what is the sum of all proper fractions with denominators less than ten?
help me plssss
sum
= 1/2 + (1/3+2/3) + (1/4+2/4+3/4) + (1/5+2/5+3/5+4/5) + ... (1/9+2/9+... +8/9)
= 1/2 + 3/3 + 7/4 + 10/5 + ... 36/9
= 1/2 + 1 + 3/2 + 2 + ... + 4
looks like an arithmetic series with
a = 1/2, d = 1/2, and n = 8
sum = (8/2)(2(1/2) + 7(1/2))
= 4( 1 + 7/2)
= 4 + 14
= 18
My name Jeff
To find the sum of all proper fractions with denominators less than ten, we can follow these steps:
Step 1: List all the proper fractions with denominators less than ten.
The proper fractions with denominators less than ten are:
1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9,
2/3, 2/5, 2/6, 2/7, 2/8, 2/9,
3/4, 3/5, 3/7, 3/8, 3/9,
4/5, 4/6, 4/7, 4/8, 4/9,
5/6, 5/7, 5/8, 5/9,
6/7, 6/8, 6/9,
7/8, 7/9,
8/9.
Step 2: Find the sum of these fractions by adding them up.
1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9
+ 2/3 + 2/5 + 2/6 + 2/7 + 2/8 + 2/9
+ 3/4 + 3/5 + 3/7 + 3/8 + 3/9
+ 4/5 + 4/6 + 4/7 + 4/8 + 4/9
+ 5/6 + 5/7 + 5/8 + 5/9
+ 6/7 + 6/8 + 6/9
+ 7/8 + 7/9
+ 8/9.
Step 3: Simplify the fractions and add them together.
The sum is equal to:
119/252.
Therefore, the sum of all proper fractions with denominators less than ten is 119/252.
To find the sum of all proper fractions with denominators less than ten, you need to add up all these fractions. Let's break down the steps to solve this problem:
1. Determine the range of denominators: To find fractions with denominators less than ten, we consider fractions with denominators from 1 to 9, since proper fractions have a numerator smaller than the denominator.
2. Calculate the sum for each denominator: For each denominator from 1 to 9, you need to find the sum of all the fractions with that denominator.
For example, when the denominator is 2, the fractions are 1/2, 1/4, and 1/6. So, you add them together: 1/2 + 1/4 + 1/6 = 11/12.
Similarly, for the denominator 3, the fractions are 1/3, 1/6, and 1/9. Adding them together: 1/3 + 1/6 + 1/9 = 5/6.
Repeat this process for each denominator from 1 to 9.
3. Sum up the results: Finally, add up the sums of each denominator to get the overall sum.
To make the calculation easier, you can use a calculator or an online tool. Input each fraction for a specific denominator, sum them, and repeat for all denominators. Then sum up the results.