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FOURTH GRADE / MATHEMATICS / ADDITION of NATURAL NUMBERS 1.1

MATHEMATICS IS EASY. PROVIDED THAT ONE WORKS ENOUGH .
ENOUGH IS SKILL DEPENDENT, HAVE TO FIND "ENOUGH" RIGHT FOR YOU.

 (METHODS* of) ADDITION of INTEGERS (are) is

   ♦ 
METHOD(S) of DETERMINING the RESULT of
   ♦ COUNTING FORWARD from a given INTEGER  NAMED as ADDEND (SUMMAND)


 by an AMOUNT of  ANOTHER  GIVEN  INTEGER  NAMED AGAIN as ADDEND (SUMMAND).

    Ref.-1
 * There are diff. methods all doing the same .
y
1.
!

In the text above yellow digits indicate the number of letters within the words. How are then the blank spaces in the followings to be filled in ?
I- The red-colored "k" within the text is the ...th letter.
II- Counting forward from the red "k" by 82 one gets as the 118th letter a "..." .


In case of need for tips or help, please get the mouse on "y" or click on "ry1/2" .

 
RY1
RY2



a) 36 and k
b) 36 and m
c) 36 and ı
d) 36 and a


Digits are not counted as letters, make use of the yellow digits.
There are 6 words with a total of 29 letters before the red "k".
There are 6 letters before the red "k" within the same word.
As you see in ry1, the two sentences have some very important properties in common.
Counting forward without taking care of these commons is waste of time.
Both sentences have 82 letters.
2.   The equality 88A + 77B = 1662 is given. (A and B any digits with A ≠ B)

I- What should be the value of the sum A + B ?
II- How many possibilities exists there for A ?

a) 12 , 7
b) 12 , 5
c) 12 , 6
d) 10 , 9
7+8 = 15 and the sum has 6 at tens. This implies A+B = 12 .
A can not be 0 , 1 and 2. Otherwise it is not posible to satisfy A+B = 12 .
A can not be 6 , otherwisw B is forced to be also 6.
3.  What is the sum of consequent numbers from 1 to 5 ?



a) 10
b) 15
c) 20
d) 25
3 solutions are possible:
1st is to find the sum 1+2+3+4+5 = ... directly.
When the number of consequents increases to 10 000, addition is hard to perform, better drive a formula.
The formula was given in :"http://www.test-dr.com/education/4th-Grade-Mathematics/fourth-grade-natural-numbers-1.2.html" Question 1.
For the geometrical way look at the figure carefully:
The sum of first five integers is the colored region.
The big square has 5x(5+1) small squares .
The no. of white and colored areas are equal.
4. How can we prove that the validity of 1+2+3+4+5 = 5 x (5+1) / 2 implies the validity of 1+2+3+4+5+6 = 6 x (6 + 1) / 2 ?

a) 1 + 2 + 3 + 4 + 5 + 6 = 21 = 6/3 * 7
b) 1 + 2 + 3 + 4 + 5 + 6 = 21 = 6/3 * 7 = 6 x (6+1) / 2
c) 5 x (5+1) / 2 + 6 = 5/2 x (5+1) + 12/2 = 5/2 x (6+6) = 1/2 x (12 + 5)
d) 5 x (5+1) / 2 + 6 = 1/2 x 5 x (5+1) + 1/2 x 2 x (5+1) 1/2 x 7 x (5+1)
An equation valid for the natural number 5 should nor necessaryly be valid for the natural number 6.
If there is an implication then we look for the general case:.
"http://www.tech.plym.ac.uk/maths/resources/PDFLaTeX/induction.pdf"
5. What is the sum of the greatest 4-digit and 5-digit numbers ?

a) 100998
b) 999998
c) 109998
d) 109999
Sum the least 5 and 6-digit number and count backward by 2 !
6. What is the sum of the non multiples of 5 between 1 and 50 ?

a) 500
b) 1000
c) 1275
d) 275
Sum of all numbers between 1 and 10 = 25 x 51
Sum of multiples of 5 between 1 and 50 = 5+10+...+45+50 = 5x(1+2+...+9+10) = 5 x 10/2 x 11
7. What is the sum of the digits "5" we have to use as we write the number from 1 to 65 down ?

a) 75
b) 80
c) 85
d) 90
5, 15 , ... 50,51, ..., 55 , ... , 65
y
8. What is the sum of all odd numbers from 1 to 100 ?

In case of need for tips or help, please get the mouse on "y" or click on "ry1/2".

RY1 RY2


a) 2500
b) 2525
c) 2550
d) 3000
One possible solution is to subtract the sum of even numbers from tha sum of all.
Or else we may make all elemnets of the sequence even by subtracting 1 from each and considering this subtraction within the equation somehow.
9. If the sum of 6 consequtive numbers is 333, the least of them is supposed to be ... .

a) 56
b) 55
c) 54
d) 53
333/6 = 55.5 implies that the middle two terms should be 55 and 56.
10. Which of the choices should replace B in the figure below ?



a) 1
b) 2
c) 3
d) 4
A=4 is a must!
11. What is the sum of squares of numbers from 1 to 5 ?

a) 50
b) 55
c) 60
d) 65
Square of a number is the number multiplied by itself.
Examples : square of 1 = 1x1 = 1 ; square of 5 = 5x5 = 25
12.   The factorial of an integer n, is the product of all integers less than or equal to n.
For example, factorial of 4 = 4 × 3 × 2 × 1 = 24 ; factorial of 5 = 5 x factorial of 4 = 120 ; factorial of 7 = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040


With the definion above, what will be the sum of factorials of natural numbers from 1 to 5 ?

a) 120
b) 153
c) 175
d) 75
13.   The blue colored numbers (1,3,6,10,15,21) in the figure below are called triangular numbers. Why of it is self explanatory.

Which triangular number should be expected to appear at the bottom right if 4 more rows are added ?



a) 66
b) 45
c) 49
d) 55
Take care that their difference goes as 2,3,4,5, ... .
14. In the figure of the previous question the triangular numbers are sequenced .... as even and odd numbers .

a) 2 by 2
b) 1 by 1
c) 3 by 3
d) irregular
15. In the figure below the triangular numbers within the 12 by 12 multiplication table are indicated.

Which one of them is not on the diagonal but can still be factored as the product of a natural number by itself ?



a) 12
b) 24
c) 36
d) 45
y
16. Which interesting properties of triangular numbers are to be observed in the figure below ?



In case of need for tips or help, please get the mouse on "y" or click on "ry1/2".

RY1 RY2



a) Triangular numbers share the same rows/columns with odd multiplicands.
b) In half-triangles the triangular numbers are on rows/columns having an even number of "squares" if the diagonal "squares" are excepted.
c) The sum of two consecutive triangular numbers is equal to the square of the number on the corresponding diagonal.
d) All of the above.
All of them seem to be true .
17. The two-digit decimal number AB has the property A+B = 1/2 x AB.
Which of the following numbers has this property ?

a) 16
b) 22
c) 18
d) 14
18. The following figure refers to triangular numbers.
What will be the number of the spheres on the whole circumference when there are 15 spheres on a side ?



a) 45
b) 42
c) 39
d) 48
19. In case of 15 spheres on a side, what would be the number of non circumferential spheres ?

a) 120 - 45 = 75
b) 120 - 45 = 65
c) 120 - 42 = 78
d) 120 - 42 = 68
The solution of the previous question solves this q. also .
If we had different choices it would help to know: The 15th triangular number is the sum of no. from 1 to 15, i.e. 15 x 16/2.
20. How many digits are there in the seventh row of Pascal's triangle and what is the middle term of this row ?



a) 7 , 20
b) 10 , 10
c) 10 , 20
d) 7 , 10
21. Which of the choices gives the 10. element of the blue transverse row ?



a) 65
b) 35
c) 45
d) 55
They are the triangular numbers.
22. What may the white region in the following figure be containing ?



a) All of Pascal Triangle's content.
b) Even numbers of Pascal's Triangle.
c) Odd numbers of Pascal's Triangle.
d) Integers of Pascal's Triangle.
All numbers of PT are integers ...
23. How many additions in the following row are correct ?

1845+3456=4301    3552+7347=10899    14769+72674=87443    10551+54=10605


a) 1
b) 3
c) 4
d) 2
Use the followings :
Addition of an even and an odd numbers gives an odd number .
If the digit-sum of the summands is a multiple of 3, the digit-sum of the sum is also ...
24. Which one is incorrect ?

a) 1845+3456=4301
b) 14769+72677=87446
c) 10551+554=11105
d) 3552+7342=10894
25. Putting two "+" signs easily make out of the middle row in the figure below an equation.
Which one of the choices makes the same for the other two ?



a) (1 + 2) x 2 = 6 ve (5 / 5) + 1 = 6
b) (1 + 1) x 3 = 6 ve 5 + (5 / 1) = 6
c) (1 + 1 + 1)! = 6 ve (5 / 5) + 1 = 6
d) (1 + 1 + 1)! = 6 ve (5 - 0) + 1 = 6
Remember the def. of factorial and make use of it .
This is the feedback!
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© 2011 , İsmail GERMAN. ALL RIGHTS RESERVED
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