IKMC 2027 - Grades 7-8 - Self-study SET 10 - Thematic Batch 3

Question 1

Tank I has base area 2 dm² and initially contains water 5 cm deep. Empty tank II has base area 1 dm² and height 7 cm. Tank II is placed inside tank I, and water spills into it. Find the water depth inside tank II.

Model: tank II is held upright with its bottom on the bottom of I; wall thickness and floating are ignored.

1. 1 cm
2. 2 cm
3. 3 cm
4. 4 cm
5. 5 cm

Question 2

A young deer is sent by its mother to pick ten apples from the Numbers tree. Each apple bears a number, and the deer picks ten labelled with consecutive natural numbers. On the way home, one apple is dropped. The labels of the remaining nine sum to 508. What number was on the dropped apple?

1. 58
2. 57
3. 56
4. 55
5. 54

Question 3

A closed box contains 3 red, 5 yellow and 7 white balls. Without looking, what is the smallest number of balls that must be drawn to guarantee at least THREE balls of one colour?

Original printed choices: A) 2; B) 3; C) 4; D) 5; E) 6

Editorial task: Explain the literal model, the source method and any necessary correction.

Question 4

Four children, Max, Alex, Swan and Emma, become lost on a mysterious island and are pursued by its inhabitants. They reach a bridge that holds at most two people. They need 1,2,5,8 minutes respectively to cross. Whenever two cross, one person must return to guide the others. What is the shortest total time to get everyone across?

1. 12
2. 13
3. 14
4. 15
5. 16

Question 5

In triangle ABC, angle A is three times angle B and half of angle C. Find angle A.

1. $30^\circ$
2. $36^\circ$
3. $54^\circ$
4. $60^\circ$
5. $72^\circ$

Question 6

Last week 10% of a class were ill and absent. This week 10% of those ill pupils have recovered, but 10% of those who attended last week are now ill. All healthy pupils attend and all ill pupils stay away. What percentage of the class now attend?

1. 18%
2. 80%
3. 82%
4. 93%
5. Cannot be determined

Question 7

Each time a clock's minute hand reaches 9, it falls back to 6. The same happens to the hour hand at 9. Otherwise the clock runs normally. It now shows 8:32. Which listed time cannot be the correct current time?

Original printed choices: A) 2h17; B) 5h52; C) 11h02; D) 5h17; E) 2h32

Editorial task: Explain the literal model, the source method and any necessary correction.

Question 8

Square ABCD lies inside right triangle OEF as shown. OA=48 and OB=36. Find EF.

1. 176
2. 180
3. 185
4. 188
5. 190

Question 9

Squares ABCD,EFGH,IJKL are arranged as shown, with AE=FD=CG=BH=AB/3. I,J,K,L are respectively the midpoints of EH,EF,FG,GH. Find the ratio of the area of IJKL to the area of ABCD.

1. $\frac5{18}$
2. $\frac13$
3. $\frac7{18}$
4. $\frac29$
5. $\frac14$

Question 10

Find the last digit of $1\times3+1\times3\times5+1\times3\times5\times7+1\times3\times5\times7\times9+1\times3\times5\times7\times9\times11$.

1. 1
2. 3
3. 5
4. 8
5. All the above answers are wrong

Question 11

Four friends, Bob, Alex, Ginny and Flora, are invited to a party hosted by the Lord of the Forest. Each knows exactly two of the languages bird, fish, squirrel and cricket. Alex knows bird but not fish. Bob knows squirrel but not fish and can interpret between Alex and Ginny. Flora knows neither bird nor cricket but can talk directly with Alex. Which statement is true?

Interpretation: Bob is needed to interpret because Alex and Ginny have no shared language.

1. Alex knows bird and cricket
2. Ginny knows fish and squirrel
3. Bob knows squirrel and cricket
4. Flora knows squirrel and bird
5. None of the statements is true

Question 12

Special paving pieces are needed at a bend. Each illustrated piece has a 135° angle. How many such pieces form a closed path?

Interpretation: a simple convex loop using identical bends, all turning the same way.

1. 6
2. 8
3. 9
4. 10
5. 12

Question 13

Ann and Ben are identical twin sisters; one always lies and the other always tells the truth. John married one of them but cannot distinguish them by appearance. Meeting one sister, which listed question lets him identify whether she is his wife?

1. Are you my wife?
2. Do you always lie?
3. Is your sister my wife?
4. Does my wife always tell the truth?
5. Does your sister always lie?

Question 14

Every unit edge in the strip is coloured red, blue or green, with three different colours around each unit triangle. Some green edges are marked g and blue edges b. What colour must edge x have?

1. Either blue or green
2. Green only
3. Blue only
4. Any of red, green or blue
5. None of these colours is possible

Question 15

Which of the five pictures shows the illustrated pyramid viewed directly from above?

1. Figure A from the source
2. Figure B from the source
3. Figure C from the source
4. Figure D from the source
5. Figure E from the source

Question 16

A rectangular fish pond is divided by nets meeting at an interior point, with the lengths shown. Find x.

1. 10$\sqrt{11}$
2. 11$\sqrt{11}$
3. 110
4. 50
5. 60

Question 17

In square ABCD, M is the midpoint of AB and triangle AMD has area 7 cm². Find the area of ABCD.

1. $14 cm^2$
2. $21 cm^2$
3. $25 cm^2$
4. $28 cm^2$
5. $36 cm^2$

Question 18

Hak folds this net into a cube and adds the dot counts on each pair of opposite faces. What is the smallest such sum?

1. 5
2. 6
3. 7
4. 8
5. 9

Question 19

Six houses float on a river. Nearby houses are connected by the six bridges shown for convenient travel. How many unordered pairs of houses do not have a direct bridge between them?

1. 6
2. 7
3. 8
4. 9
5. 10

Question 20

Michael and his father play a shooting game. They agree that Michael is allowed five shots. Each hit earns two additional shots. He takes 23 shots in total. How many hits does he make?

Assume Michael uses every shot he earns.

1. 5
2. 6
3. 7
4. 8
5. 9