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FOURTH GRADE / MATHEMATICS / DIVISION 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 DIVISION of INTEGERS   is  (are)
 ♦ 
METHOD(s) of DETERMINING the RESULT  (QUOTIENT) of
 ♦ HOW MANY TIMES A NATURAL NUMBER  CALLED a  DIVISOR

 ♦ CAN BE SUBTRACTED FROM / IS CONTAIN-  ED IN ANOTHER NATURAL NUMBER CALLED
 a DIVIDEND


 and WHICH NATURAL NUMBER (REMAINDER,  LESS THAN DIVISOR) STAYS BEHIND.
  Ref.-1
1. Assess both of the operations in the figures below and sign the correct choice .



I. The operation above is correct.
II. The operation below is correct.

a) Only I
b) Only II
c) Both I and de II
d) none
The way we define division permits decompositions in the numerator but not in the denominator .
2/4 = 1/4 + 1/4 (one half two quarters) but 2/4 =! 2/2 + 2/2 (one half can not be two , =! denotes inequality)
2. Compare the following two methods of calculating 21873/23 and find the correct answer.



I. The method above is correct.
I. The method below is correct.

a) Only I
b) Only II
c) Both I and de II
d) none
The second method is calculating the result by subtraction. Subtractions begin from the higher place values, are block subtractions and result can be found in plausible time.
3. Have a look at the operations given in the figures below and find the correct statements.



I. The one above is the well known division method.
II. In the figure below the numerator is decomposed.
III. The decomposition below begins from the lower place values and will be a long operation.

a) Only I is true.
b) Only II is true.
c) Only III is true.
d) All statements here are true.
Look especially at the operation below. It will certainly be a long way to go.
4. Which of the following numbers should respectively be in the dotted places from above to below ?



a) 502 , 263 and 1638
b) 251 , 217 and 1638
c) 502 , 251 and 1757
d) 251 , 251 and 1717
The first should be 251 .
5. If you can, find the 3 digits after the decimal dot for 633512/863 ?

a) 810
b) 81
c) 081
d) 11
Since 3 digits are asked, b and d are out of question. Just look at the first digit after the dot and answer if it is an 8 or a zero .
6. A number is divided by 28 with a remainder and the quotient is 61. Which of the followings can this number not be ?

a) 1735
b) 1717
c) 1707
d) 1727
The number should be more than 1708 (61x28) but it can at the most be 27 more .
7. A division with remainder has 61 and 34 as divisor and quotient respectively. What will the result be if the number obtained by subtracting 1900 from the maximum possible dividend is divided by 8 ?

a) 24
b) 25.875
c) 24.875
d) 16.875
61x34=2074. Adding max. rem. we get 2107.
2107 - 1900 = 207. 207/8 is not a natural number.
Calculate 2008/8 = 26. The result must be 1/8 less then 26.
8. The divisor of a division is 9. What will the product of all possible remainders be ?

a) 720
b) 40320
c) 5040
d) 120
1x2x3x4x5=120 ... (and 3 more multiplications ...) .
9. A train from Istanbul to Ankara has stopped only in Izmit, Eskisehir and Polatli and waited 9 , 42 and 9 minutes respectively. If the distance between is 640 km. and the total trip took 9 hours, what should the average driving speed of this train has been ?

a) 71
b) 71.1
c) 80
d) 71.111
The distance is 640 km and it should be divided by 8 because of underlined ... Not the average speed is being asked .
10. What will the result of 707707.707 / 707 be ?

a) 1001.01
b) 11.1
c) 101.101
d) 1001.001
Think as 707000 + 707 + 0.707 .
0.707/707 = 1/1000 x (707/707) = 1/1000 = 0.001 since 1 percent is 0.01 .
11. One thousandth of a millimeter is ... meter .

a) 1 m.
b) 1 deci milli meter
c) 1 micro meter
d) 1 nano meter
micro is 1/(1000x1000) of .
12. What will the result be if the gereatest 4-digit even number having 7 at tens is divided by the greatest 3-digit number ?

a) 9.987...
b) 9.970...
c) 9.978...
d) 9.998...
Dividend is 9978 , and divisor is 999 .
Do not try to calculate, try to estimate: 9978/999 = 9990/999 - 12/999 = 1 - 12/999 .
If 12/999 were 12/100 it would be pretty easy. And their results should really be very close to each other. Their difference should be around one thousandth.
13. Correponding to the 0 in the upper blue background there is a comma (decimal point in US notation) added to quotient just after just after the second 9. What is the reason behind ?



a) so is the rule.
b) because there is no (full) 99 in 78.
c) the aim is to determine the rato of 78 to 99.
d) 78 being greater than 100, is multiplied by 10 to make it greater (and to find how many 100s in this greater nu.) and there should be a corrsponding act at the quotient side.
The decomposition is 9900/100 + 70/100 + 8/100 . The first element of this sum is 99.
The second is 7/10 and coded decimally it is 7 just after a comma (decimal point).
The third is 8/100 and its the 8's place shows it is the coefficint of hunderths.
14. Assuming that the given data in the following figure is correct, what should be the result of 6445/63 calculated up to the thousandths?



a) 102,300
b) 102,301
c) 100,230
d) 101,230
The comma is the decimal dor of the US notation.
c is out because 100x63 is 6300 and the diff. with dividend is greater than the divisor.
d is out bacauese 101x63 = 6363 " " " " " " .
In ccase of a and b : Just after the decimal point we have (6445 - 6426)/63 =19/63 .
Since the decimal point is already there we may calculate 19000/63 , which we know how to do.
15. Referring to the figure below, which of the following statements are true ?



a. 1/63 of 64/63 having "100" as multiplier is transferred as 10/63 to the element having "10" as multiplier.
b. Some part of 45/63 having the multiplier "1" might be transferred to the element having the multiplier "10".
c. If the operations mentioned in a and be are performed, they contribute 14/53 to the element with multiplier "10".
d. 14/63 contains no natural number so a 0 is written to its place and it is transferred to multiplier "1" group as 140/63.
e. After the decimal point we do not need to use 0s for divisor > partial dividend.

a) only e is not correct.
b) all are correct.
c) a and b are correct, the others not.
d) a, b and c are correct, the others not.
The situation after the decimal point can best be handled if we multiply te remainder by "10...0" the number of zeros being equal to the number of digits after the decimal dot.
Add so many zeros to the right of the remainder and divide this by divisor, place the result after the dot.
16. Referring to the figure below, which of the following choices is true (expresses the full truth) ?



a) 6445 is thought and handled as 6000+45
b) 145 is thought and handled as =126+19
c) both a and b are true
d) all operations in figure are meaningless.
The question contains clue to the answer .
17. What will be the remainder of 10 000 / 7 if only natural numbers are to be used ?

a) 5
b) 8
c) 4
d) 3
7000 = 100x7, 2800 = 400x7 140 = 20x7 and 56 = 8x7 .
18. Calculate 14 592/528 and keep the digits (of quotient)
Which of the following words can be constructed using only the letters of the alphabet having order numbers containing these digits at ones ?

a) bravo
b) excess
c) chance
d) duke
1.(a) , 4. (d), 11. (k) , 14. (n) , 21. (u) ve 24. (x) harfler .
19. 235 person from Purdue University want to visit The Chicago Museum of Science and Industry together and traveling by bus. How many buses will be needed if 42 persons may travel with a bus ?

a) 4
b) 5
c) 6
d) 7
235/42 = 5 with remainder and even if it is only a 1-digit number, a bus is needed.
20. The great architect Mimar Sinan of Ottoman Empire has built The Suleymaniye Mosque in 6 years. What should be the average number of workers if he had paid a salary of 1200 man x month ?

a) 16,67
b) 16
c) 17
d) 15,70
He should have paid 100 man x years of salary in 6 years .
21. 27 lemons are needed for 3 liters of lemonade. How many lemons are needed for 500 guests drinking a glass (200 ml.) of lemonade each ?

a) 750
b) 500
c) 600
d) 900
500 guests drink 100 liter and 9 lemons are needed ffor a liter .
22. Mrs. Rose went to market, bought 8 different items and paid $ 34 .
What an average amount has she paid for an item ?

a) 4 dollars and 25 cents
b) 3 dollars
c) 3 dollars and 25 cents
d) 3 dollars and 75 cents
34/8 = ?
23. How much a distance runs a marathon runner in a minute (average speed m. and cm. per minute) if he has taken 42 km. and 300 m. in 2 hours and 20 minutes ?

a) 181 m. 28 cm.
b) 302 m. 14 cm.
c) 30 m. 20 cm.
d) 18 m. 12 cm.
42 300 m. distance in 140 minutes.
24. What is the average beat per minute of an healty heart that beats 100 800 times a day ?

a) 80
b) 65
c) 75
d) 70
A day is 24x60=1440 minutes .
y
25. The member of the mehter team shown in the figure below walks 5 steps forward and then 3 steps backward during an exhibition.
Walking this way , if he gets 38 steps, how many steps shall he has advanced. ?


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

RY1 RY2



a) 8
b) 14
c) 13
d) 12


This is a division problem. The member will get forth only 2 steps in each 8 .
We also need to know which step after a definite multiple of 8 because it makes a difference .
The first five will be added and the consequents up to 8 will be subtracted.
If it can not be solved by perceiving it as a division problem, you can simulate the 38 steps as seen in "ry1" .
This is the feedback!
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