Of the following fractions: 9/19, 5/11, 7/15, and 11/23, which is the largest?

Note that each fraction is of the form

n/(2n+1)

This can be written as

1/2 - 1/(2(2n+1))

clearly, as n gets larger, a smaller amount is subtracted, so the fraction with the largest denominator is the largest -- closer to 1/2

1/19 2/19 3/19 7/19. 10/19 11/19

To determine the largest fraction among 9/19, 5/11, 7/15, and 11/23, we can compare their denominators.

1. Start by finding a common denominator:
- The denominators in the given fractions are 19, 11, 15, and 23.
- The least common multiple (LCM) of these denominators is 165.

2. Convert each fraction to have the common denominator of 165:
- 9/19 * (11/11) = 99/209
- 5/11 * (15/15) = 75/165
- 7/15 * (11/11) = 77/165
- 11/23 * (7/7) = 77/161

3. Now, compare the numerators of these fractions:
- 99/209 > 75/165
- 99/209 > 77/165
- 99/209 > 77/161

Therefore, 9/19 (99/209) is the largest fraction among the given options.

To determine which fraction is the largest among 9/19, 5/11, 7/15, and 11/23, we can compare them using a common denominator.

To find a common denominator, we need to identify the least common multiple (LCM) of the denominators. In this case, the denominators are 19, 11, 15, and 23.

To find the LCM, we can use several methods, such as prime factorization or the shortcut method. Let's use the shortcut method:

- Start by listing the multiples of each denominator:
- Multiples of 19: 19, 38, 57, 76, 95, etc.
- Multiples of 11: 11, 22, 33, 44, 55, etc.
- Multiples of 15: 15, 30, 45, 60, 75, etc.
- Multiples of 23: 23, 46, 69, 92, 115, etc.

- Look for the smallest number that appears in all the lists. In this case, we see that 115 is the smallest number that appears in all the lists.

Therefore, the LCM of 19, 11, 15, and 23 is 115.

Now that we have a common denominator, we can convert all the fractions to have a denominator of 115.

- Converting 9/19 to an equivalent fraction with a denominator of 115, we multiply both the numerator and denominator by 115/19: (9/19) * (115/19) = 1035/2185.
- Converting 5/11 to an equivalent fraction with a denominator of 115, we multiply both the numerator and denominator by 115/11: (5/11) * (115/11) = 575/1265.
- Converting 7/15 to an equivalent fraction with a denominator of 115, we multiply both the numerator and denominator by 115/15: (7/15) * (115/15) = 805/1725.
- Converting 11/23 to an equivalent fraction with a denominator of 115, we multiply both the numerator and denominator by 115/23: (11/23) * (115/23) = 1265/2645.

Now we can compare numerators since they all have the same denominator.

- Comparing the numerators: 1035, 575, 805, 1265.

Thus, the largest fraction is 1265/2645, which corresponds to 11/23.