Say we have a population of n llamas. Each year, n / 3 new llamas are born, and n / 4 llamas pass away.

For example, if we were to start with n = 1200 llamas, then in the first year, 1200 / 3 = 400 new llamas would be born and 1200 / 4 = 300 llamas would pass away. At the end of that year, we would have 1200 + 400 – 300 = 1300 llamas.

#include <math.h>
#include <stdio.h>

int main(void)
{
    // TODO: Prompt for start size
    int a, b;
    do
    {
        printf("Start Size: ");
        scanf("%i", &a);
    }
    while (a < 9);
    // TODO: Prompt for end size
    do
    {
        printf("End Size: ");
        scanf("%i", &b);
    }
    while (b < a);
    // TODO: Calculate number of years until we reach threshold
    int y = 0;
    while (a < b)
    {
        a = round((float)a + (a / 4) - (a / 3));
        y++;
    }

    // TODO: Print number of years
    printf("Years: %i\n", y);
}

The Input I tried and output I expected:

1. Start size: 1200
   End size: 1300
   Years: 1

2. Start size: -5
   Start size: 3
   Start size: 9
   End size: 5
   End size: 18
   Years: 8

3. Start size: 20
   End size: 1
   End size: 10
   End size: 100
   Years: 20

4. Start size: 100
   End size: 1000000
   Years: 115

>Solution :

The formula is incorrect: you swapped the number of newly born llamas and the number of deceased ones.

Write this instead:

a = round((float)a + (a / 3) - (a / 4));

Note that you should use plain integer arithmetics as the number of new llamas and dead llamas are whole numbers. Since a in an int, so are a / 3 and a / 4, no need for a (float) cast or a round() function call:

a = a + a / 3 - a / 4;

or simpler:

a += a / 3 - a / 4;  

Note also that you should test the return value of scanf() to avoid undefined behavior in case of invalid or missing input.

Here is a modified version:

#include <stdio.h>

int main(void)
{
    // Prompt for start size
    int a, b;

    for (;;) {
        printf("Start Size: ");
        if (scanf("%i", &a) != 1) {
            int c;
            printf("invalid input\n");
            while ((c = getchar()) != EOF && c != '\n')
                continue;
            if (c == EOF)
                printf("unexpected end of file\n");
                return 1;
            }
            continue;
        }
        if (a < 9) {
            printf("value must be at least 9\n");
            continue;
        }
        break;
    }

    // Prompt for end size
    for (;;) {
        printf("End Size: ");
        if (scanf("%i", &b) != 1) {
            int c;
            printf("invalid input\n");
            while ((c = getchar()) != EOF && c != '\n')
                continue;
            if (c == EOF)
                printf("unexpected end of file\n");
                return 1;
            }
            continue;
        }
        if (b < a) {
            printf("value must be at least %d\n", a);
            continue;
        }
        break;
    }

    // Calculate number of years until we reach threshold
    int y = 0;
    while (a < b) {
        a += a / 3 - a / 4;
        y++;
    }

    // Print number of years
    printf("Years: %i\n", y);
    return 0;
}

As you are taking the CS50 course, you should use the function get_int() from <cs50.h> to simplify the error handling in a convenient manner:

#include <cs50.h>
#include <stdio.h>

int main(void)
{
    // Prompt for start size
    int a, b;

    while ((a = get_int("Start Size: ")) < 9) {
        printf("value must be at least 9\n");
    }

    // Prompt for end size
    while ((b = get_int("End Size: ")) < a) {
        printf("value must be at least %d\n", a);
    }

    // Calculate number of years until we reach threshold
    int y = 0;
    while (a < b) {
        a += a / 3 - a / 4;
        y++;
    }

    // Print number of years
    printf("Years: %i\n", y);
    return 0;
}