LCM of 9 and 12 → 36 minutes

A: 36 ÷ 9 = 4; B: 36 ÷ 12 = 3. The ratio of buses, 4 : 3, is the reverse of the ratio of intervals, 9 : 12.

GCD of 24 and 36 is 12 → side 12 cm. (24÷12) x (36÷12) = 2 x 3 = 6 squares

LCM of 8 and 12 is 24 → 24, 48, 72, 96

GCD of 42 and 28 is 14 → 14 people; each gets 42÷14 = 3 pencils and 28÷14 = 2 notebooks

Multiples of 4: 4,8,…,36 / of 6: 6,12,…,36 / of 9: 9,18,27,36 → 36

Product = 6 x 72 = 432. Writing them as 6a and 6b with a, b coprime: 36ab = 432 → ab = 12 → (1,12) or (3,4).

6 and 72, or 18 and 24

8 and 12: GCD 4, LCM 24. 4 x 24 = 96 and 8 x 12 = 96 → equal ✓

84 = 2x2x3x7 and 126 = 2x3x3x7; shared are 2x3x7 = 42

LCM of 8 and 20 is every 40 minutes. 7 am to noon is 300 minutes, and 0, 40, 80 … 280 gives 8 times — not at noon itself.

LCM of 18 and 30 is 90 → side 90 cm. (90÷18) x (90÷30) = 5 x 3 = 15 rectangles

Question 3 is the GCD, question 11 the LCM.

Cutting up needs a number that divides into the originals; laying out needs a number the originals divide into. A child who can say that sentence is done worrying about September.

LCM of 12, 18, 24 → 72 teeth

60 x ¹⁄₂ = 30 cm

60 x ²⁄₃ = 40 cm

60 x ³⁄₄ = 45 cm

LCM of 3 and 4 is 12. ²⁄₃ = ⁸⁄₁₂, ³⁄₄ = ⁹⁄₁₂. 8 < 9, so G is shorter (40 cm < 45 cm confirms it)

G is higher

90 x ²⁄₃ = 60 cm

Accept anything with this sense: 'when the denominators differ, one piece is a different size, so the numerators alone tell you nothing. Only after making both twelfths can they be compared.'

60 x ⁴⁄₅ = 48 cm

Over 60 → ⁴⁰⁄₆₀, ⁴⁵⁄₆₀, ⁴⁸⁄₆₀ → E (48) > F (45) > G (40)

Low to high — a longer string sounds lower. Length and pitch running opposite ways is the whole interest of this costume.

¹⁄₃ = ¹⁄₂ x ²⁄₃ → G, one octave up

G, one octave higher. Halving the length always raises the note by an octave.

⁸⁄₁₂ ＋ ³⁄₁₂ = ¹¹⁄₁₂

¹⁵⁄₂₄ − ⁴⁄₂₄ = ¹¹⁄₂₄

⁹⁄₃₀ ＋ ²⁄₃₀ = ¹¹⁄₃₀ (11 is prime, so no cancelling)

⁵⁄₁₂ ＋ ³⁄₁₂ = ⁸⁄₁₂ → ²⁄₃

²¹⁄₂₄ − ⁸⁄₂₄ = ¹³⁄₂₄ m

⁴⁄₁₀ ＋ ³⁄₁₀ = ⁷⁄₁₀ L

⁶⁄₁₂ ＋ ⁴⁄₁₂ ＋ ³⁄₁₂ = ¹³⁄₁₂ = 1¹⁄₁₂

LCM of 7 and 10 is 70. ⁵⁰⁄₇₀ against ⁴⁹⁄₇₀ → ⁵⁄₇ is larger

'Because answers go in simplest form' is acceptable; ' ⁸⁄₁₂ and ²⁄₃ are the same size, and the shorter one is easier to read' is better.

Over 12 → ⁹⁄₁₂ − ²⁄₁₂ + ⁴⁄₁₂ = ¹¹⁄₁₂ L

Over 36 → ¹⁶⁄₃₆, ¹⁸⁄₃₆, ¹⁵⁄₃₆ → ⁵⁄₁₂ < ⁴⁄₉ < ¹⁄₂

¹⁄₃ = ⁴⁄₁₂ and ¹⁄₂ = ⁶⁄₁₂, so there is only ⁵⁄₁₂.

What matters is noticing there is just one. Over 24 there would also be ⁹⁄₂₄ and ¹¹⁄₂₄: how many there are depends on the denominator you allow.

⁸⁄₁₆＋⁴⁄₁₆＋²⁄₁₆＋¹⁄₁₆ = ¹⁵⁄₁₆. Noticing it is ¹⁄₁₆ short of 1 is enough.

2H₂ ＋ O₂ → 2H₂O

O: the left has 2; each H₂O on the right has 1, so two H₂O are needed. That makes 4 H on the right, and each H₂ has 2, so two H₂.

CH₄ ＋ 2O₂ → CO₂ ＋ 2H₂O

C: 1 = 1 ✓　H: 4 on the left → two H₂O　O: 2＋2 = 4 on the right → two O₂

N₂ ＋ 3H₂ → 2NH₃

N: 2 on the left → two NH₃　H: 6 on the right → three H₂

4Fe ＋ 3O₂ → 2Fe₂O₃

O arrives in 2s on the left and 3s on the right. The LCM of 2 and 3 is 6, so three O₂ and two Fe₂O₃. That puts 4 Fe on the right, so 4 Fe on the left.

Sense required: 'the left can only make multiples of 2 and the right only multiples of 3, so they match at a number divisible by both — and the smallest such number is the LCM, 6.'

60 ÷ 4 = 15 gaps. There is one more tree than gaps → 16 trees.

From one end to the other means both ends get a tree, so add 1. Anyone who drew it will not get this wrong.

60 ÷ 5 = 12 gaps → 13 trees

60 ÷ 4 = 15 trees. A loop has no ends, so there is no +1.

A straight road has two ends, and a tree stands at each, so there is one more tree than gap. A loop has no ends, so trees and gaps come out equal.

Saying whether you are counting gaps or counting points is enough.

One side: 48 ÷ 6 = 8, plus 1 is 9. Both sides → 18 trees

Multiples of 3: 33. Of 4: 25. Of 12: 8. So 33 + 25 − 8 = 50 are a multiple of one or the other, and 100 − 50 = 50

21 trees means 20 gaps. 60 ÷ 20 = 3 m apart