Unit 3 · Topic 06 · Playing with Numbers
Two ropes, 24 metres and 36 metres long, need to be cut into equal pieces of the greatest possible length, with nothing left over.
Kabir wanted to know: is this an HCF problem or an LCM problem? He couldn't tell just from the numbers.
Anaya taught him a trick: look at the WORDS, not the numbers, to decide which tool to reach for.
"Cutting the greatest possible equal length, nothing wasted — this is an HCF problem," said Anaya. "Whenever a question asks for the BIGGEST size that fits evenly into several amounts, or the FEWEST groups you can split things into, reach for HCF." She found the HCF of 24 and 36: factors of 24 are 1,2,3,4,6,8,12,24; factors of 36 are 1,2,3,4,6,9,12,18,36; the largest shared factor is 12. "So each rope piece should be 12 metres, giving 2 pieces from the first rope and 3 from the second."
Then Ms. Rao posed a different puzzle: "Two lighthouses flash every 15 seconds and every 20 seconds. If they flash together now, when will they next flash together?" Kabir noticed the word 'together again' — a repeating, cyclic event. "That's an LCM problem," he said, "because we want the next SHARED moment, not a shared SIZE." He found the LCM of 15 and 20: multiples of 15 are 15,30,45,60; multiples of 20 are 20,40,60; the smallest shared one is 60. "They'll flash together again in 60 seconds."
"Here's the pattern to remember," said Ms. Rao. "HCF problems usually involve CUTTING, GROUPING, or SPLITTING into the largest equal pieces or fewest groups. LCM problems usually involve REPEATING events, cycles, or things happening 'together again.'" Anaya tried one more: "A bakery has 18 cupcakes and 30 cookies to arrange into identical trays with nothing left over, using the fewest trays." Fewest trays, nothing left over — HCF again. HCF(18,30)=6, so each tray holds 6 items: 3 trays of cupcakes, 5 trays of cookies.
HCF word problems involve splitting or grouping quantities into the LARGEST equal size, or the FEWEST possible groups, with nothing left over.
LCM word problems involve events that REPEAT on a cycle, asking when they will next happen TOGETHER again.
Clue words for HCF: 'greatest', 'largest equal', 'fewest groups', 'maximum size that divides evenly'.
Clue words for LCM: 'next together', 'repeat', 'same time again', 'smallest number that works for all cycles'.
A florist has 20 roses and 30 lilies and wants the fewest identical bouquets with nothing left over. Is this HCF or LCM? Solve it.
Two clocks chime every 12 minutes and every 18 minutes, starting together. Is this HCF or LCM? Find when they next chime together.
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