12 + 7 + 0 + 15 + 6 = 40, 40 ÷ 5 = 8. Watch for 40 ÷ 4 = 10 — leaving out the person with 0.

Total of tests 1 to 4: 6.5 × 4 = 26. An average of 7 over five needs 7 × 5 = 35. 35 − 26 = 9. Watch for an answer from just averaging 6.5 and 7.

"There need not be." For example two people with 40 and 80: the average is 60 but nobody scored 60. An average is a number made by evening out the total; it need not be any of the original numbers. Either "evening out" or "not the middle" in the reason is enough.

Boys' total 50 × 4 = 200, girls' 60 × 6 = 360, total 560 ÷ 10 people = 56. (50 + 60) ÷ 2 = 55 is wrong because the groups differ in size (more girls, so closer to 60).

Diagram: in a 10 × 10 square, 4 rows and 3 columns → 3 × 4 = 12 cells = 12/100 = 0.12. If he can say why from the cells, that is enough.

Times 10: 12 × 15 = 180; it is 100 times too big, so ÷ 100 = 1.8. Estimate: a bit more than 1 × 1.5 = 1.5.

25 × 8 = 200, ÷ 100 = 2 (2.0). Estimate: a bit under 2.5 × 1 = 2.5, so 20 and 0.2 are out.

A and D. Multiplying by a number below 1 (0.9, 0.05) makes it smaller. B (1.1) makes it bigger; C (1) leaves 6.

0.06 times 100 → 6, 0.7 times 10 → 7. Together 1,000 times. 6 × 7 = 42, ÷ 1,000 = 0.042.

36 × 25 = 900, ÷ 100 = 9 m². Look for the diagram (a rectangle 3.6 by 2.5).

0.45 × 3.2: 45 × 32 = 1,440; 100 times and 10 times is 1,000 times → 1.44 L. Estimate: close to 0.5 × 3 = 1.5.

0.2. 0.4 is below 1, so 0.5 × 0.4 is less than 0.5; 2 is far too big. Area diagram: 4 × 5 = 20 cells = 20/100 = 0.2. Saying the point was put one place out is enough.

822.0 ÷ 30 = 27.4 °C. (total of 30 years ÷ 30)

28.9 − 27.4 = +1.5 °C. That is +1.0 or more, so hot.

26.8 − 27.4 = −0.6 °C. (below normal; above −1.0, so "about normal")

2022: +0.5, 2023: +0.9, 2024: −0.2, 2025: −0.6, 2026: +1.5. Above normal: 2022, 2023, 2026 — 3 years.

27.9 + 28.3 + 27.2 + 26.8 + 28.9 = 139.1, ÷ 5 = 27.82 → 27.8 °C. 27.8 − 27.4 = 0.4 °C above.

28.3 − 27.4 = +0.9 °C. That is short of +1.0, so by the rule it is about normal. "It felt hot" is a feeling, and free; in numbers, this is what it is.

27.4 + 2.1 = 29.5 °C. (run "difference = this year − normal" backwards)

(1) 186 ÷ 31 = 6 mm　(2) 155 ÷ 31 = 5 mm　(3) 6 ÷ 5 = 1.2 times. Putting it per day lets us compare even when the day counts differ.

P: (29.8 + 30.2 + 30.0 + 29.9 + 30.1) ÷ 5 = 150.0 ÷ 5 = 30.0 °C. Q: (24.5 + 35.5 + 30.0 + 33.0 + 27.0) ÷ 5 = 150.0 ÷ 5 = 30.0 °C. The same. Hottest − coolest: P 30.2 − 29.8 = 0.4 °C, Q 35.5 − 24.5 = 11.0 °C.

(1) 822.0 − 273.0 + 281.0 = 830.0, 830.0 ÷ 30 = 27.666… → 27.7 °C. (2) 28.9 − 27.7 = +1.2 °C (against the old normal it was +1.5). The new 10 years were warmer, so the normal itself rises and the same 28.9 °C gives a smaller difference.

Example: it need not be about 27 °C every day. As in question 9, the same average can hide a very different gap between the hottest and coolest days (the spread). An average is an evened-out number, so it cannot say how many hot days there were or how hot the hottest one was.

(1) 155 ÷ 9 = 17.22… → 17.2 mm　(2) 186 ÷ 11 = 16.909… → 16.9 mm　(3) The total is bigger this year (186 > 155), yet per rainy day it is smaller (16.9 < 17.2): there were more rainy days (11), so each rainy day brought less. The conclusion can change depending on whether we compare totals or "per one".

A and C. 0.8 and 0.09 are below 1. B (1.5) gives 6; D (1.01) gives about 8.9, less than 9. Watch for D (1.01): above 1 means smaller, even if only slightly.

Both times 100 → 780 ÷ 65 = 12. (65 × 12 = 780 ✓)

42 ÷ 8 = 5 remainder 2 → 20 ÷ 8 = 2 remainder 4 → 40 ÷ 8 = 5. 5.25.

95 ÷ 6 = 15.83… the second decimal is 3, so round down → 15.8.

89 ÷ 4 = 22 remainder 1. Line the remainder's point up with 8.9: 22 remainder 0.1. Check: 0.4 × 22 + 0.1 = 8.9. The week's trap is "remainder 1": can he notice it from the check giving 9.8?

Diagram: A = 1, B = 1.6 (3.84 kg). A = 3.84 ÷ 1.6 = 38.4 ÷ 16 = 2.4 kg. (2.4 × 1.6 = 3.84 ✓) Watch for 3.84 × 1.6.

(1) 3.6 ÷ 2.4 = 1.5 m²　(2) 2.4 ÷ 3.6 = 0.666… → 0.67 kg. In (1) 1 kg is "1"; in (2) 1 m² is "1". If he can point to the "1" in his diagram, that is enough.

320 ÷ 35 = 9 remainder 5. Line the point up with 3.2: 9 bottles, 0.05 L left. Check: 0.35 × 9 + 0.05 = 3.2.

Last year is 1, this year is 1.25. Last year = 5.5 ÷ 1.25 = 550 ÷ 125 = 4.4 kg. (4.4 × 1.25 = 5.5 ✓)

area ÷ width = 8.64 ÷ 2.4 = 86.4 ÷ 24 = 3.6 cm. (2.4 × 3.6 = 8.64 ✓)

12.5 km on 1 L, and 7.5 km is shorter than 12.5 km → it must need less than 1 L. 7.5 ÷ 12.5 = 75 ÷ 125 = 0.6 L.

(1) 120 × 2.5 = 300 yen　(2) 120 ÷ 2.5 = 48 yen. In (1) the price of 1 m is known → ×. In (2) it is what we want → ÷. If he can say whether the ? sits at 1 or at 2.5, that is enough.

300 ÷ 240 = 1.25 times.

Times: milk 230 ÷ 200 = 1.15, eggs 300 ÷ 240 = 1.25, rice 2,400 ÷ 2,000 = 1.2, bread 154 ÷ 140 = 1.1. (1) In yen: milk +30, eggs +60, rice +400, bread +14 → rice. (2) In times → eggs (1.25 times). That the two answers differ is the point of the day.

228 + 235 + 230 + 224 + 238 = 1,155, ÷ 5 = 231 yen. 231 ÷ 200 = 1.155 → 1.16 times. Exactly 5 rounds up (third place 5 → 1.16).

165.5 × 1.2: 1,655 × 12 = 19,860, 100 times too big so ÷ 100 = 198.6 yen. Estimate: 165.5 plus a fifth is about 198.

(1) 360 ÷ 0.45 = 36,000 ÷ 45 = 800 yen (diagram: 0.45 kg → 360 yen, 1 kg → ?). (2) 800 ÷ 700 = 1.1428… → 1.14 times.

Last year 200 + 240 + 2,000 + 140 = 2,580; this year 230 + 300 + 2,400 + 154 = 3,084. 3,084 ÷ 2,580 = 1.1953… → 1.20 times.

1.02 + 1.03 + 1.01 + 1.04 + 1.05 = 5.15, ÷ 5 = 1.03.

In "times": A is 230 ÷ 200 = 1.15, B is 150 ÷ 120 = 1.25. Not the same — B rose more. The cheaper the starting price, the heavier the same 30 yen.

5 kg bag: 2,400 ÷ 5 = 480 yen; 10 kg bag: 4,500 ÷ 10 = 450 yen (per kg). The 10 kg bag is cheaper (by 30 yen a kg). Watch for comparing the bag prices themselves (4,500 > 2,400); only per 1 kg is the comparison fair.

Check against the notebook. Done when a reason is circled and the redo is right. If the reason was "no diagram", has it been redone with one?

Going from 6 to 9 each adds 3 per child. The whole gap, from 15 over to 3 short, is 15 + 3 = 18. 18 ÷ 3 = 6 children. Sweets: 6 × 6 + 15 = 51. Check: 9 × 6 − 3 = 51 ✓. Does the bar diagram show both "15 left over" and "3 short"?