UPSC Prelims 2009 Q101: Science & Technology – Quantitative Aptitude
Four metal rods of lengths 78 cm, 104 cm, 117 cm and 169 cm are to be cut into parts of
equal length. Each part must be as long as possible. What is the maximum number of pieces that
can be cut?
- (a) 27
- (b) 36
- (c) 43
- (d) 480
Answer: (b) 36
Explanation
Cutting rods into the longest possible equal pieces
Four rods — 78, 104, 117, and 169 cm — must be cut into equal-length pieces, each piece as long as possible.
| Step | Calculation |
|---|---|
| Find the GCD of all four lengths | 78 = 2×3×13; 104 = 2³×13; 117 = 3²×13; 169 = 13² |
| Common factor | 13 is the greatest common divisor shared by all four |
| Pieces per rod | 78/13=6; 104/13=8; 117/13=9; 169/13=13 |
| Total pieces | 6+8+9+13 = 36 |
Why the GCD gives the longest possible equal piece:
| Point | Detail |
|---|---|
| Equal-length requirement | Piece length must divide all four rod lengths exactly |
| 'As long as possible' | Means using the greatest common divisor (13 cm) |
Answer: (b) 36.
Why each option is right or wrong
- (a) Incorrect. 27 does not match the correct total using a 13 cm piece length.
- (b) CORRECT. Using the GCD of 13 cm, the rods yield 6+8+9+13 = 36 pieces total.
- (c) Incorrect. 43 overstates the actual total number of pieces.
- (d) Incorrect. 480 results from multiplying rather than correctly summing the piece counts.
Key point
The longest equal piece length must be the GCD of all four rod lengths (13 cm here); dividing each rod by 13 and summing gives 6+8+9+13 = 36 total pieces.
Remember
Rods of 78, 104, 117, 169 cm, GCD=13 cm: total pieces = 6+8+9+13 = 36.
Common mistakes
- Using a smaller common factor than the true GCD, understating the piece length and overcounting pieces.
- Forgetting to sum the individual piece counts from each rod.
- Making an arithmetic error while finding the GCD of four different numbers.
Exam tip
'Cut into equal parts, each as long as possible' = divide by the GCD of all given lengths. A standard HCF/GCD application.
Sources
- NCERT Class X Mathematics — Real Numbers
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (H.C.F. and L.C.M. of Numbers)