7/25/2019 2015 Level4 Review

1/35

Enrichment Classes CMK, 2015

Canada

Grade 4, Review Class

7/25/2019 2015 Level4 Review

2/35

Q1Find the sum of all numbers which when divided by 10 give 10 as a quotient

7/25/2019 2015 Level4 Review

3/35

Q1 - SolutionFind the sum of all numbers which when divided by 10 give 10 as a quotient

Remember, when dividing two numbers we have a quotient and a reminder

What are the possible reminders when dividing by 10?

They are: 0,1,2,3,4,5,6,7,8, and 9

So, the numbers that give 10 as a quotient are:

100, 101, 102, 103, 104, 105, 106, 107, 108, and 109

Their sum is

1045

7/25/2019 2015 Level4 Review

4/35

Q2

The kings palace has 4 floors. There are 15 rooms on each floor, and 6

windows in each room. For decoration, there are two statues in every

other window, and every fifth statue is made out of marble, the restof them are made out of granite. How many granite statues are there

in the palace?

7/25/2019 2015 Level4 Review

5/35

Q2 - Solution

The kings palace has 4 floors. There are 15 rooms on each floor, and 6

windows in each room. For decoration, there are two statues in every

other window, and every fifth statue is made out of marble, the restof them are made out of granite. How many granite statues are there

in the palace?

There are two statues in every other window

That is as many statues as when there is one statue on every window

Thus the total number of statues is 4 x 15 x 6

Every fifth statue is made out of marble: (4 x 15 x 6) / 5 = 4 x 3 x 6

The rest are made out of granite:

4 x 15 x 6 - 4 x 3 x 6 = 4 x (15-3) x 6 = 4 x 12 x 6 = 288

7/25/2019 2015 Level4 Review

6/35

Q3

One side of a rectangle is three times the other side and its area is 48

square meters. What is the perimeter?

7/25/2019 2015 Level4 Review

7/35

Q3 - Solution

One side of a rectangle is three times the other side and its area is 48

square meters. What is the perimeter?

Let the longer side be a meters and the shorter b meters

One side of a rectangle is three times the other side: a = 3 x b

Its area is 48 square meters: a x b = 48 or 3 x b x b = 48 or b x b = 16

Thus, b = 4 and a = 12

The perimeter is 2 x (a + b) = 2 x 16 = 32

7/25/2019 2015 Level4 Review

8/35

Q4John bought a square shaped piece of land. He used 240 m of wire to

build a fence around it. On the third of this land he built a house (as seen

on the image). How many meters of fence does he need to deconstruct ifthere is no need to have fence where the house is?

7/25/2019 2015 Level4 Review

9/35

Q4 - Solution

The perimeter of the square piece of land is 240m

So, its side is 60m

The house cuts off one third from the left and the right sides: 20m each

So the removed fence is: 20m+60m+20m+60m = 160m

John bought a square shaped piece of land. He used 240 m of wire to

build a fence around it. On the third of this land he built a house (as seen

on the image). How many meters of fence does he need to deconstruct ifthere is no need to have fence where the house is?

7/25/2019 2015 Level4 Review

10/35

Q4 - Solution

The perimeter of the square piece of land is 240m

So, its side is 60m

The house cuts off one third from the left and the right sides: 20m each

So the removed fence is: 20m+60m+20m+60m = 160m

John bought a square shaped piece of land. He used 240 m of wire to

build a fence around it. On the third of this land he built a house (as seen

on the image). How many meters of fence does he need to deconstruct ifthere is no need to have fence where the house is?

Ooops, we made a mistake, there is no fence below the house!

7/25/2019 2015 Level4 Review

11/35

Q4 - Solution

The perimeter of the square piece of land is 240m

So, its side is 60m

The house cuts off one third from the left and the right sides: 20m each

So the removed fence is: 20m+60m+20m = 100m

John bought a square shaped piece of land. He used 240 m of wire to

build a fence around it. On the third of this land he built a house (as seen

on the image). How many meters of fence does he need to deconstruct ifthere is no need to have fence where the house is?

7/25/2019 2015 Level4 Review

12/35

Q5

If Paul goes to school by walking and comes home by bus, it will take him

an hour and a half. If he takes the bus to go and come back, it takes him

half an hour. How long it would take him to go and come back walking?

7/25/2019 2015 Level4 Review

13/35

Q5 - Solution

If Paul goes to school by walking and comes home by bus, it will take him

an hour and a half. If he takes the bus to go and come back, it takes him

half an hour. How long it would take him to go and come back walking?

Let w denote the time it takes him to walk in one direction

Let b denote the time it takes him to travel by bus in one direction

What is given is: w + b = 90 mins and b + b = 30 mins hour

The question asks us to find w + w = ?

The second equation tells us that the bus in one direction takes 15 mins

From the first equation, we find that walking in one direction is 75 mins

Thus, walking in both directions is 2x75 = 150 mins or 2 hours and 30 mins

7/25/2019 2015 Level4 Review

14/35

Q6

A year before Masha's birth the sum of her parent's ages was 40. In 2

years from now the sum of Masha's age and the ages of her parents

will be 90. How old is Masha now?

7/25/2019 2015 Level4 Review

15/35

Q6 - Solution

A year before Masha's birth the sum of her parent's ages was 40. In 2

years from now the sum of Masha's age and the ages of her parents

will be 90. How old is Masha now?

In the year of Mashas birth, each parent was a year older and the sum

of their ages was 42

Today Masha is x years old and her parents are x years older each

So, now the sum of Masha's age and the ages of her parents is

x + 42 + x + x = 90 or 3x = 48 or x = 16

7/25/2019 2015 Level4 Review

16/35

Q7

A pattern of squares is made from toothpicks, as shown. If a total of

94 toothpicks are used, how many squares have been formed?

7/25/2019 2015 Level4 Review

17/35

Q7 - Solution

A pattern of squares is made from toothpicks, as shown. If a total of

94 toothpicks are used, how many squares have been formed?

The first square uses 4 toothpicks

The second, third, fourth, and so on square uses 3 toothpicks each

If x denotes the number of squares after the first one, then we have

4 + 3x = 94 or 3x = 90 or x = 30

7/25/2019 2015 Level4 Review

18/35

Q8

Observe the following pattern

At the first step, we have 1 triangle, at the second we have 4

triangles, at the third step 9 trianglesHow many triangles will we

have at step 6?

7/25/2019 2015 Level4 Review

19/35

Q8 - Solution

Observe the following pattern

3 triangles are added between the first and the second step

5 triangles are added between the second and the third step

7 triangles will be added between the third and the fourth step

the pattern should now be clear

9 triangles will be added between the fourth and the fifth step

11 triangles will be added between the fifth and the sixth step

So at the sixth step there will be: 9 + 7 + 9 + 11 = 36 triangles

7/25/2019 2015 Level4 Review

20/35

Q9

A digital clock displays 21:03. After how many minutes will these four

digits appear again on the screen for the first time?

7/25/2019 2015 Level4 Review

21/35

Q9 - Solution

A digital clock displays 21:03. After how many minutes will these four

digits appear again on the screen for the first time?

The digits 2 and 1 change slower than 0 and 3

Soon, 0 and 3 will exchange their places

That will happen after 27 minutes at 21:30

7/25/2019 2015 Level4 Review

22/35

Q10

What is the smallest multiple of 12 having its sum of digits equal to 12?

l i

7/25/2019 2015 Level4 Review

23/35

Q10 - Solution

What is the smallest multiple of 12 having its sum of digits equal to 12?

Let us list several multiples of 12 and see if we can quickly find an answer12, 24, 36, 48, 60, 72, 84, 96,

The first multiple having sum of its digits 12 is 48

Q11

7/25/2019 2015 Level4 Review

24/35

Q11Paul, Zara and Hanna play a game. Paul multiplies a number by 4, Zara

adds to it 23 and Hanna multiplies it by 11. Each child will perform

their operation on the result of the previous operation. However, wedo not know the order of the operations. They started with a single

digit number and obtained in the end 331. What was the beginning

number?

Q11 S l ti

7/25/2019 2015 Level4 Review

25/35

Q11 - SolutionPaul, Zara and Hanna play a game. Paul multiplies a number by 4, Zara

adds to it 23 and Hanna multiplies it by 11. Each child will perform

their operation on the result of the previous operation. However, wedo not know the order of the operations. They started with a single

digit number and obtained in the end 331. What was the starting

number?

Have to see in what order it is possible to apply the reversed operations

In reverse, Paul divides by 4; Zara subtracts 23; and Hanna divides by 11

Let us start investigating the possible orders

331 / 4 = 82.75, not an integer, so Paul was not last331 / 11 = 30.09, not an integer, so Hanna was not last

331 - 23 = 308; 308/4 = 77; 77/11 = 7

331 - 23 = 308; 308/11 = 28; 28/4 = 7

The starting number was 7. Then, it could be 7*4*11+23 = 7*11*4+23 = 331

Q12

7/25/2019 2015 Level4 Review

26/35

Q12

The sum of the points on the opposite faces on a standard dice is 7.

Bennie makes a tower of seven dice such that on the top face we can see

5 points. What is the sum of points on all visible faces?

Q12 S l ti

7/25/2019 2015 Level4 Review

27/35

Q12 - Solution

The sum of the points on the opposite faces on a standard dice is 7.

Bennie makes a tower of seven dice such that on the top face we can see

5 points. What is the sum of points on all visible faces?

From each dice, we can see fours faces (two pairs of opposite faces)

Thus, with a total sum of 14

But we can also see the top face of the top dice, which shows 5 points

The total number of visible points is 7*14+5 = 103

Q13

7/25/2019 2015 Level4 Review

28/35

Q13

Zara wants to organize her books on three shelves. One third of her books

goes on the top shelf, and the remaining 24 books are divided evenly

between the other two shelves. How many books does she have in all?

Q13 S l ti

7/25/2019 2015 Level4 Review

29/35

Q13 - Solution

Zara wants to organize her books on three shelves. One third of her books

goes on the top shelf, and the remaining 24 books are divided evenly

between the other two shelves. How many books does she have in all?

The remaining books are two-thirds of all 24 books

If we divide those into two equal parts, we get that one-third of all books

is 24/2 = 12

Thus, all books are 3*12 = 36

Q14

7/25/2019 2015 Level4 Review

30/35

Q14

In a year, three consecutive months have exactly 4 Sundays.

Which months could these be?

Q14 S l ti

7/25/2019 2015 Level4 Review

31/35

Q14 - Solution

January 31 90

February 28 89

March 31 92

April 30 91

May 31 92

June 30 92

July 31 92

August 31 92

September 30 91

October 31 92

November 30

December 31

This is a fantastic problem that allows a very elegant solution

In the table I listed the months of the year with the number of days in each

Then I added the number of days in all three consecutive months

For example: Jan, Feb, and March have

90 days together (non-leap year)

For example: Feb, March, and April have

89 days together (non-leap year)

Now, note that if three consecutive months have91 days or more, then since 91 = 13*7, they will

cover fully, at least 13 weeks. Thus, they cannot

have all 4 Sundays. One of the three months will

have 5 Sundays.

In a year, three consecutive months have exactly 4 Sundays.

Which months could these be?

Q14 Sol tion

7/25/2019 2015 Level4 Review

32/35

Q14 - Solution

January 31 90

February 28 89

March 31 92

April 30 91

May 31 92

June 30 92

July 31 92

August 31 92

September 30 91

October 31 92

November 30

December 31

Only Jan+Feb+March and Feb+March+April have less than 91 days

So, these are the only candidates for such three consecutive months

The year 2016 is an example when Feb+March+April all have 4 Sundays

The year 2007 is an example when Jan+Feb+March

all have 4 Sundays

In a year, three consecutive months have exactly 4 Sundays.

Which months could these be?

Q15

7/25/2019 2015 Level4 Review

33/35

Q15

What is the biggest product that can be obtained by multiplying positive

numbers that add up to 16?

Q15 Solution

7/25/2019 2015 Level4 Review

34/35

Q15 - SolutionWhat is the biggest product that can be obtained by multiplying positive

numbers that add up to 16?

We need to find positive numbers such that a+b+c+d+ = 16 and theproduct a*b*c*d* to be as large as possible

Clearly each of these numbers has to be > 1 since the product will not

increase when multiplied by 1.

Next, for numbers , it is always true that

This is, if one of the summands, say a, is a = m + n for some ,

then we can only increase the product if we replace a in the sum by

[m+n]+b+c+d+ = 16[m*n]*b*c*d*and change the product to

This shows that all the numbers in the sum have to be 2 or 3

A little experimentation shows that 3+3+3+3+2+2 = 16 gives the largest

product 3*3*3*3*2*2 = 324

m,n 2 m n m+ n

m,n 2

Q15 Solution

7/25/2019 2015 Level4 Review

35/35

Q15 - SolutionWhat is the biggest product that can be obtained by multiplying positive

numbers that add up to 16?

This shows that all the numbers in the sum have to be 2 or 3

So, 3+3+3+3+2+2 = 16 gives the largest product 3*3*3*3*2*2 = 324

2+2+2+2+2+2+2+2 = 16 gives 2*2*2*2*2*2*2*2 = 256

2+2+2+2+2+3+3 = 16 gives 2*2*2*2*2*3*3 = 288

2+2+3+3+3+3 = 16 gives 2*2*3*3*3*3 = 324

There are only three possibilities