Ancient Number Systems - CEMC · Ancient Number Systems 1.What is the ancient Roman symbol for 14?XIV 2.What is the following Babylonian numeral in decimal?10 3.What base did the

Faculty of Mathematics Centre for Education in

Waterloo, Ontario N2L 3G1 Mathematics and Computing

Grade 6 Math Circles

March 29 & 30 2016

Review

Ancient Number Systems

1. What is the ancient Roman symbol for 14? XIV

2. What is the following Babylonian numeral in decimal? 10

3. What base did the ancient Mayans use? Base 20

4. What is the following in decimal?

There is a zero symbol in the 200 place and three 1 symbols in the 201 place, so this

number in decimal is (0× 200) + (3× 201) = 60

5. Write 603 in Babylonian numerals.

There are 10 sixties and 3 tens in this number, so the answer is:

We can check this answer. Remembering that the Babylonians used base 60, we have

3 in the 600 place and a ten in the 601 place: (3× 600) + (10× 601) = 603 in decimal.

1

Kinematics

1. What is the difference between a scalar and a vector?

Scalars have a size whereas vectors have both a size and a direction

2. Name three vector quantities.

The following are vector quantities: displacement, velocity, acceleration, force

3. When does size of velocity = speed?

When size of displacement = distance (movement in a straight line)

4. I walk 2 km [E] in 4 hrs and 700 m [N] in 45 mins. What is my velocity and my speed?

Give two decimal places.

5. Give the formula for average acceleration.

a =v2 − v1

t

Where v2 is final velocity, v1 is initial velocity, and t is time.

2

Dynamics

1. What is Newton’s second law?

F = ma

2. Determine the acceleration of the block.

The unbalanced force is

F = 50 N [right] + 30 N [left]

= 50 N [right]− 30 N [right]

= 20 N [right]

Using Newton’s second law F = ma, we get that the acceleration is

a =F

m

=20 N [right]

5 kg

= 4m

s2[right]

3. What is the difference between weight and mass?

Weight is relative to your environment; it is the force of gravity measured in (N).

Mass is constant. It’s how much “stuff” you have inside you, measured in (kg).

4. What is the purpose of the strong nuclear force?

To keep protons together in the nucleus

5. What are the four fundamental forces in order from weakest to strongest?

Gravitation, Weak Nuclear, Electromagnetism, Strong Nuclear

3

Modular Arithmetic

1. Was the year 1500 a leap year?

No, since 1500 (mod 4) = 0 and 1500 (mod 100) = 0 and 1500 (mod 400) 6= 0.

2. Convert 1110001 to decimal.

(1×26)+(1×25)+(1×24)+(0×23)+(0×22)+(0×21)+(1×20) = 64+32+16+1 = 113

3. Convert 341 to binary.

341 = 2(170) + 1

170 = 2(85) + 0

85 = 2(42) + 1

42 = 2(21) + 0

21 = 2(10) + 1

10 = 2(5) + 0

5 = 2(2) + 1

2 = 2(1) + 0

1 = 2(0) + 1

⇒ 101010101

4. Anna was facing East and rotated 3195◦ counterclockwise. Which direction is she

facing now?

3195 (mod 360) = 315

Therefore, Anna rotated 315◦ counterclockwise from East.

Therefore she is now facing SE.

5. You have 7 goblets one of which is real gold. When you align them and count (back

and forth starting with A, B, C, D, E, F, G, F, E, D, ) then the golden goblet would

be the 1000th one that you count. Which one is the golden goblet?

1 full round (A,B,C,D,E, F,G, F,E,D,C,B) has 12 goblets. So 1000 (mod 12) = 4.

Therefore the 4th goblet made of real gold.

4

Number Theory

1. What its 98× 11? 1087

2. Is 23659254 divisible by 9?

Yes, since 2 + 3 + 6 + 5 + 9 + 2 + 5 + 4 = 36 which is divisible by 9.

3. Is 59874036 divisible by 11?

No, since 5− 9 + 8− 7 + 4− 0 + 3− 6 = −2 which is not divisible by 11.

4. Is 3652420 divisible by 15?

Since the unique prime factorization of 15 is 15 = 5× 3, we need to check if 3652420 is

divisible by 3 and 5 individually, and we will be guaranteed that it is divisible by 15.

First of all, since the last digit is 0, thus 3652420 is divisible by 5.

However, the sum of its digits is 3 + 6 + 5 + 2 + 4 + 2 + 0 = 22 which is not divisible

by 3.

Therefore, no 3652420 is not divisible by 15.

5. Is 9008868 divisible by 66?

Since the unique prime factorization of 66 is 66 = 2 × 3 × 11, we need to check if

9008868 is divisible by 2, 3 and 5 individually, and we will be guaranteed that it is

divisible by 66.

First of all, since the last digit is 8, we know that 9008868 is divisible by 2.

Secondly, the sum of its digits is 9 + 0 + 0 + 8 + 8 + 6 + 8 = 39 which is divisible by 3.

Lastly, the alternating sum of its digits is 9− 0 + 0− 8 + 8− 6 + 8 = 11 which is clearly

divisible by 11.

Therefore, yes, 9008868 is divisible by 66.

5

Geometry

1. In the diagram, find the length AD.

By Pythagorean Theorem, AC =√

32 + 42 =√

9 + 16 =√

25 = 5.

Again by Pythagorean Theorem, AD =√

52 + 122 =√

25 + 144 =√

169 = 13.

Therefore AD = 13.

2. In the diagram, AB = 15 cm, DB = 6 cm, BC = 8 cm and ∠B = 90◦. Find the

perimeter of 4ABC.

By Pythagorean Theorem, AC =√AB2 + BC2 =

√152 + 82 =

√225 + 64 =

√289 =

17.

Therefore, P4ABC = 15 cm + 8 cm + 17 cm = 40 cm .

3. In the diagram, determine the measure ∠AFE.

Since 4ABE is an isosceles triangle and ∠ABE = 90◦ then

∠BAE = ∠BEA =90◦

2= 45◦.

6

Since ∠BEF and ∠AFE are supplementary,

∠AFE = 180◦ − ∠BEF = 180◦ − (45◦ + 86◦) = 180◦ − 131◦ = 49◦

Therefore, ∠AFE = 49◦

4. A metrestick leans against a vertical wall with 28 cm between the foot of the metrestick

and the base of the wall. If the top of the metrestick slips 16 cm down the wall, how

far does the foot of the metrestick slide?

By Pythagorean Theorem, the initial height of the meter stick was

√1002 − 282 =

√10000− 784 =

√9216 = 96 cm

Therefore the new height would be 96 cm - 16 cm = 80 cm.

Since the length of the meterstick does not change during the shift, the hypotenuse

remains at 100 cm.

Therefore, the foot of the meterstick is now√

1002 − 802 =√

10000− 6400 =√

3600 =

60 cm.

Thus, it shifted 60 cm - 28 cm = 32 cm.

5. Two circles with equal radii are enclosed by a rectangle, as shown. The distance

between their centers is 2x/3. What is the value of x?

Since x is the diameter of each circle, we know that the radius is x/2. We are also told

that the distance between the centers of the 2 circles is 2x/3.

10 =x

2+

2x

3+

x

2

10 = x +2x

3

10 =5x

3

5x = 30

x = 6

7