How many golf balls?

a) How many golf balls can be stacked in a triangular pyramid with a triangular base of ten balls on each side? 

b) How many golf balls can be stacked in a triangular pyramid with a triangular base of n balls on each side? 

How many rectangles are there in an mxn grid?

Let us consider the following grid with m rows and n columns: 

Number of rectangles with dimension 1X1 = m*n

Number of rectangles with dimension 2X1 = (m-1)*n

Number of rectangles with dimension 3X1 = (m-2)*n

....................

Number of rectangles with dimension mX1 = [(m-(m-1)]*n

Similarly,

Number of rectangles with dimension 1X2 = m*(n-1)

Number of rectangles with dimension 2X2 = (m-1)*(n-1)

Number of rectangles with dimension 3X2 = (m-2)*(n-1)

..............

Number of rectangles with dimension mX2 = [(m-(m-1)]*(n-1)

Increasing the number of columns incrementally by 1, the last set of number of rectangles with n columns are

Number of rectangles with dimension 1Xn = m*1

Number of rectangles with dimension 2Xn = (m-1)*1

Number of rectangles with dimension 3Xn = (m-2)*1

.....................

Number of rectangles with dimension mXn = [(m-(m-1)]*1

Sum (1X1, 2X1, 3X1…..mX1) = n * [m + (m-1) + (m-2) +……1]                = n * m(m+1)/2

Sum (1X2, 2X2, 3X2…..mX2) = (n-1) * [m + (m-1) + (m-2) +……1]         = (n-1) * m(m+1)/2

Sum (1X3, 2X3, 3X3…..mX3) = (n-2) * [m + (m-1) + (m-2) +……1]         = (n-2) * m(m+1)/2

.....................

Sum (1Xn, 2Xn, 3Xn…..mXn) = [(n-(n-1)] * [m + (m-1) + (m-2) +……1]  = 1 * m(m+1)/2

And total number of rectangles will be sum of the sums of rectangles of various sizes

So, S = [m(m+1)/2] * [n + (n-1) + (n-2) +…….1]

S = [m(m+1)/2] * S = [n(n+1)/2]

S = mn(m+1)(n+1)/4

Welcome to Expert Math! Do you have a child who needs a little extra help with high school Math? Or do you want your child to get ahead of the curve? If you want your child to catch up or get ahead, give Expert Math a call! We have a team of Math Experts, each with more than a decade of experience. Our rates are reasonable and become all the more attractive as the group grows. So, if you need a helping hand, don't be afraid to reach out to the Math Experts! We're here to help your child learn and grow. 

Visit http://www.expertmath.ca

Service offered in: Richmond Hill, Thornhill, North York, Toronto, Markham, Vaughan

Subjects: Calculus, Vectors, Probability, Advanced Functions
Specialized tutoring in IB Mathematics, AP Mathematics, Academic Mathematics, Applied Mathematics

Sum of an infinite geometric sequence (|r| < 1)

Let us first find the sum of following geometric sequence: 1 + 1/2 + 1/4 + 1/8 + 1/16 + .........

Let S = 1 + 1/2 + 1/4 + 1/8 + 1/16 + .........
then S - 1 = (1 + 1/2 + 1/4 + 1/8 + 1/16 + .........) - 1
which implies S - 1 = 1/2 + 1/4 + 1/8 + 1/16 + .........
So S - 1 = 1/2 times (1 + 1/2 + 1/4 + 1/8 + 1/16 + .........)
Since 1 + 1/2 + 1/4 + 1/8 + 1/16 + ......... = S
It follows that S - 1 = 1/2 * S
Therefore S - 1/2 * S = 1
So 1/2 * S = 1
Therefore S = 1 * 2 = 2

In general if we have a geometric sequence a, ar, ar^2, ar^3........ where a is the first term and r (common ratio) is such that |r| < 1

S = a + ar + ar^2 + ar^3........
S - a = ar + ar^2 + ar^3........
S - a = r * (a + ar + ar^2 + ar^3........)
S - a = r * S
S - r * S = a
S * (1 - r) = a
S = a/(1 - r) which denotes the sum of an infinite geometric sequence where |r| < 1

Example:
In the sequence 3 + 3/10 + 3/100 + ....
a = 3, r = 1/10
S = 3/(1 - 1/10) = 3/0.9 = 3.33333333... or 10/3


Welcome to Expert Math! Do you have a child who needs a little extra help with high school Math? Or do you want your child to get ahead of the curve? If you want your child to catch up or get ahead, give Expert Math a call! We have a team of Math Experts, each with more than a decade of experience. Our rates are reasonable and become all the more attractive as the group grows. So, if you need a helping hand, don't be afraid to reach out to the Math Experts! We're here to help your child learn and grow. 

Visit http://www.expertmath.ca

Service offered in: Richmond Hill, Thornhill, North York, Toronto, Markham, Vaughan

Subjects: Calculus, Vectors, Probability, Advanced Functions

Specialized tutoring in IB Mathematics, AP Mathematics, Academic Mathematics, Applied Mathematics

Question of the day - Measuring 4 and 4 Liters

Measuring 4 and 4 Liters

A milkman has an 8-liter bowl which is full of fresh milk.

He needs to divide this milk equally into exactly two portions (4 + 4 liters).

But he only has two empty bowls: a 5-liter and a 3-liter bowl. 

How can he divide the 8 liters in half in as few steps as possible.


Welcome to Expert Math! Do you have a child who needs a little extra help with high school Math? Or do you want your child to get ahead of the curve? If you want your child to catch up or get ahead, give Expert Math a call! We have a team of Math Experts, each with more than a decade of experience. Our rates are reasonable and become all the more attractive as the group grows. So, if you need a helping hand, don't be afraid to reach out to the Math Experts! We're here to help your child learn and grow. 

Visit http://www.expertmath.ca

Service offered in: Richmond Hill, Thornhill, North York, Toronto, Markham, Vaughan 
Subjects: Calculus, Vectors, Probability, Advanced Functions

Specialized tutoring in IB Mathematics, AP Mathematics, Academic Mathematics, Applied Mathematics

Linear equations in 2 variables

Various cases of system of linear equations in two variables:

ax    +    by  =  c

a'x    +   b'x  =  c'

Case 1
a/a' is NOT = b/b'
The system of equations are consistent and represent intersecting lines. The system has exactly one solution.

e.g. 2x + 3y = 5, 3x - y = 2 (equations are consistent and have exactly one solution which is x=1, and y=1)

Case 2:
When a/a' = b/b' = c/c'
The two equations are essentially the same and represent the same line. The system has infinite solutions.

e.g. 2x + 3y = 5, 4x + 6y = 10 (equations are consistent and have infinite solutions which are x, (5-2x)/3)

Case 3 
a/a' = b/b' is NOT = c/c'
The system of equations are inconsistent and represent parallel lines. The system has no solution. 

e.g. 2x + 3y = 5, 4x + 6y = 8 (equations are inconsistent and have no solution

Welcome to Expert Math! Do you have a child who needs a little extra help with high school Math? Or do you want your child to get ahead of the curve? If you want your child to catch up or get ahead, give Expert Math a call! We have a team of Math Experts, each with more than a decade of experience. Our rates are reasonable and become all the more attractive as the group grows. So, if you need a helping hand, don't be afraid to reach out to the Math Experts! We're here to help your child learn and grow. 

Visit http://www.expertmath.ca

Service offered in: Richmond Hill, Thornhill, North York, Toronto, Markham, Vaughan 
Subjects: Calculus, Vectors, Probability, Advanced Functions

Specialized tutoring in IB Mathematics, AP Mathematics, Academic Mathematics, Applied Mathematics

Binomial Expansion & Pascal's Triangle

(a + b)^1 = a + b
(a + b)^2 = a^2 + 2a^1b^1 + b^2
(a + b)^3 = a^3 + 3a^2b^1 + 3a^1b^2 + b^3
(a + b)^4 = a^4 +  4a^3b^1 + 6a^2b^2 +  4a^1b^3 + b^4
(a + b)^5 = a^5 +  5a^4b^1 + 10a^3b^2 + 10a^2b^3 + 5a^1b^4 + b^5
(a + b)^6 = a^6 +  6a^5b^1 + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6a^1b^5 + b^6

The coefficients in these binomial expansions follow what is called a Pascal's Triangle:

1 1

\/

1 2 1

\/ \/

1 3 3 1

\/ \/ \/

1 4 6 4 1

\/  \/ \/  \/

1 5 10 10 5 1

\/  \/  \/  \/  \/

1 6 15 20 15 6 1

Interesting Products

1 x 1 = 1

11 x 11 = 121

111 x 111 = 12321

1111 x 1111 = 1234321

11111 x 11111 = 123454321

111111 x 111111 = 12345654321

1111111 x 1111111 = 1234567654321

11111111 x 11111111 = 123456787654321

111111111 x 111111111 = 12345678987654321