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FOURTH GRADE / MATHEMATICS / MULTIPLICATION of NATURAL NUMBERS 1.3

MATHEMATICS IS EASY. PROVIDED THAT ONE WORKS ENOUGH .
ENOUGH IS SKILL DEPENDENT, HAVE TO FIND "ENOUGH" RIGHT FOR YOU.
 Method(s) of MULTIPLICATION of INTEGERS   is  (are)
 ♦ 
METHOD(s) of DETERMINING the RESULT of
 ♦ COUNTING FORWARD from ZERO by BLOCKS containing so many NATURAL NUMBERS called a MULTIPLICAND

 ♦ by an AMOUNT of  ANOTHER  GIVEN  INTE-  GER  NAMED again a MULTIPLICAND.

 IN CASE BOTH MULTIPLICANDS EQUAL, IT IS JUST A CHANGE OF UNITS.                     Ref.-1
1. Investigate the multiplication algortihm (method with well-defined steps) below and choose the choice that gives its properties.



a) all digit multiplications are recorded, no need for carry during multiplications.
b) the number of recorded, written characters is over the number of our usual algorithm.
c) during summation carry may be needed.
d) all choices are right.
An example of cage/lattice method. 247 x 38 = 9386 is calculated.
The multiplication of ones, 3x7 , is written into the upper left cell as 21. The division of the cell into two parts is to easily recognize the place values and make corresponding summations.
2. Why are blue numbers in white backgrounds inserted ?




a) they are step's order numbers.
b) they indicate that the sums are to be multiplied with them.
c) they are carries from the previous queer region's sums.
d) they have no special implications.
The operation is the multiplication 876 x 765 = 675 149 .
Since the results of digits multiplications are placed into cells having two divisions that permit two-digit numbers to be written into, no need for carry in multiplications.
Queer regions are regions of definite place values. Their sum may be a 2-digit number and there is then need for carry. It is plausible to write these carried digits somewhere, otherwisw they are to be forgotten.
The ceels of the upper row may have three regions, one being for carries from the previous regional sum. This is especially proper not to forget carries.
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3. In the figure below an algorithm which is similar to the one we use is shown. What is being done in this algorithm ?

..

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


RY1 RY2

a) First the digit of greatest place value is multiplied with the digits of the other multiplicand.
b) If the multiplication of digits gives a two-digit number, carry is noted with small font.
c) Both 'a' ve 'b' are true.
d) The method is not correct, carries are not considered.
The algortihm we use is given in ry1.
4. Which one is the explicit form of multiplication of 538 by 217 ?


a) 538x200 + 538x10 + 538x7
b) 538x200 + 538x100 + 538x7
c) 500x200 + 30x200 + 8x200 + 500x10 + 30x10 + 8x10 + 500x7 + 30x7 + 8x7
d) 500x200 + 30x200 + 8x200 + 500x100 + 30x100 + 8x100 + 500x7 + 30x7 + 8x7
The explicit form should contain 9 summands.
5. Both methods mentioned in Q. 4, both the one in the figure and the one in ry1, give the same results. What is the reason of attaining the same result ?

a) The operands are the same.
b) Change of place of operands in summation does nor affect the result.
c) Both methods are carried with special care.
d) They are bound to give the same result.
One of them begins from the least place value, the other from the greatest; just the order of summands changes.
6. What is the implication of the figure below ?

...


a) it is not understandable.
b) it is easier to move up-down.
c) the equation 3 x 5 = 5 x 3 should be valid.
d) the figure has symbolic meaning.
The order of multiplicands does not affect the result of multiplication ...
7. The statements given in choices refer to the figure below. Which one is correct?


a) On the left, first the hundreds of the multiplicand below takes part in multiplication its place value taken into account.
b) On the right, our usual method is given and uses lesser digits, needs care for placement.
c) The operation given at the left of the figure is no multiplication.
d) The statements in "a" and "b" are both correct.
In the figures the only difference is the order of multiplication of the digits of the multiplicand below .
8. Which method is used in the multiplication in the figure below ?

a) The usual method but the results of digit-multiplicand multiplications are transversely written.
b) The multiplicand above takes the role of the one below.
c) Both "a" and "b" are correct.
d) The opereation is not done, just the correct result is written.
The first summand is the the multiplication of "ones" of the multiplicand above by the other multiplicand (6x271=1626) .
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9. If one of the multiplicands is consecutively divisible by 2 (till 2/2 = 1) , the multiplication of the other multiplicand by 2 (as many as the division) gives the result.
In the figure below is the generalized form of the statement above is used as a method of multiplication. This method was largely used in history.
Referring to this method, which statements in the choices are correct ?


1. The generalization is not successful, method works only if one multiplicant is consecutively divisible by 2.
2. Division by two may have only one remeinder:1 . If any stage, any step has remainder one, the corresponding number should be included in final sum.
3. If remainder 0, the other multiplicand is multiplied by 2 and the corresponding number will somehow be represented in the final sum .


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

... RY1 RY2



a) 1 , 2 and 3
b) 1 and 2
c) 1 and 3
d) 2 and 3
ry1 explains explicitly the case of consequent divisibility.
ry2 is an explicit example of the generalization.
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10. The squence 1,2,4,8 ve 16, ... is obtained by consecutively doubling the numbers, its generator being 1. The first element of the sequence is 1 and the fourth is 8.
What will be the result of multipilication of the 10th and 12th elements ?


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

! RY1 RY2



a) 262144
b) 524288
c) 1048576
d) 2097152
The tenth element is given in ry1 and the 12th in ry2.
You should be able to easily recognize this among the choices; it is 1024x1024 .
11. 4096 x 512 = ?

a) 2097154
b) 2097152
c) 2097162
d) 2097252
It is twice 1024 x 1024 !
12. Which multiplication is done correctly ?

a) 786 x 951 = 747486
b) 555 x 742 = 411812
c) 741 X 355 = 263050
d) 449 x 888 = 443113
If one of the multiplicands has a 5 at ones, the multiplication has either 5 or 0 at ones.
Both multiplicands odd , multiplication is odd.
One multiplicand even, multiplication even.
13. Which is the easiest way of multiplying 777 by 9 ?

a) Multiplication of 9 by 700 , 70 and 7 and summation of the results.
b) Multiplication of 9 by 7 and add/put 0 and 00 to the right and summation of the results.
c) Apply the usual formal multiplication method.
d) To add/put a 0 to the right of 777 and subtract 777 from.
10x777 = 7770 & 7770 - 777 = 7000 - 7 .
14. 9873 x 653 = ?


a) 6346059
b) 6347069
c) 6447069
d) 6040090
In the figure the first multiplicand (9873) is first multiplied by hundreds of the second (6); take care.
15. 707707707 x 707 = ?

a) 500348348849
b) 400349348849
c) 500349348849
d) 500349338849
To calculate 707x707 and add "000" and "000 000" to the right, thento sum suffices .
A schhol boy with self confidence may try 707x707 mentally.
707 x 707 = 7 x 7 x 101 x 101 = 49 x (10 000 + 200 + 1) .
16. 1999 x 1998 = ?

a) 4000000 - 12000 + 2 = 3988002
b) 4000000 - 6000 + 2 = 3994002
c) 4000000 - 12000 = 3888000
d) 4000000 - 6000 = 3894000
(2000 - 1)x(2000 - 2)= 4 000 000 - 4 000 - 2 000 + 2 = ...
17. Calculate 1999 x 2001 by making use of the equation given below .

(A - B) x (A + B) = A x A - B x B

a) 3 999 999
b) 3 998 000
c) 3 999 998
d) 3 999 009
(2000 - 1)x(2000 + 1)= 4 000 000 - 1 .
18. What should be the result of a multiplication if one of the multiplicands is 3 lesser then the other and the sum of both is 37 ?

a) 280
b) 340
c) 170
d) 450
Since the difference is small (3) , the mutiplicands should be around 37/2 .
20 and 17 satisfy the demand.
19. What will be the result if 127 is added to multiplication of the least and the greatest 4-digit numbers with 7 at hundreds and 8 at tens ?

a) 18 562 647
b) 17 424 420
c) 17 424 547
d) 18 562 640
Nr. format : . 7 8 .
greatest : 9789
least : 1780
20. The multiplication of a 7-digit and an 8-digit number may at the most be a(n) ...-digit number .

a) 16
b) 9
c) 10
d) 15
There are only two plausible choices:15 and 16
Think of the multiplication of the smallest 8-digit and 9-digit numbers. It is 100 000 000 x 10 000 000 = 1 000 000 000 000 000 i.e. a 16-digit nr. and the least of them.
Ours should be lesser .
21. If A1 x B0 = 1240 as A and B are digits, what should be their respective values ?

a) 2 and 3
b) 3 and 4
c) 4 and 6
d) 7 and 3
There are two possible choices:
1. Making use of the good old trial and error.
2. Recognizing B=4 (1240 is a multiple of 40) .
22. If the equation A21 x B11 = 109 931 is valid as A and B are digits, what values may A and B respectively take ?

...


a) 4 and 3
b) 5 and 2
c) 1 and 6
d) 0 and 7
From the yellow transverse region : B+A=7 or 17.
From the green transverse region : A plus the ones of 2xB is equal to nine.
23. An equivalent of the previous question is finding out the values A and B via (100 x A + 21) x( 100 x B + 11) = 109931 or the multiplication in the figure given below .

This is ...

...


a) easily done from the sum A21 + A210 + (AxB)(2xB)B00 given in the fig. below .
b) easily done from 10000 x (AB) + 1100 x (A+B) + 1000 x B .
c) mission impossible.
d) pretty hard to calculate.
Hundreds of the sum gives A+B = 7 .
And the thousands gives A+2xB = 9 . Thes are enough to find A and B out.
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24. Which A<5 and B<5 values respectively satisfy the equality A9 x B7 = 1833 ?

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

! RY1 RY2



a) 3 and 4
b) 1 and 4
c) 2 and 3
d) 2 and 5
One possible method is , as always, trial and error.

We can easily get 100 x A x B + 70 x A + 90 x B = 1770 from (10xA + 9) x (10 x B + 7) = 1833, but this one equation is nor enough to determine both A and B.
From cage method we see ones of 9xB plus ones of 7xA may be ether 7 or 17..
Taking care of the fact that both numbers are less than 5 and making a table ... .
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25. If A<5 and B<5 satisfy the equality 1A9 x 1B7 = 20433, what should AxB be equal to ?

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

! RY1 RY2



a) 18
b) 15
c) 12
d) 24
Not A and B each but AxB is asked, take care !
Please have a look at the table in ry2. Which values for A and B gives which AxB and is this to be found among the choices ?
This is the feedback!

THANK YOU FOR YOUR VISIT; HOPE WE COULD BE OF USE.
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© 2011 , İsmail GERMAN. ALL RIGHTS RESERVED
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