Answer: A 1 – 2 + 3 + 4

Explanation:

We need to determine which of the five calculations yields the largest result. Let’s compute each option step-by-step, following the standard order of operations (left to right for addition and subtraction, as there are no parentheses or higher-priority operations like multiplication).

• Option A: 1 – 2 + 3 + 4

  Start with 1 – 2 = -1. Then, -1 + 3 = 2. Finally, 2 + 4 = 6. So, the result is 6.

• Option B: 1 + 2 – 3 + 4

  Begin with 1 + 2 = 3. Then, 3 – 3 = 0. Lastly, 0 + 4 = 4. So, the result is 4.

• Option C: 1 + 2 + 3 – 4

  First, 1 + 2 = 3. Then, 3 + 3 = 6. Finally, 6 – 4 = 2. So, the result is 2.

• Option D: 1 + 2 – 3 – 4

  Start with 1 + 2 = 3. Then, 3 – 3 = 0. Finally, 0 – 4 = -4. So, the result is -4.

• Option E: 1 – 2 – 3 + 4

  Begin with 1 – 2 = -1. Then, -1 – 3 = -4. Lastly, -4 + 4 = 0. So, the result is 0.

Now, let’s list the results:

• A: 1 – 2 + 3 + 4 = 6
• B: 1 + 2 – 3 + 4 = 4
• C: 1 + 2 + 3 – 4 = 2
• D: 1 + 2 – 3 – 4 = -4
• E: 1 – 2 – 3 + 4 = 0

Comparing these values: 6, 4, 2, -4, and 0. Clearly, 6 > 4 > 2 > 0 > -4. Thus, 6 is the largest.

Official Solution: The UKMT solutions leaflet states: “The values of the expressions are: A 6, B 4, C 2, D -4, E 0.” This matches our calculations exactly, confirming that Option A gives the largest answer, 6.

Alternative Approach: Notice that each expression uses the numbers 1, 2, 3, and 4 once, with varying signs. The sum of the numbers without signs is 1 + 2 + 3 + 4 = 10. The final result depends on how many numbers are subtracted (preceded by a minus sign):

• A: Only 2 is subtracted → 10 – 2 × 2 = 10 – 4 = 6 (subtract 2 twice because one negative sign affects one number).
• B: Only 3 is subtracted → 10 – 2 × 3 = 10 – 6 = 4.
• C: Only 4 is subtracted → 10 – 2 × 4 = 10 – 8 = 2.
• D: 3 and 4 are subtracted → 10 – 2 × (3 + 4) = 10 – 14 = -4.
• E: 2 and 3 are subtracted → 10 – 2 × (2 + 3) = 10 – 10 = 0.

The fewer numbers subtracted (or the smaller the subtracted values), the larger the result. Option A subtracts the smallest amount (just 2), yielding the highest value, 6.

Verification: Let’s recompute A: 1 – 2 = -1, -1 + 3 = 2, 2 + 4 = 6. Correct. For B: 1 + 2 = 3, 3 – 3 = 0, 0 + 4 = 4. Correct. The pattern holds across all options.

Thus, among the options, A: 1 – 2 + 3 + 4 gives the largest answer, which is 6.

Answer: E 98

Explanation:

We need to calculate the number of minutes from 22:22 to midnight (00:00). Midnight marks the start of the next day, so we compute the time difference.

• From 22:22 to 23:00: There are 60 minutes in an hour, so from 22:00 to 23:00 is 60 minutes. Since it’s 22:22, we subtract 22 minutes: 60 – 22 = 38 minutes.
• From 23:00 to 00:00: This is one full hour, or 60 minutes.
• Total: 38 + 60 = 98 minutes.

Official Solution: “At 22:22, there are 60 – 22 = 38 minutes to 23:00. There are then a further 60 minutes to midnight. So the number of minutes which remain until midnight is 38 + 60 = 98.”

Alternative Method: Convert times to minutes past midnight: 22:22 = (22 × 60) + 22 = 1320 + 22 = 1342 minutes. Midnight = 24 × 60 = 1440 minutes. Difference: 1440 – 1342 = 98 minutes.

Thus, the answer is E: 98.

Answer: D 823

Explanation:

First, compute the denominator: 1 + 2 + 3 + 4 + 5 = 15. Now, evaluate the expression: \(\frac{12345}{15}\).

Perform the division: 12345 ÷ 15 = 823 (since 15 × 823 = 12345; check: 15 × 800 = 12000, 15 × 23 = 345, 12000 + 345 = 12345).

Official Solution: “The value of \(\frac{12345}{1 + 2 + 3 + 4 + 5} = \frac{12345}{15} = \frac{2469}{3} = 823\).” (Note: \(\frac{12345}{15} = 823\) directly, and the intermediate step is an alternative factorization.)

Verification: 12345 is divisible by 15 (ends in 5, sum of digits 1+2+3+4+5=15 divisible by 3). Result 823 is an integer, and 823 × 15 = 12345 confirms correctness.

Thus, the answer is D: 823.

Answer: A 3

Explanation:

Since the pyramid diagram isn’t provided, we rely on the solution’s steps. Each number is the sum of the two below it. The official solution gives: \(p = 105 – 47 = 58\), \(q = 58 – 31 = 27\), \(r = 47 – 27 = 20\), \(s = 20 – 13 = 7\), \(t = 13 – 9 = 4\), \(x = 7 – 4 = 3\).

Working backwards: Bottom row has 9 and 13 (since 13 – 9 = 4). Next row includes 13 and \(s = 20 – 13 = 7\), so \(s = 7\) and \(t = 4\), and \(x = 7 – 4 = 3\).

Official Solution: “The calculations required to find the value of \(x\) are: \(p = 105 – 47 = 58\); \(q = p – 31 = 58 – 31 = 27\); \(r = 47 – q = 47 – 27 = 20\); \(s = r – 13 = 20 – 13 = 7\); \(t = 13 – 9 = 4\); \(x = s – t = 7 – 4 = 3\).”

Thus, the answer is A: 3.

Answer: B 36

Explanation:

Let the number be \(x\). Then, \(\frac{1}{3}x – \frac{1}{4}x = 3\). Find a common denominator (12): \(\frac{4x}{12} – \frac{3x}{12} = \frac{x}{12} = 3\). Thus, \(x = 3 \times 12 = 36\).

Official Solution: “Let the required number be \(x\). Then \(\frac{1}{3}x – \frac{1}{4}x = 3\). Multiplying both sides by 12 gives \(4x – 3x = 36\). So \(x = 36\).”

Verification: Check: \(\frac{1}{3} \times 36 = 12\), \(\frac{1}{4} \times 36 = 9\), 12 – 9 = 3. Correct.

Thus, the answer is B: 36.

Answer: B 50

Explanation:

Without the diagram, use the solution: Exterior angles sum to 360°. Given angles 110° and 120°, \(y = 360 – (110 + 120) = 130\). Then, \(x = 180 – y = 180 – 130 = 50\).

Official Solution: “The sum of the exterior angles of any polygon is 360°. So \(y = 360 – (110 + 120) = 130\). The sum of the angles on a straight line is 180°, so \(x = 180 – y = 180 – 130 = 50\).”

Thus, the answer is B: 50.

Answer: A 3

Explanation:

Compute \(123456789 \times 8 = 987654312\). Compare with 987654321: digits 1 and 2 are swapped (last two positions). Sum: 1 + 2 = 3.

Official Solution: “The units digit of \(123456789 \times 8\) is 2, since \(9 \times 8 = 72\). So, the two digits which are in a different order are 1 and 2, whose sum is 3. Check: \(123456789 \times 8 = 987654312\).”

Thus, the answer is A: 3.

Answer: C 7

Explanation:

Check remainders: 3 (2:1, 3:0), 5 (2:1, 3:2), 7 (2:1, 3:1), 9 (2:1, 3:0), 11 (2:1, 3:2). Only 7 has remainder 1 for both.

Official Solution: “All options are odd (remainder 1 when divided by 2). Check by 3: 7 gives remainder 1.”

Thus, the answer is C: 7.

Answer: D 2000

Explanation:

Time: 63 – 50 = 13 years. Miles per year: 25048 ÷ 13 ≈ 1926. Closest option is 2000.

Official Solution: “Approximately 25000 miles in 13 years: 25000 ÷ 13 ≈ 1923, roughly 2000.”

Thus, the answer is D: 2000.

Answer: E 25

Explanation:

Maximize by using × between larger numbers: \(1 + 2 × 3 × 4 = 1 + 24 = 25\). (Order: × before +.)

Official Solution: “Maximize with \(1 + 2 × 3 × 4 = 25\), not \(1 × 2 × 3 × 4 = 24\).”

Thus, the answer is E: 25.

Answer: D 19

Explanation:

Primes: 2, 3, 5, 7, 11, etc. Smallest three odd primes: 3 + 5 + 7 = 15 (not prime). Next: 3 + 5 + 11 = 19 (prime).

Official Solution: “Smallest sum of three odd primes is 3 + 5 + 11 = 19.”

Thus, the answer is D: 19.

Answer: B 3

Explanation:

Let weight = \(w\). Then, \(w = 2 + \frac{1}{3}w\). So, \(w – \frac{1}{3}w = 2\), \(\frac{2}{3}w = 2\), \(w = 3\).

Official Solution: “2 kg is two-thirds of the weight, so one-third is 1 kg, total 3 kg.”

Thus, the answer is B: 3.

Answer: A 10

Explanation:

Without the figure, use solution: Sums are (1+2), (1+5), (1+8), (2+4), (2+7), (3+6), (4+5), (4+8), (5+7), (7+8). Multiples of 3: 3, 6, 9, 6, 9, 9, 9, 12, 12, 15. Total: 10.

Official Solution: “Labels: (1+2), (1+5), (1+8), (2+4), (2+7), (3+6), (4+5), (4+8), (5+7), (7+8). 10 are multiples of 3.”

Thus, the answer is A: 10.

Answer: E 4

Explanation:

Prime digits: 2, 3, 5, 7. Bars (7-segment): 2:5, 3:5, 5:5, 7:3. All have prime bars (3, 5). Total: 4.

Official Solution: “Primes and bars: 2→5, 3→5, 5→5, 7→3. All four are prime.”

Thus, the answer is E: 4.

Answer: C \(45 \times 56\)

Explanation:

Need LCM of 1-10: \(2^3 \times 3^2 \times 5 \times 7\). Check: \(45 \times 56 = 5 \times 9 \times 8 \times 7 = 2^3 \times 3^2 \times 5 \times 7\).

Official Solution: “\(45 \times 56 = 2^3 \times 3^2 \times 5 \times 7\), divisible by 1-10.”

Thus, the answer is C: \(45 \times 56\).

Answer: D 150

Explanation:

Triangle: 60° each. Square: 90° each. Using solution’s logic: \(p = 120 – x\), \(q = 120 – y\), \(p + q + 90 = 180\), so \(x + y = 150\).

Official Solution: “From geometry, \(330 – (x + y) = 180\), so \(x + y = 150\).”

Thus, the answer is D: 150.

Answer: B 2

Explanation:

If H tells truth, C lies, D tells truth, S lies. If H lies, C tells truth, D lies, S tells truth. Always 2 truths.

Official Solution: “In both consistent cases, exactly two Knaves tell the truth.”

Thus, the answer is B: 2.

Answer: B \(\frac{1}{7}\)

Explanation:

\(\frac{5274}{36918} = 2 \times \frac{2637}{18459} = 2 \times \frac{1}{7} = \frac{2}{14} = \frac{1}{7}\).

Official Solution: “\(\frac{5274}{36918} = \frac{2637}{18459} = \frac{1}{7}\).”

Thus, the answer is B: \(\frac{1}{7}\).

Answer: D 216

Explanation:

Cubes: 1, 8, 27, 64, 125, 216. Check: \(27 + 64 + 125 = 216\). Smaller cubes fail.

Official Solution: “216 = 27 + 64 + 125, smallest such cube.”

Thus, the answer is D: 216.

Answer: C \(72 \text{m}^2\)

Explanation:

Base: 4×4 = 16 m². Top: 16 m². Vertical: \(4 \times (1 + 2 + 3 + 4) = 40\). Total: 16 + 16 + 40 = 72 m².

Official Solution: “Base 16 m², top 16 m², vertical 40 m², total 72 m².”

Thus, the answer is C: \(72 \text{m}^2\).

Answer: C 730 mm

Explanation:

Wall: 4800 mm. Picture width: 420 mm. Distance between outer centers: 4800 – 420 = 4380 mm. Three gaps, so each gap = 4380 ÷ 3 = 1460 mm. Center of wall: 2400 mm. Distance from center to inner picture centers: 1460 ÷ 2 = 730 mm.

Official Solution: “Distance between outer centers is 4380 mm. So \(2x + 2x + 2x = 4380\), \(x = 730\).”

Thus, the answer is C: 730 mm.

Answer: E \(\frac{1}{2}\)

Explanation:

Hexagon: 6 triangles. Solution shows 3 shaded, 3 unshaded. Fraction: \(\frac{3}{6} = \frac{1}{2}\).

Official Solution: “Hexagon divided into 6 triangles, 3 shaded, fraction \(\frac{1}{2}\).”

Thus, the answer is E: \(\frac{1}{2}\).

Answer: D \(4 \text{cm}^2\)

Explanation:

Total square area: 1 + 4 + 9 + 16 = 30 cm². Rectangle: 5 × 5.5 = 27.5 cm². Unshaded: 1.5 cm². Covered: 26 cm². Overlap: 30 – 26 = 4 cm².

Official Solution: “Sum of squares 30 cm², rectangle occupied 26 cm², overlap 4 cm².”

Thus, the answer is D: \(4 \text{cm}^2\).