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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

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Q: What is the sum of all the positive proper fractions with denominators less than or equal to 100?

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Related questions

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.

Equivalent fractions.

You multiply the fractions until the denominators are equal. Then, you subtract.

They are known as equivalent fractions

Improper fractions have numerators that are greater than or equal to their denominators.

When the denominators are relatively prime, that is, they have a GCF of 1.

equal fractions

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.

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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