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FOURTH GRADE / MATHEMATICS /SUBTRACTION 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 BACKWARDS TENDS TO BE  ADDING END-TO-END, THE NEGATIVE ONE POINTING BACKWARDS, SUBTRACTION.
 ♦ HOWEVER BIG/SMALL THEY ARE, WE SHOULD BE ABLE TO SUM & SUBTRACT.

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).
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1. We have seen that fractions being kind of "two-dimensional" have two degrees of freedom, which makes calculations with them little bit harder.
The numerator must change but the denominator might be a chosen constant, e.g. 1 000 000, without not giving up much from the accuracy.
What would be the nearest fraction to 1/3 and its absolute accuracy if we had chosen 1 000 000 a constant unique denominator ?


In case of need for tips get the cursor on "y" or for help click at "ry1/2" .
RY1 RY2



a) 333333/1000000 and 1/3000000
b) 333333/100000 and 1/3000000
c) 333333/1000000 and 1/300000
d) 333333/1000000 and 1/3


As numerator we would have any natural number and the denominator could be only 1 000 000 .
The nearest fraction to 1/3 would be 333 333/1 000 000 .
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2. What would be the relative accuracy of the fraction 333 333/1 000 000 used for 1/3 ?

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RY1 RY2



a) 3/300000
b) 3/1000000
c) 1/3000000
d) 1/1000000
Take the relative accuracy as the difference divided by the greater of the fractions .
3. Which choice would correspond to the figure if we had adopted the constant denominator of question 1 ?

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a) 1100000/1000000 + 1200000/1000000 - 1600000/1000000 = 800000/1000000
b) 1200000/1000000 + 1200000/1000000 - 1600000/1000000 = 800000/1000000
c) 1000000/1000000 + 1200000/1000000 - 1600000/1000000 = 800000/1000000
d) 1200000/1000000 + 1200000/1000000 - 1400000/1000000 = 800000/1000000
We should write long numerators and denominators, but denominators could be avoided .
4. How could we avoid the denominators of question 1 and still recognize fractions ?

a) We could write 2300/ instead of 2300/1000000
b) We could write 2300Fraction instead of 2300/1000000
c) We could write Fraction2300 instead of 2300/1000000
d) We could write 2300/D instead of 2300/1000000
The easiest is the best .
5. Which of the followings would be the advantage of adopting a constant denominator ?

a) We could easily have an idea about the order of magnitudes .
b) Multiplication would be easy, denominator would be scalable .
c) We would be able to apply all the tricks of 4 basic operations with natural numbers.
d) Fractions would be easily understood .
The emphasis is on "the" .
6. Which of the followings would be the a disadvantage of adopting a constant denominator ?

a) Addition would be impossible .
b) Numerators would be longer/greater .
c) Results , all results, would not be error free .
d) Multiplication could not be done .
Since the denominator needs to be relatively great , numerators should be correspondingly great .
7. What will total area to total red colored area ratio be ?

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a) 200/239
b) 100/139
c) 200/139
d) 200/339
In the first square : total area = 1 and red area = 3/4 .
in the second square : total area = 1 and red area = 1-9/25 .
The total area is 2 and red colored total is 139/100 .
The questioned ratio is 2/(139/100) and you should be able to judge that it should be between 1 and 2 .
8. What will total area to total red colored area ratio be ?

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a) 2000/139
b) 1000/139
c) 500/139
d) 200/139
It gives the same result as in the previous question because (6/10) x (9/15) = 3/5 x 3/5 = 9/25 .
9. What will total colored area to total red colored area ratio be ?

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a) 7/3
b) 3/7
c) 10/7
d) 7/10


Consider just half of the smallest squares .
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10. Per Capita soil, our source of food, gets scarce due to population increase. The image below shows the situation just before the end of the second millennium.
Which of the following choices may be about per capita land decrease in N&C America during the epoch shown in the figure ?

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a) 1 - 4/5 = 2/10
b) 3/10
c) 1 - 3/10 = 7/10
d) 1/4
The decrease is about 3/10 ths of the original .
11. If the red line shown in the image below approximates per capita crop land decrease within thirty years in N&C America, what will be the difference of the decreases in the first and second halves ?

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a) 1
b) 0
c) 3
d) 1/10
30 years' decrease is taken as 3/10 hectares and first half's (linear) decrease is 3/20.
12. What will be the difference of the fractions corresponding to deforestation and industrialization ?

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a) 1/100
b) 33/50
c) 17/100
d) 29/100
30 % means 30/100 .
13. Which of the following subtractions are true ?

I: 44/77 - 77/154 - 1/28 = 1/27
II: 66/77 - 44/77 - 11/67 = 11/77


a) only I
b) only II
c) both
d) none
1/14 - 1/28 ?=? 1/27
22/77 - 11/67 ?=? 11/77
14. Uncle Jim's dog is 3 1/2 years old. His cat is 2 1/4 .
How many years is his dog older than his cat ?

a) 1 1/4
b) 1 1/2
c) 2/1/2
d) 2 1/4
1/2 - 1/4 = 1/4 .
15. Suppose the distance between two cities shown in the figure 2200 km. The red and grey cars begin to travel simultaneously in opposite directions from one of the cities to the other.
With grey car having the average speed 5/6 ths of the average speed of the red car they met after 10 hours.
What is the average speed of the red car in km./hour ?

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a) 110
b) 120
c) 240
d) 220
Their total speed is 11 unit fraction of speed .
In 10 hours they make 110 unit fraction of speed and the corrsponding distance is the whole distance i.e. 2200 km...
In an hour they together make 220 km. 220/11 = 20 so 5x20 and 6 x 20 are the speeds ...
16. Which of the following operations gives the minimal result ?

I :2/8 - 1/7 + 2/12 - 1/11 = ?     II : 2/10 - 1/9 + 2/14 - 1/13 = ?


a) both are equal
b) I
c) II
d) both are 0
Try to think all of them with a common denom. of 100 .
We then have (approx. ) 25 - 14 + 16 1/3 - 9 = 18 1/3 for the first .
20 - 11 + 14 - 7 1/2 = 15 1/2 for the second .
17. Janet is 8 years old and her age is 2/9 ths of her mothers .
What will be the sum of their ages after 8 years ?

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a) 44
b) 52
c) 60
d) 66
Now her mother is 36 years old. 36+8 = 44 but be careful : 2x8 will be added to this sum !
18. Jane and Judy were counting the days until their birthdays. Jane said 4/73 of the year would pass before it was her birthday. Judy said she only needed to wait 1/73 of the year. How many days more did Jane have to wait than Judy? ?

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a) 10
b) 15
c) 12
d) 27
5x73 = 365 which means unit fraction corresponds to 5 days in this case .
19. 17 18/29 - 3 10/29 + 7 8/15 - 4/15 = ?

a) 11 236/435
b) 11 236/425
c) 10 236/435
d) 10 236/425
First sum the ones with denom. 29 :14 8/29 . Then : 3 4/15
The wholes give : 11
8/29 + 4/15 = (120 + 116)/435 = 236/435
20. 5 3/5 - 1 1/4 + 4/5 - 9/16 = ?

a) 4 49/70
b) 4 45/80
c) 4 47/70
d) 4 47/80
The same way as in the prev. question .
1 1/4 = 1 4/16
21. Which of the following expansions is wrong ?

a) 1/7 - 1/8 = 5/280
b) 3/9 - 4/16 = 7/102
c) 1 1/2 - 1/4 = 20/16
d) 3 4/5 - 2 1/5 = 40/25
3/9 = 1/3 and 4/16 = 1/4 .
1/3 - 1/4 = 1/12 = 7/84
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22. Which natural number is to replace the question mark ?

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

RY1 RY2



a) 4
b) 1
c) 3
d) 2


See the ry1 .
23. We are given a fraction whose simplest form is 3/7 and if we add 2 to the numerator and denominator we get an equivalent fraction to 1/2.
Which of the chices may be the denominator of the fraction given to us ?

a) 16
b) 14
c) 12
d) 8
3/7 = 6/14 and if we add 2 to the num. and den. we get 8/16 .
24. We are given the 5 smallest fractions which happen to be equivalent to 1/2 after addition of 1 both to their numerators and to their denominators.
What will be the maximal difference between any two of them ?
a) 5/34
b) 2/11
c) 4/33
d) 1/11
The fractions should be 1/3 , 2/5 , 3/7 , 4/9 and 5/11 .
5/11 - 1/3 = (15 - 11)/33 = ...
25. 5 1/7 - 3 - 1 1/14 - 1 1/14 = ?

a) 0
b) 1
c) 1/27
d) 1/99
1/7 - 1/14 - 1/14 = ? .
This is the feedback!

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