after-change/). Difficulty: standard.

### Q.1 · General Aptitude · Arithmetic · MCQ, 1 mark (choose one option)

In a hostel, every resident prefers exactly one of tea and coffee, and the ratio of tea drinkers to coffee drinkers is $5 : 3$. Then 12 new residents, all coffee drinkers, move in and 5 tea drinkers move out. The ratio of tea drinkers to coffee drinkers is now $5 : 4$.

How many tea drinkers were there in the hostel originally?

(A) 60
(B) 75
(C) 80
(D) 128

**Answer:** (C) 80

**Solution.** **Method.** Give the original ratio a unit: tea $= 5k$, coffee $= 3k$. After the change, tea $= 5k - 5$ and coffee $= 3k + 12$, and these are in the ratio $5 : 4$:

$$\frac{5k-5}{3k+12} = \frac{5}{4} \;\Rightarrow\; 20k - 20 = 15k + 60 \;\Rightarrow\; k = 16.$$

So there were $5 \times 16 = 80$ tea drinkers and $48$ coffee drinkers. Check: after the change, $75 : 60 = 5 : 4$.

**The trap.** Because the tea side of the ratio is 5 both before and after, it is tempting to say "coffee went from 3 parts to 4 parts, so one part is 12" and answer $60$. That shortcut is valid only when the tea count is unchanged. Here both groups changed, so a "part" before and a "part" after are different sizes; you must write one equation in $k$.

**What was tested.** Turning a ratio into counts with a single unknown and re-imposing the second ratio on the changed counts, and then answering the exact quantity asked (original tea, not new tea or the total).

**Remember:** When a ratio changes because both groups change, a 'part' before and a 'part' after are different sizes. Write the original counts as multiples of one unknown, apply the changes, and impose the new ratio as one equation.

**Why each option:**
- (A) Treats the change as one ratio part equal to 12, which only works if the tea count stayed fixed; here 5 tea drinkers also left.
- (B) This is the number of tea drinkers after the change (80 - 5), not the original number asked for.
- (C) With 5k tea and 3k coffee, (5k - 5)/(3k + 12) = 5/4 gives k = 16, so 5k = 80.
- (D) This is the original total of residents (80 + 48), not the number of tea drinkers.