Consider a denominator of r;
It has proper fractions:
1/r, 2/r, ...., (r-1)/r
Their sum is: (1 + 2 + ... + (r-1))/r
The numerator of this sum is
1 + 2 + ... + (r-1)
Which is an Arithmetic Progression (AP) with r-1 terms, and sum:
sum = number_of_term(first + last)/2
= (r-1)(1 + r-1)/2
= (r-1)r/2
So the sum of the proper fractions with a denominator or r is:
sum{r} = ((r-1)r/2)/r = ((r-1)r/2r = (r-1)/2
Now consider the sum of the proper fractions with a denominator r+1:
sum{r+1} = (((r+1)-1)/2
= ((r-1)+1)/2
= (r-1)/2 + 1/2
= sum{r) + 1/2
So the sums of the proper fractions of the denominators forms an AP with a common difference of 1/2
The first denominator possible is r = 2 with sum (2-1)/2 = ½;
The last denominator required is r = 100 with sum (100-1)/2 = 99/2 = 49½;
And there are 100 - 2 + 1 = 99 terms to sum
So the required sum is:
sum = ½ + 1 + 1½ + ... + 49½
= 99(½ + 49½)/2
= 99 × 50/2
= 2475
It is 2475.
Because to add fractions, the denominators must be equal.
Improper fractions have numerators that are greater than or equal to their denominators.
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There are an infinite number of proper fractions equal to any other fraction - as long as you multiply the numerator and denominator by the same number, the fractions will be equivalent.
Proper fractions are when the numberator is less than or equal to the denominator. So, the only three fractions that can be proper with the denominator as three are: 1/3, which equals approx. 0.333333333 2/3, which equals approx. 0.666666667 3/3, which equals 1.
If the denominators (bottom numbers) are not equal, making equivalent fractions of them so that the denominators are equal. With the denominators equal if the numerators (top numbers) are equal, then the fractions are equal.
Because to add fractions, the denominators must be equal.
They are known as equivalent fractions
Equivalent fractions.
You multiply the fractions until the denominators are equal. Then, you subtract.
Yes, because for any x and y that are positive fractions (y not equal to zero), x/y is also a positive fraction. Note that whole numbers are considered fractions with denominators of 1 -- otherwise it doesn't work.
Improper fractions have numerators that are greater than or equal to their denominators.
equal fractions
In actual problems, we convert two (or more) fractions so that they have equal denominators.
You can represent the two fractions with one fraction with a numerator equal to the sum of the two individual numerators (with sign) and a denominator equal to just one of the two denominators.
Only if the numerator is zero,
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