Mathematics Magazine for Grades 1-12  

Grade 8

Theory:

 Volume Formulas

Note: "ab" means "a" multiplied by "b". "a²" means "a squared", which is the same as "a" times "a". "b³" means "b cubed", which is the same as "b" times "b" times "b".

Be careful!! Units count. Use the same units for all measurements. Examples

Cube = a ³

Rectangular prism = a b c

Irregular prism = b h

Cylinder = b h = pi r ² h

Pyramid = (1/3) b h

Cone = (1/3) b h = 1/3 pi r ² h

Sphere = (4/3) pi r ³

Ellipsoid = (4/3) pi r₁ r₂ r₃

  Solutions from the Previous Issue:

1.        How many of the first 160 positive whole numbers are divisible (leaving a remainder of 0) by all of the three smallest prime numbers?

Solution:

The smallest three prime numbers are 2, 3, 5

2 x 3 x 5 = 15

15 x 2 = 30

15 x 3 = 45

15 x 4 = 60

15 x 5 =75

15 x 6 =90

15 x 7 = 105

15 x 8 =120

15 x 9 = 135

15 x 10 = 150

15 x 11 = 165 > 160

There are 10 whole numbers divisible by all three smallest prime numbers.

2.        The evil warlord, Grossout, locked Princess Stunning in the dungeon. Before he lets her out, she must find the first perfect square number larger than 100 where the sum of the digits is not a perfect square. Help the princess by finding the number.

Solution:

11 x 11 = 121                  1 + 2 + 1 = 4 perfect square

12 x 12 = 144                  1 + 4 + 4 = 9 perfect square

13 x 13 =169                                   1 + 6 + 9 = 16 perfect square

14 x 14 = 196                  1 + 9 + 6 = 16 perfect square

15 x 15 = 256                  2 + 5 + 6 = 13 it is not a perfect square

The first perfect square number larger than 100 where the sum of the digits is not a perfect square is 256

3.        What number must be added to both the numerator and denominator of ¹/₄ to make the resulting fraction ⁷/₈?

Solution:

8(1 + x) = 7(4 + x)

8 + 8x = 28 + 7x

x = 20

20

4.        Six numbers between 3 and 30 are either squares or cubes ? 4, 8, 16, 25, 27. How many numbers between 30 and 350 are either squares or cubes? Note: If a number is both a square and a cube, it should only be counted once.

Solution:

18 – 4 = 14 square numbers between 30 and 350,

7 – 2 = 5 cubes between 30 and 350

There are two numbers that are in the same time square and cube and are smaller that 350.

14 + 5 – 2 = 17 numbers between 30 and 350 are either squares or cubes.

Proposed Exercises:

Calculate:

1        

2        

3        

4