Class 4 Maths Olympiad Hard Problems – 28 Advanced Questions with Worked Solutions

Correct answer: (b) 1

1. Number the places 1 to 5 from the left. Second from the left is place 2 and second from the right is place 4.

2. Only place 3 lies between place 2 and place 4.

3. So 1 child stands between them.

Usual mistake: Counting positions moved rather than positions in between.

Correct answer: (b) Every child who plays cricket also plays badminton.

1. A child who plays cricket also plays football.

2. Anyone who plays football also plays badminton.

3. Chaining the two, a cricket player must play badminton. So every child who plays cricket also plays badminton.

Usual mistake: Reversing a chain of conditions instead of following it forwards.

Correct answer: (b) 2653

1. For the largest number, put the bigger of the two free digits in the highest place that is still free, which is the hundreds place.

2. So the hundreds digit is 6, and the remaining 3 goes in the ones place. The tens digit must stay 5.

3. The largest number is 2653.

Usual mistake: Maximising a number by putting the biggest digit first, regardless of which place is fixed.

Correct answer: (a) 57

1. Each remainder is 1 less than the divisor, so adding 3 to the number makes it divisible by 4, by 5 and by 6.

2. The least common multiple of 4, 5 and 6 is 60, so the number is 60 − 3, which is 57.

3. Checking: 57 = 4 times 14 + 1, 57 = 5 times 11 + 2, and 57 = 6 times 9 + 3, so all three conditions hold.

4. So the number is 57.

Usual mistake: Not checking a candidate against every one of the conditions it has to satisfy.

Correct answer: (a) 49

1. 1, 4, 9, 16, 25 and 36 are 1 squared, 2 squared, 3 squared, 4 squared, 5 squared and 6 squared.

2. 36 is 6 squared, so the next number is 7 squared.

3. 7 times 7 is 49.

Usual mistake: Continuing a sequence of differences without noticing that the differences are themselves increasing by 2.

Correct answer: (d) ₹35

1. The cost of one apple is 140 divided by 20, which is Rs.7.

2. The 5 apples that are left cost 5 times 7.

3. 5 times 7 is Rs.35.

Usual mistake: Finding the cost of the wrong group of apples once the total is known.

Correct answer: (a) 34

1. Work backwards. The last step was a division by 3 giving 14, so multiply: 14 times 3 is 42.

2. Before that step 8 had been added, so subtract 8: 42 − 8 is 34.

3. So the number was 34.

Usual mistake: Reversing a chain of steps in the same order rather than in the opposite order.

Correct answer: (c) 6

1. One 4 cm piece together with one 7 cm piece makes a group 11 cm long.

2. The rope is 66 cm, and 66 divided by 11 is 6.

3. So there are 6 groups, and the number of 4 cm pieces is 6.

Usual mistake: Counting all the pieces when only one kind of piece was asked about.

Correct answer: (d) 32

1. Along the length there is room for 8 squares of side 1 cm.

2. Along the width there is room for 4 squares.

3. So the total is 8 times 4 = 32 squares.

Usual mistake: Counting one direction and forgetting the other.

Correct answer: (c) 8

1. The three numbers are 3, then 7, then 11, because each step adds 4.

2. There are two steps from the smallest to the largest.

3. Two steps of 4 is 2 times 4 = 8.

Usual mistake: Reporting one of the intermediate values rather than the requested difference.

Correct answer: (d) 31

1. The sum of the two numbers is 99 and the larger one is 68.

2. The smaller one is 99 − 68.

3. 99 − 68 is 31, and 68 written backwards is 86, which is not a two-digit reverse of 68, so the pair that matters is 68 and 31.

Usual mistake: Getting tangled in the reversal detail instead of using the sum that is given.

Correct answer: (b) 100

1. Any rectangle is fixed by choosing 2 of the 5 vertical lines and 2 of the 5 horizontal lines.

2. Choosing 2 from 5 gives 10 ways, and that can be done independently in each direction.

3. So the total is 10 times 10 = 100 rectangles.

Usual mistake: Counting squares when the question asks for rectangles, which must include squares.

Correct answer: (b) 300 km

1. The car covers 180 km in 3 hours, so in 1 hour it covers 180 divided by 3, which is 60 km.

2. In 5 hours it covers 5 times 60 km.

3. 5 times 60 is 300 km.

Usual mistake: Adding distance for the extra hours instead of multiplying the hourly rate by the new time.

Correct answer: (a) 4

1. In the worst case the first three marbles are one red, one blue and one green, so all different.

2. There is no fourth colour, so the fourth marble must match one of the first three.

3. So 4 marbles are enough to be certain.

Usual mistake: Counting the objects rather than the number of colour choices available.

Correct answer: (c) 81 square cm

1. The perimeter of a square is 4 times its side, so the side is 36 divided by 4, which is 9 cm.

2. The area of a square is the side multiplied by itself, so it is 9 times 9.

3. 9 times 9 is 81 square cm.

Usual mistake: Confusing perimeter and area, or dividing the perimeter by 2 instead of by 4.

Correct answer: (d) 22

1. The steps are 1, 2, 3, 4 and 5, so each step grows by 1.

2. The next step after 5 is 6.

3. So the next number is 16 + 6 = 22.

Usual mistake: Repeating the last step instead of increasing it.

Correct answer: (c) ₹15

1. Selling price is cost price plus profit, so cost price is selling price minus profit.

2. So the cost price is 18 − 3.

3. 18 − 3 is Rs.15.

Usual mistake: Adding the profit when working backwards from the selling price.

Correct answer: (b) 2.5 marks

1. The four scores add up to 12 + 15 + 9 + 14, which is 50.

2. The mean is 50 divided by 4, which is 12.5. The highest score is 15.

3. The difference is 15 − 12.5 = 2.5 marks.

Usual mistake: Subtracting from the second highest score, or using a different central value than the mean.

Correct answer: (a) 21

1. Every number is the sum of the two before it, so the next number is 8 + 13.

2. Adding the two gives 21.

3. So the next number is 21.

Usual mistake: Looking for a difference pattern when the rule is a sum of the previous two terms.

Correct answer: (d) 21

1. The rows hold 1 + 2 + 3 + 4 + 5 + 6 blocks.

2. Adding a run of whole numbers from 1 to 6 means multiplying 6 by 7 and halving.

3. 6 times 7 is 42, and 42 halved is 21 blocks.

Usual mistake: Leaving out the first row, or adding the same number repeatedly instead of counting up.

Correct answer: (b) 24

1. Let the tens digit be a and the units digit be b. The number is 10a + b and the product of the digits is a multiplied by b.

2. So 10a + b = 3ab. Testing the digits from 1 to 9 leaves only two answers: 15, because 3 times 1 times 5 is 15, and 24, because 3 times 2 times 4 is 24.

3. The largest of the two is 24.

Usual mistake: Assuming a property must survive reversing the digits.

Correct answer: (c) 28

1. Work backwards. The last step doubled the number to give 40, so halve 40 to get 20.

2. Before that step 6 had been added, so subtract 6: 20 − 6 is 14.

3. Before that the number had been halved, so double it: 14 times 2 is 28. The number was 28.

Usual mistake: Halving a chain of steps too many times and forgetting to undo the addition.

Correct answer: (a) 59

1. Each remainder is 1 less than its divisor, so adding 1 to the number makes it divisible by 3, by 4 and by 5.

2. The least common multiple of 3, 4 and 5 is 60, so the number is 60 − 1, which is 59.

3. Checking: 59 = 3 times 19 + 2, 59 = 4 times 14 + 3, and 59 = 5 times 11 + 4, so all three conditions hold.

4. So the number is 59.

Usual mistake: Not testing a candidate against all of the given conditions at once.

Correct answer: (b) 8 cm

1. The square has perimeter 4 times 12, which is 48 cm, and that is the whole length of the wire.

2. Half of the rectangle’s perimeter is therefore 48 divided by 2, which is 24 cm. That is the length plus the width.

3. So the width is 24 − 16 = 8 cm.

Usual mistake: Confusing the whole perimeter with half the perimeter when a wire is rebent.

Correct answer: (a) 75 minutes

1. He covers 4 km in 50 minutes, so the same speed covers 6 km in the same proportion of the time.

2. 6 km is 6 divided by 4, which is 1 and a half times as far, so the time is 1 and a half times 50 minutes.

3. That gives 75 minutes.

Usual mistake: Adding a time per kilometre to the original total instead of scaling it.

Correct answer: (d) 2

1. Six equal squares in one row make a long shape that is 6 squares wide and 1 square tall, which is a rectangle rather than a square.

2. A rectangle that is not a square has one line of symmetry along its length and one across the middle, in opposite directions.

3. So the shape has 2 lines of symmetry.

Usual mistake: Counting the symmetries of the parts instead of the symmetries of the whole joined shape.

Correct answer: (b) 2750 millilitres

1. 3/4 of 3 litres is 2 and a quarter litres, which is 2250 millilitres.

2. Bika adds half a litre, which is 500 millilitres, giving 2250 + 500.

3. The jug now holds 2750 millilitres.

Usual mistake: Converting units partway through instead of at the end, or halving the wrong quantity.

Correct answer: (c) 125

1. The first dial has 5 settings and so does the second and the third.

2. The number of codes is 5 times 5 times 5, because the dials are chosen one after another.

3. 5 times 5 times 5 is 125 codes.

Usual mistake: Adding instead of multiplying when there is one choice to make for each of several positions.