Scholarship Paper 3 (ACER Level 2/3 form)

84 ÷ 7 = 12, so Ben gets 4 × 12 = $48. ($36 is Ava's share; $42 splits the money equally; $63 takes three quarters of $84, reading 3 : 4 as the fraction 3/4.)

2 916: 16 is divisible by 4 and 2 + 9 + 1 + 6 = 18 is divisible by 9, so it passes both (2 916 = 36 × 81). 3 726 and 4 518 pass the 9 test but end in 26 and 18, which 4 does not divide; 5 732 passes the 4 test but its digit sum is 17.

All five add to 5 × 72 = 360. The four shown add to 283, so the fifth is 360 − 283 = 77. (70.75 is the mean of the four shown; 72 assumes the missing score equals the mean.)

3.4² − 1.6² = 11.56 − 2.56 = 9, and √9 = 3 m. (1.8 m just subtracts the lengths, 5 m adds them, and 9 m forgets the square root.)

4 + 3 × 39 = 3 × 40 + 1 = 121. (160 gives every square its own 4 sticks, counting each shared stick twice; 124 adds 3 for all 40 squares on top of the first 4.)

Per kilogram: $4.80, $4.40, $4.20 and $4.30. The cheapest is 1 kg for $4.20. (The biggest pack is not always the best buy: 2 kg works out at $4.30 a kilogram.)

x − 2 = 13 − 2x gives 3x = 15, so x = 5 and y = 3: (5, 3). (2, 0) lies on the first line only; (3, 7) and (4, 5) lie on the second line only.

The two-digit multiples of 7 with digit sum 10 are 28 and 91. Only 28 is even. (46 and 64 have digit sum 10 and are even but are not multiples of 7; 91 is odd.)

75% of $240 is $180, and 90% of $180 is $162. ($156 takes 35% off in one go; $180 stops after the first discount; $216 applies only the 10%.)

The diagonal is √(24² + 10²) = √676 = 26 cm, which is the diameter, so the radius is 13 cm. (26 cm is the diameter; 17 cm averages the sides; 34 cm adds them.)

The pairs are 1 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12 and 8 × 9: 12 factors. (Missing a pair such as 8 × 9 gives 10.)

Pia and Quinn must share Japanese and Spanish between them, so Rosa (not French, not Japanese) has German and Sam has French. Rosa. (Pia was ruled out directly; Sam is excluded by his clue.)

The fixed fee is $4 and the line rises $10 over 5 km, so $2 per km. 4 + 2 × 12 = $28. ($24 forgets the fixed fee; $14 is the fare for 5 km.)

Height = √(17² − 8²) = √225 = 15 cm, so the area is ½ × 16 × 15 = 120 cm². (136 cm² uses the sloping side 17 as the height; 240 cm² forgets the half.)

2 parts = $40, so 1 part = $20 and the total of 8 parts is $160. ($200 is Cara's share if one part were $40; $320 takes the $40 itself as one part.)

Running totals 3, 8, 14 put the 10th and 11th scores both at 3, so the median is 3 (1 ✓). The total is 3 + 10 + 18 + 16 + 10 = 57, and 57 ÷ 20 = 2.85, which is less than 3 (2 ✗). Score 3 is the most frequent (3 ✓). 1 and 3.

80% of the price is $96, so 10% is $12 and the full price is $120. ($115.20 adds 20% of $96; $76.80 takes a second 20% off.)

Row 2: 2 + x, x + 3, 8. Row 3: 2x + 5, x + 11. Top: 3x + 16 = 40, so x = 8. (30 adds the bottom row once; 13.5 weights the middle bricks by 2 instead of 3; 24 forgets to divide by 3.)

The scale factor from triangle ABC to triangle EDC is 9 ÷ 6 = 1.5, so DE = 1.5 × 8 = 12. (11 adds the difference 9 − 6 = 3 instead of multiplying; 13.5 applies the factor to 9; 16 doubles AB.)

Only perfect squares have an odd number of factors. The squares up to 200 are 1², 2², …, 14² = 196: 14. (13 leaves out 1, which has one factor; 10 stops at 100; 12 stops at 144.)

20 + 0.1m = 0.3m gives 0.2m = 20, so m = 100 minutes (both cost $30). (200 divides $20 by 10c; 50 divides it by 40c.)

5 friends × 4 cards each = 20. (10 counts each pair of friends once, as if one card did for both; 25 has each friend post a card to themselves too.)

The total rises from 112 to 160, so the two numbers add to 48 and their mean is 24. (48 is their sum, not their mean; 15 averages 14 and 16.)

If only Ari is truthful, Bea lies, so Cal tells the truth — two truth-tellers. If only Cal is truthful, Ari lies, so Bea tells the truth — again two. If only Bea is truthful, Cal lies (correct, since Bea is truthful) and Ari lies (correct) — it works, so it can be decided.

Base diagonal² = 12² + 4² = 160; rod² = 160 + 3² = 169, so the rod is 13 cm. (12 cm is only the longest edge; 19 cm adds the three edges.)

Ana : Bo : Cy = 8 : 12 : 15, which is 35 parts of $6. Cy gets 15 × 6 = $90. ($48 and $72 are Ana's and Bo's shares; $105 joins the ratios as 2 : 3 : 5 without matching Bo.)

The gap starts at $1000 and grows by 120 − 100 = $20 a year. It must grow by $500: 500 ÷ 20 = 25 years. (50 needs the gap to grow by the full $1000, ignoring the head start.)

The gradient is 6 ÷ 2 = 3: y rises 3 for each step of 1 in x. From y = 10 to y = 19 is a rise of 9, so x goes up 3, from 3 to 6. (9 takes the rise of 9 as the step in x.)

With n balls at first there are n/3 red and 2n/3 blue. Afterwards red = blue: n/3 + 6 = 2n/3, so n = 18 — that was before. Now there are 18 + 6 = 24 (12 red, 12 blue). (18 is the number before the red balls were added; 12 is the number of blue.)

20 multiples of 3 and 12 of 5; 4 of them (15, 30, 45, 60) are multiples of both. 20 + 12 − 2 × 4 = 24. (28 is "3 or 5", which still includes the 4; 32 counts them twice.)

11(a + b) = 132 gives a + b = 12; 9(a − b) = 36 gives a − b = 4. So a = 8, b = 4: 84. (75 and 66 also add to 132 with their reversals, but the differences are 18 and 0; 48 is the reversal.)

Multiples of 5 up to 40: eight of them, each giving one 5, and 25 = 5 × 5 gives an extra one. That is 9 fives, so 9 zeros. (8 misses the second 5 in 25; 4 counts only 10, 20, 30 and 40.)

Unfolded, P and Q are opposite corners of an 8 by 4 rectangle: √(8² + 4²) = √80 = 4√5 cm. (4√3 cm goes through the inside of the cube; (4 + 4√2) cm runs along an edge and then a face diagonal, which is longer; 12 cm follows three edges.)

The total is 35. 10 must appear twice, so d = e = 10, leaving a + b = 7 with a < b < 8 (a = b, or b = 8, would make a second mode). The pairs are 1 and 6, 2 and 5, 3 and 4, so the smallest can be at most 3. (4 would need b = 3, smaller than a; 3.5 and 3.5 are not whole and would repeat.)

The shorter sides add to 30 − 13 = 17 and their squares add to 169: they are 5 and 12. Area = ½ × 5 × 12 = 30 cm². (60 cm² forgets the half.)