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FOURTH GRADE / MATHEMATICS / ADDITION of NATURAL NUMBERS 1.2

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. Aykhan is the youngest son of a family living in Central Asia around 7 000 B.C. The division of labor within the family requires that he cares for the family sheep.
He leads the sheep to pastureland to graze and back home. And he tries to provide security to them, since each sheep is valuable for the family. Life without them would be really hard ...
In order to be sure that he brought the same number of sheep back into the sheep fold, he puts one pebble into his bag for each sheep he brings to the pasture and he gets out one pebble for each sheep on return. He keeps these pebbles in front of the stall.
One evening after all sheep entered the fold, he observes that he has still some pebbles in the bag.
What is the meaning of this situation ?


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RY1
RY2



a) A sheep is lost in pasture.
b) There are sheep that have not returned.
c) Some of the sheep have gone prey.
d) Some sheep were added to the herd.
This is a trivial correspondence problem. 1-1 correpondence is desired and is there in case of no problem.
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.   Aykhan is need of communicating his father to inform the situation by signing
What do you expect him to do in case the number of the excess pebbles is less than 5 ?

a) He indicates the bag.
b) He builds fire and uses smoke.
c) For each pebble he opens a finger of his fist and shows.
d) For each excess pebble he closes a finger.
Rational is the common in this case.
3.   Aykhan searched for simpler methods of indicating the number of the sheep and has decided to draw figures on ground.
First he drew a line segment for each sheep and gradually he began to group these segments.
Referring to the figure he drew as he went to pasture, how many sheep should he have ?

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RY1 RY2



a) 10
b) 11
c) 16
d) 20
Each line segment from corner to corner of the outer rectangle corresponds to a sheep.
4.  Aykhan has seen the feasibility of drawing a circle instead of drawing two full rectangles each one representing 8 sheep. Leaving the fold with 18 sheep in the morning, which of the followings is more likely to be the figure he drew ?

a) !    b) !    c) !    d) !   
The rightmost figure should be easily copmleted to a rectangle/square.
5. Being a clever young boy, Aykhan has noticed that (he has less than 32 sheep) he can code the number of the sheep he brings to pasture, even if some of them stays in the fold, by using some minimal amount of symbols only once.
Which of the followings may be the minimal number of symbols he can use independent of all possible changes of the number of the sheep he brings to pasture ?

a) !    b) !    c) !    d) !   
In order to be able to represent 1 to 31 sheep, use of symbols representing 1,2,4,8 and 16 sheep will be enough. By summing these numbers he may reach to any amount between 1 and 31 !
6. After a while Aykhan was aware that he can make use of only one symbol and place/position of this symbol in order to be able to represent all amounts of sheep he has.
He made a permanent table as seen below and for 16 sheep he places just below the figure representing 16 an "I" ; for 8 sheep he places just below the figure representing 8 an "I" etc. . Using this table the representation is easier and he uses.

!


Which figure he may have drawn/got after a morning leave with 27 sheep?

a) !   b) !   c) !   d) !
Now (for Aykhan's case) only one simple figure and use of place/position information is made to represent a definite amount up to 31.
7. Under which condition will it be possible to eliminate the symbols representing 1,2,4,8 and 16 and the table and making use of place information only ?

a) Under all conditions.
b) By placing symbols like "." , "," , "0" etc. into blank places.
c) The table is mandatory.
d) By placing the symbol "I" into blank places .
Just making use of place information and no other symbols .
y
8. In winter Aykhan's family had 27 sheep and with spring each gave birth to a lamb. In summer Aykhan has to bring twice as many animals to pasture.
What should he adopt now to idicate this many animals ?



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RY1 RY2


a) I I . I I .
b) I . I I I
c) I I I . I .
d) I I I . I
Just here a new symbol of value 32 is needed.
9. In the previous questions we used two different symbols and we managed to code every possible natural number within an interval.
How come is this possible ?

a) No need for a special reason.
b) Addition is commutative.
c) It is always possible to take a group of numbers in parenthesis during adition.
d) Every place has the value of a power of two, in a definite order.
10. Which of the following properties for addition of two equal numbers (multiplication by 2) can be claimed for the kind of coding we have seen above ?

1: It is just a left shift of the codes and a "0" is added at the end to indicate the shift.
2: All natural number can be coded this way.
3: Besides place values we have only two digits: "0" and "1".
4: We can make use of coefficients in front of the digits.

a) 1, 2, 3 ve 4
b) 1 ve 2
c) 1, 2 ve 3
d) Yalnız 1
11. In case of addition of natural numbers with base-10 coding we have digits at places and the sum may require carry. The amount to be carried may easily be forgotten and incorrect results may be obtained.
Which precautions may be taken against carelessness to prevent a possible failure in an exam ?


1: To remember and check that the sum of two even or odd numbers is an even number.
2: To remember and check that the sum of an even and an odd numbers is an odd number.
3: To remember and check that when digit sums of both sides (of the equation) are divided by 3 , the remainders should be equal.
4: To remind the student of the consequences of carelesness.


a) 1 and 2
b) 1, 2 and 3
c) all
d) 3
There is no need for advice on crelesness, a student taking an exam concentrates normally.
Odd-even check is of no use in case of carry.
12. Find which of the additions below is incorrect by calculating the digit-sums (the sum of all digits) on both sides of the equations, dividing them by 3 (just disregard any 3 or multiples of 3 in the sum) and and observing if the remainders are equal.

a) 526+155+2134=2815
b) 703+204+3205=4112
c) 303+1011+1101=2415
d) 999+100+1001=2000
13. Using division by 3 and comparing remainders may not always be right.
Under which condition it may fail to give a successful result ?

a) It has always success.
b) In case the error is a multiple of 3.
c) It is of no use.
d) In case even numbers are being added.
The probability of success is 66 % ...
14. Which is the shortest way of adding 99 and 999 ?

a) The normal method of addition.
b) To subtract 1 from one of them and add 1 to the other, then addition.
c) To subtract 1 from one of them and add 1 to the other.
d) Both choices in b and c .
15. A and B are digits and the equation 7457 + AB = 4347 + 4459 is valid.
What should then be the value of the sum A+B ?

a) the equality given can not be right.
b) 1349
c) 62
d) The sum of A and B can not be evaluated because they are not known.
y
16. While A and B are digits the following addition is given.

tot12s16


What should be the value of the sum A+B ?
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RY1 RY2



a) 12
b) 13
c) 18
d) 14
The ones imply that this sum should be either 3 or 13.
17. Which of the followings can not be made in order to check 548+472=1020 ?

a) Subtraction of 472 from 1020 and observation of 548 as the result.
b) To evaluate the remainders of both sides after division by 3 (dismiss all 3's and multiples of 3 in digit sums)
c) Subtraction of 548 from 1020 and observation of 472 as the result.
d) Subtraction of 472 from 548 .
18. 1040 + 1040 + 10 + 10 + 3000 + 3938 + 100 + ( 1040 + 1040 + 10 + 10 + 3000 + 3938 + 100 + 100 ) = ?

a) 20176
b) 18376
c) 18876
d) 18276
19. 1888324 + 9222786 = ?

a) 11111110
b) 11111060
c) 11111111
d) 11111120
1+9=10, 8+2=10, 3+7=10, 2+8=19 ...
20. Which choice gives the sum of the biggest 3-digit prime number and and the biggest 3-digit number with 3 at ones ?

a) 1986
b) 1990
c) 1994
d) 9977
21. How many pair of natural numbers are there with a sum of 12 ?

a) 12
b) 10
c) 8
d) 6
Order is not counted, 6 nad 5 is equivalent to 5 and 6 .
0 is not counted as natural
22. In La Liga if Barcelona and Real Madrid tie, points difference will continue to be +6 for Barcelona .
If Barcelona wins by which amount will the points difference change in comparison with a defeat ?

a) 6
b) 3
c) 9
d) 2
Computation should be relative to victory-defeat.
23. a + b + c = 11 is given .

tot12s16


What will be the result of the addition given in the figure below ?

a) 1221
b) 1333
c) 1232
d) 1266
The last three terms contribute an 11 .
24. A small school library has 1044 short stories books and 44 more novels than these books. What will be the total amount of books in this library ?

a) 2088
b) 2044
c) 2132
d) 2122
The number of novels is 1088 .
The sum is 2000 + 44 + 88 .
25. How many set of natural numbers (without 0) have the sum of 5 , if all numbers in a set are added ?

a) 4
b) 2
c) 6
d) 5
e.g. 5 alone is such a set. 1 and 4 is another set which is equivalent to 4 and 1, so 1&4 and 4&1 couples form only one set.
This is the feedback!
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