Ribbon Cuts and Blinking Boards
During a school reading week, Meera volunteers to prepare the main hall. She has two ribbon rolls, one 156 cm long and the other 234 cm long, and must cut both into equal pieces with no ribbon left over. The pieces must be identical and as long as possible for a display. She also wants to know the total number of pieces. A separate display uses two notice boards that blink at fixed intervals: board A every 18 seconds and board B every 24 seconds. Both are switched on together at 9:00:00. The hall coordinator claims that cutting 13 cm pieces uses both rolls exactly and is therefore the best choice. He also asks how many times the boards blink together from 9:00:00 to 9:12:00 inclusive, because he wants to synchronise a photograph. The boards continue blinking without interruption, and no piece of ribbon may be joined to another.
1. Find the largest identical length into which both ribbons can be cut, and the total number of pieces. Then evaluate the coordinator's claim about 13 cm pieces.
Read the worked answer
Both lengths must be divisible by the piece length, so the largest possible length is HCF(156, 234). Now 156 = 2² × 3 × 13 and 234 = 2 × 3² × 13, so HCF = 2 × 3 × 13 = 78 cm. The 156 cm roll gives 156 ÷ 78 = 2 pieces; the 234 cm roll gives 234 ÷ 78 = 3 pieces; total = 5 pieces. A 13 cm piece does divide both rolls exactly (12 and 18 pieces), so it is valid, but it is not the largest identical length. The coordinator has confused 'no waste' with 'best': the best choice for identical, maximum-length pieces is 78 cm, not 13 cm.
2. How many times do the boards blink together from 9:00:00 to 9:12:00 inclusive? Explain whether the start instant is counted.
Read the worked answer
Board A blinks every 18 s = 2 × 3² and board B every 24 s = 2³ × 3. A simultaneous blink recurs after LCM(18, 24) = 2³ × 3² = 72 s. The interval from 9:00:00 to 9:12:00 is 12 × 60 = 720 s. Since 720 ÷ 72 = 10 intervals, the common instants are at 0, 72, 144, ..., 720 s. Counting both endpoints gives 11 simultaneous blinks. If the wording were 'after 9:00:00', the start would be excluded and the count would be 10; the phrase 'inclusive' means 11.