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FOURTH GRADE / MATHEMATICS /ADDITION of FRACTIONS 1.3

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

   WITH INCLUSION of FRACTIONS
 ♦ 
COUNTING FORWARD TENDS TO BE  ADDING END-TO-END, SUMMING.
 ♦ HOWEVER BIG/SMALL THEY ARE, WE SHOULD BE ABLE TO SUM.

THERE ARE TWO GOLDEN RULES:
1. DENOMINATORS SHOULD BE EQUAL, OR EXPANDED/REDUCED AND MADE EQUAL.
2. THE MOST EFFICIENT WAY GOES THROUGH THE LEAST COMMON DENOMINATOR (LCD).
y
1. Let us remember the figure of question 2, from our 3rd addition test, given below. Which one is almost impossible in case of addition of fractions that are sole additions with 2 operands ?

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

..


>
RY1 RY2



a) tricks
b) omission of multiples of 3 or 9 in digit sums
c) subtraction to check
d) mental calculations


We may have some formulas to remember and use.
Even and odd has lost sense. mod 3 and mod 9 will not work any more due to division.
Subtraction is always inverse of addition and may be used as a check.
2. Mathematics has its own language and is not easy with words but our thinking is mostly with words; we sometimes need a happy medium in between. We certainly have
a decimal base for numbers that are between natural numbers. We use 1/10 , 1/100 , 1/1000 etc. and we multiply them with coefficients like 1, 2, 3 etc.
Why are these numbers not enough to expand every number we might think of ?

a) natural numbers are countable, fractions are not
b) fractions are queer.
c) natural numbers are finite, fractions are not
d) fractions are two dimenisonal in a sense
Every time we divide by 10 and specify a coefficient, a whole lot, an infinite amount in between are skipped.
This is because points have no dimension.
3. If we consider the 9 fractions having 1 as numerators and the digits as denominators we observe there are 2 groups:
1/1=1 , 1/2=0.5 , 1/4=0.25 , 1/5=0.2 and 1/8=0.125 (which are equal to a finite coefficient times negative powers of ten) expressable in decimal system with finite elements.
1/3=0.3333... , 1/6= 0.16666... , 1/7=..... and 1/9=0.11111... not expressable with finite number of elements
The value of 1/7 is not given; which of the followings may it be ?

a) 0.185234...
b) 0.123123...
c) 0.137137...
d) 0,142857 ...
It should be between 0.1666... and 0.125 .
4. Is being not expressable with finite elements base dependent, can this situation be avoided by exchange of base ?

a) yes, of course,
b) it depends on number.
c) no, it is a property which is base independent.
d) for very special bases, yes.
There are conversion formulas among different bases, a number in coded base 2 may be converted to a number in base 10 .
5. As an example of tricks being still posssible, calculate the following sum:
22/7 + 1/2 + 5/2 = ?

a) 6 1/7
b) 5 3/8
c) 7 1/2
d) 4 7/8
22/7 is almost π which is 3.1416...
1/2 + 5/2 = 3 So the sum should be around 6.14
6. 1/3 + 1/6 + 1/9 = ?

a) 11/18
b) 7/9
c) 13/21
d) 9/14
5.5 / 9 since 1/6 = 3/18 = 3/2 x 1/9. We write it as 11/18
7. To increase our understanding let us look at the distribution of finite number of elements after the decimal dot for 457/561 through 457/569 using Windows' calculator:

457/561 = 0,81461675579322638146167557932264...
457/562 = 0,81316725978647686832740213523132...
457/563 = 0,81172291296625222024866785079929...
457/564 = 0,81028368794326241134751773049645...
457/565 = 0,80884955752212389380530973451327...
457/566 = 0,8074204946996466431095406360424
457/567 = 0,80599647266313932980599647266314...
457/568 = 0,80457746478873239436619718309859...
457/569 = 0,80316344463971880492091388400703...

So we have only one fraction that can be expressed with finite number of elements in the decimal coding (and in every other).

Choosing two of these numbers, answer: 457/565 + 457/566 = ?

a) 516867/319799
b) 516869/319790
c) 516867/319790
d) 516868/319798
We are trying to drive you to the concept of denseness that you will later learn. We want you to be curious.
8. Which result gives the omission of multiples of 3 in digit sums, for the addition in the previous question, for the right answer ?

a) 3=3
b) 2=2
c) 1=1
d) 0=0
Remember this is not a valid check anymore. Just sum all digits and omit any mutiple of 3 at both sides of the equation.
9. Which result gives the omission of multiples of 9 in digit sums, for the addition in the previous question, for the right choice ?

a) 0=0
b) 2=8
c) 7=8
d) 4=4


Remember this is not a valid check anymore. Just sum all digits and omit any mutiple of 9 at both sides of the equation.
10. The Swansee region has 700 cranes that will soon immigrate. At 10.05 in the morning first came four fifths and in 5 minutes gathered with fifteen twenty eighths of the rest.
What is the total number of them which arrived and gathered till 10.10 ?



a) 575
b) 675
c) 635
d) 615
First came 560 of them . Then another 75 .
11. Calculation by hand using at most abacus or slide rule belongs to the past, nobody does it any more. Now we have pocket calculators even in our cellular phones and a lot more. All began in early seventies with "HP and "TI" pocket calculators and developed pretty fast.
Why then are we taught the basics of operations with fractions ?

a) mathematics is like arts
b) machines do need operators
c) no sense, it is not needed anymore
d) teachers should teach
The basics must be known by some people in the society and we need to know how to operate machines, must have sense of it.
12. Given that M is a natural number (different from 0) and the fraction (6+M)/9 is not to be an improper fraction, what will be the average value of all possible fractions ?

a) 1
b) 8/9
c) 7/8
d) 9/10
(7/9+8/9+1)/3 is desired and 1 + 7/9 = 16/9
13. Which of the following additions are correct ?

II: 333/577 + 482/577 = 815/577   II: 44/64 + 32/256 = 13/16
III:7 27/96 + 43 8/48 = 51 45/96


a) only I
b) all
c) I and II
d) only III
I is trivial.
II. 11/16 + 2/16 after reductions.
III can not be correct.
14. (1 27/41 + 3 47/123) - ( 7 - 4 36/82) = ?

a) 182/123
b) 163/123
c) 157/123
d) 207/123
1+3+4-7 + 27/41 + 47/123 + 36/82 = (81 + 47 + 54)/123 .
15. 45/372 + 59/496 = ?

a) 37/124
b) 124/496
c) 67/248
d) 119/496
372 x 4/3 = 496 & 4/3 x 45 = 60 .
16. Which summation gives the greatest result ?

I : 55/11 + 111/121 = ?     II : 185/45 + 164/46 = ?      III : 427/76 + 221/77 = ?
      IV : 16/235 + 81/24 = ?

a) III
b) II
c) IV
d) I
I is obviously around 5.9
37/9 is approximately 4 and ... which means II is around 7.5 - 8.
427/76 is is around 425/75 = 17/3 and all together around 8.5
17. What will be the result of the addition in the figure below ?

..
a) 11 3/4
b) 28 1/3
c) 12 655/252
d) 28 3/4
These are mixed numbers, calculate first the integer part. Do not be afraid, go on and see.
3+4+5=12
3/5 + 6/7 = 51/35 which means total fractional part is around 92/35 (little bit less than 3)
18. What will be the result of the addition in the figure below ?

..
a) 151/60
b) 76/30
c) 81/29
d) 121/47
4/15 + 3/10 + 1/5 = 23 / 30 .
11/12 + 30/36 = 21/12 with common den. 60
19. What should be the value of U in the given equivalent fraction below ?

387/U = 552/58




a) 3565 / 77
b) 3787 / 54
c) 3699 / 99
d) 3741 / 92
Two fractions a/b and c/d are equivalent (equal) only if the product of the numerator (a) of the first fraction and the denominator (d) of the other fraction is equal to the product of the denominator (b) of the first fraction and the numerator (c) of the other fraction.
20. What will be the average of all fractions if the fraction 17/M is desired to be a mixed number and M is desired to be greater then 14 ?

a) 1 11/347
b) 1 47/720
c) 2 9/770
d) 1 21/761
Tree fractions that satisfy the given conditions: 17/15 , 17/16 and 1 .
Set the whole number part aside and find first 17/15 + 17/16 = 527/240 = 2 47/240 .
So the total whole number of the sum is 3 and the proper fractional part is 47/240. These should be divided by 3
21. Evaluate the sum 1 2/7 + 3/21 and find the corresponding point on the number line .

..
a) A
b) B
c) C
d) D
Do not disregard the whole part.
y
22. What should be the value of A ?

..


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

RY1 RY2



a) 3
b) 1/2
c) 2
d) 1


Do not get demoralized, it is simple; your self-confidence solves it.
23. Which of the sums is greater than one and three quarters ?

a) 155/110 + 45/105
b) 43/117 + 13/11
c) 23/159 + 11/145
d) 10/39 + 1/100
The answer may be a or b.
43/117 is approx. 45/125=9/25=0.36 and 13/11=1.18 less then 1.2 anyway
24. 1 + 1/10 + 1/100 + 1/1000 ?

a) 1.1
b) 1.11
c) 1.111
d) 1.1111
Just count the elements .
25. 1 + 1/2 + 1/4 + 1/9 + 1/25 ?

a) 3124/950
b) 2212/1850
c) 1821/850
d) 1711/900
Find first the sum of first 4. Common denominator is 36. Sum is (36+18+9+4)/36 = 67/36
67/36 + 1/25 = (67x25 + 36)/900 = 1711/900
This is the feedback!
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