General Education 2: Numerical Reasoning, Logical Reasoning and Data Interpretation

Percentage = compared amount ÷ base amount = 52 ÷ 80 = 0.65 = 65%. 52% just turns the head count into a percentage; 60% and 70% are miscalculations.

This year: 400 × 1.1 = 440; next year: 440 × 0.9 = 396. The second 10% is taken from 440, so it does not return to 400. 404 underestimates the decrease, and 360 takes 10% off 400 directly.

A 30% discount means paying 0.7 of the list price, so list price = 2,100 ÷ 0.7 = 3,000 yen. 2,730 yen comes from 2,100 × 1.3, using the wrong base; 2,800 and 3,150 yen are miscalculations.

List price = 1,000 × 1.4 = 1,400 yen; selling price = 1,400 × 0.8 = 1,120 yen; profit = 120 yen. 200 yen wrongly subtracts 40% − 20%; 400 yen forgets the discount; 80 yen is a miscalculation.

Let the amounts be 5x and 3x. Then 5x − 600 = 3x + 600, so x = 600 and the older sister had 3,000 yen (the younger had 1,800 yen). 1,800 yen is the younger sister’s amount; 2,400 and 3,600 yen come from wrong equations.

The salt is 300 × 0.08 = 24 g and does not change when water is added. The total is 400 g, so 24 ÷ 400 = 6%. 5%, 7% and 4% mix up the amount of salt or the total weight.

Salt = 200 × 0.1 + 100 × 0.04 = 24 g; total = 300 g; 24 ÷ 300 = 8%. 7% averages 10 and 4 without weighting by amount; 6% and 9% are miscalculations.

Cyclists = 600 × 0.4 = 240; first-years among them = 240 × 0.25 = 60. 150 uses the whole school as the base, 240 is all cyclists, and 90 is a miscalculation.

Multiply the factors: 1.2 × 0.8 = 0.96, i.e. 96%. The 20% cut is taken from the raised price, so it does not return to 100%. 104% gets the direction wrong, and 80% forgets the increase.

Make B the same using 12, the least common multiple of 3 and 4: A:B = 8:12 and B:C = 12:15, so A:B:C = 8:12:15. 2:3:5 does not match B; 2:7:5 adds 3 + 4; 6:9:10 does not fit B:C.

Base = compared amount ÷ rate = 42 ÷ 0.15 = 280. 6.3 multiplies 42 by 0.15 by mistake; 63 and 420 also mishandle the rate.

Let x g be added: 9.6 + x = 0.2 × (240 + x), so 0.8x = 38.4 and x = 48 g. 38.4 g forgets that the total weight also increases (240 × 0.2 − 9.6); 40 g and 60 g are miscalculations.

Purchase cost = 80 × 200 = 16,000 yen. With n sold, 120n − 16,000 ≥ 4,000 gives n ≥ 166.6…, so at least 167. 100 comes from 4,000 ÷ (120 − 80), ignoring the cost of unsold items; 134 and 160 are not enough (160 gives only 3,200 yen).

25 minutes = 25/60 hour, so 12 × 25/60 = 5 km. 3, 4.8 and 6 km come from converting minutes to hours incorrectly.

One hour is 3,600 seconds, so 5 × 3,600 = 18,000 m = 18 km. 0.3 km/h multiplies by 60 only once and mistakes 300 m (0.3 km) per minute for a speed per hour (300 m per minute = 18 km/h); 3 and 30 km/h are also conversion errors.

Take the one-way distance as 12 km: 2 hours there and 3 hours back, so 24 km ÷ 5 h = 4.8 km/h (the answer does not depend on the distance). 5 km/h averages 6 and 4, ignoring that the slower trip takes longer.

Walking towards each other, the gap shrinks by 70 + 50 = 120 m per minute: 2,400 ÷ 120 = 20 minutes. 120 minutes divides by the difference in speed (20 m); 12 and 30 minutes are miscalculations.

When the older brother starts, the younger is 60 × 10 = 600 m ahead. The gap closes by 180 − 60 = 120 m per minute, so 600 ÷ 120 = 5 minutes. 2.5 minutes divides by the sum of the speeds (240 m) instead of the difference; 10 minutes confuses it with the younger brother’s head start; 15 minutes is a miscalculation.

The train travels bridge length + train length = 440 + 160 = 600 m, so 600 ÷ 20 = 30 seconds. 22 seconds forgets the train’s length (440 ÷ 20); 14 seconds uses the difference; 8 seconds uses only the train’s length.

Per day A does 1/12 and B 1/6; together 1/12 + 1/6 = 1/4, so 4 days. 9 days averages 12 and 6; 3 and 6 days add the work rates incorrectly.

Per hour A fills 1/6 and B 1/3, so together 1/2 in the first hour. The remaining 1/2 takes A 1/2 ÷ 1/6 = 3 hours. With the tank as 6 units: A 1/h, B 2/h, 3 units in the first hour, 3 units left for A = 3 hours. The other choices misjudge the remainder or the rate.

If it takes t days, B works t days and A works t − 3 days: t/12 + (t − 3)/6 = 1. Multiplying by 12: t + 2t − 6 = 12, so t = 6. 4 days ignores the break; 5 and 7 days come from wrong equations.

Use total = average × number. Six tests need 75 × 6 = 450 points; the first five total 72 × 5 = 360; 450 − 360 = 90. 78 just adds 3 to 75; 75 and 87 also ignore totals.

Total = 20 × 60 + 30 × 70 = 3,300 points for 50 students, so 66. 65 averages 60 and 70 without regard to class size; 64 and 67 are miscalculations.

With x candies and 20 − x gums: 50x + 80(20 − x) = 1,360, so 30x = 240 and x = 8 (12 gums). Check: 400 + 960 = 1,360 yen. 12 is the number of gums; 10 and 6 do not fit.

In x years: 40 + x = 3(10 + x), so 2x = 10 and x = 5 (45 and 15). Remember both ages increase. In 3 years they are 43 and 13 (more than three times); in 10 years they are 50 and 20 (2.5 times); in 20 years 60 and 30 (twice).

The contrapositive, “not cancelled → no rain”, is always true. “Cancelled → rain” is the converse and “no rain → not cancelled” is the inverse; the trip could be cancelled for another reason, so neither is guaranteed. “No rain → cancelled” cannot be concluded at all.

(a) and (b) give reading → Japanese → essays. The contrapositive of (c) is essays → not arithmetic, so reading → not arithmetic. “Essays → reading” is a converse and “not reading → arithmetic” has the form of an inverse; neither is certain. “Japanese → arithmetic” contradicts the conclusion. Checking every case confirms that only “People who like reading do not like arithmetic” is certain.

Every multiple of 4 is a multiple of 2, so the statement is true. The converse, “if even, then a multiple of 4”, is false: 6 is even but not a multiple of 4. Such an example is a counterexample. 8 is even and a multiple of 4, so it is not a counterexample.

Test each case. If A did it, B and C are telling the truth: two people, which fails. If B did it, A and C are truthful: two again. If C did it, only B is truthful: exactly one. So C broke the window, and the answer is unique.

B teaches science, so A (not Japanese, not arithmetic) teaches music. C is not Japanese and science and music are taken, so C teaches arithmetic. Japanese is left for D (consistent with D not teaching music). Checking all 24 assignments leaves only this one.

E is not first and finishes before D, so D is not first either; B is not first. So A is first and C, right after A, is second. B, D and E take 3rd–5th; D is after E and B is not last, so D is fifth. B and E can be 3rd/4th in either order (A-C-B-E-D or A-C-E-B-D). So only “D was fifth” is certain; B cannot be second.

Those with at least one pet: 40 − 5 = 35. Dog + cat − both = 35, so 25 + 18 − both = 35 and both = 8. 3 forgets the 5 with neither (25 + 18 − 40); 13 and 7 are miscalculations.

The overlap is smallest when nobody owns neither: 35 + 40 − 50 = 25. 5 and 15 subtract the wrong numbers; 35 is the maximum possible overlap (all bicycle owners).

Treat A and B as one block: 4 items can be arranged in 4 × 3 × 2 × 1 = 24 ways, and A and B can swap within the block, so 24 × 2 = 48. 24 forgets the swap; 120 is all arrangements of five people.

6 choices for the first × 5 for the second = 30, but “A and B” is the same as “B and A”, so divide by 2: 15. 30 counts ordered pairs; 12 and 36 are reasoning errors.

There are 6 × 6 = 36 equally likely outcomes. A total of 7 comes from (1,6), (2,5), (3,4), (4,3), (5,2), (6,1): 6 outcomes, so 6/36 = 1/6. 1/11 wrongly assumes the eleven totals 2–12 are equally likely; 1/12 and 7/36 are counting errors.

April has 30 days and May 31, so 1 June is 61 days after 1 April. 61 ÷ 7 = 8 remainder 5, so move 5 days on from Wednesday: Monday. Wednesday ignores the remainder; Sunday and Tuesday miscount the days (April and May have the same lengths every year).

Books per pupil = books lent ÷ pupils: East 30, West 35, South 25, North 32, Central 28. West is highest. South lends the most books in total but also has the most pupils, so it is lowest per pupil.

North: 11,520 ÷ 360 = 32; Central: 14,280 ÷ 510 = 28; the difference is 4, so this is true. South is lowest at 25 per pupil. East + Central = 26,880, short of twice South (32,500). The simple average is (30 + 35 + 25 + 32 + 28) ÷ 5 = exactly 30, not more.

Total books = 64,450 and total pupils = 2,220, so 64,450 ÷ 2,220 ≈ 29.03. About 30.0 is the simple average of the five schools and ignores the weight of South (25 per pupil), which has the most pupils.

7,650 ÷ 23,000 ≈ 0.333, about 33% (one third of 23,000 is about 7,667). About 25% is the 2015 share (6,000 ÷ 24,000); about 28% confuses it with the growth rate (27.5%).

0–14: 800 ÷ 3,200 = a 25% fall; 15–64: 1,850 ÷ 14,800 = a 12.5% fall, exactly half. The 65+ group grew by 1,650, more than the 800 fall in the 0–14 group. The total fell by 1,000 ÷ 24,000 ≈ 4.2%, not over 5%. The 15–64 share was about 61.7% in 2015 but about 56.3% in 2025.

The share was 3,200 ÷ 24,000 ≈ 13.3% in 2015 and 2,400 ÷ 23,000 ≈ 10.4% in 2025, a difference of about 2.9 points. “25” is the rate of decrease of the group itself (25%), not a difference in shares. 0.8 and 8.0 confuse the head count (800) with the share.

Growth from the previous month: May 25% (3 ÷ 12), June 20% (3 ÷ 15), August about 42.9% (6 ÷ 14), September 20% (4 ÷ 20); July fell. August is the largest. September looks large in absolute terms (4 thousand), but its base (20 thousand) is also large.

The total is 12 + 15 + 18 + 14 + 20 + 24 = 103 thousand, an average of about 17.2 thousand. June (18), August (20) and September (24) are above it: 3 months. May (15) is below.

Only July (18 → 14) fell from the previous month, so this is true. The total is 103 thousand, over 100 thousand. September (24) is twice April (12). The June–July decrease is 4 ÷ 18 ≈ 22.2%, more than 20%.

In actual numbers: 2015: 500 × 0.30 = 150; 2025: 400 × 0.25 = 100, a fall of 50. “5” confuses the 5-point difference in share with a head count. Because the totals differ, shares alone cannot compare numbers of students.

Bus: 10% → 20%, so the share doubled; students: 500 × 0.1 = 50 → 400 × 0.2 = 80, up 30. This is true. The number grew 1.6 times, not 2. Walkers fell from 300 to 220. Cyclists fell from 150 to 100, about 33%. A point change is not the same as a rate of change in numbers.

Within one library, the index can be compared: B rose 138 ÷ 115 = 1.2 times, so this is true. A fell 120 ÷ 130 ≈ 0.923, about 7.7% (a 10-point drop is not a 10% drop). Since the two libraries’ 2020 visitor numbers (the bases) are unknown, their visitor numbers and increases cannot be compared.