We can write 1/6 as the sum of two fractions, each with 1 as the numerator, as 1/7 + 1/42 .
This can be done in four other ways. Find these.
investigate "Egyptian Fractions)
1/12+1/12
To find the other four ways to write 1/6 as the sum of two fractions with 1 as the numerator, we can use the formula:
1/6 = 1/x + 1/y
where x and y are positive integers.
To solve this equation, we can cross-multiply:
6xy = 6y + 6x
Rearranging the terms, we get:
6xy - 6x - 6y = 0
Using the technique of factoring, we can rewrite the equation as:
6(x-1)(y-1) = 6
Now we can use the factors of 6 to find the possible values of (x-1) and (y-1):
1 * 1 * 6 = 6 --> x-1 = 6, y-1 = 1
1 * 2 * 3 = 6 --> x-1 = 3, y-1 = 2
2 * 3 * 1 = 6 --> x-1 = 2, y-1 = 3
2 * 1 * 3 = 6 --> x-1 = 1, y-1 = 6
Solving each of these equations, we find the possible values for x and y:
x-1 = 6, y-1 = 1 --> x = 7, y = 2 --> 1/6 = 1/7 + 1/42
x-1 = 3, y-1 = 2 --> x = 4, y = 3 --> 1/6 = 1/4 + 1/12
x-1 = 2, y-1 = 3 --> x = 3, y = 4 --> 1/6 = 1/3 + 1/4
x-1 = 1, y-1 = 6 --> x = 2, y = 7 --> 1/6 = 1/2 + 1/14
Therefore, we have found the other four ways to write 1/6 as the sum of two fractions each with 1 as the numerator:
1/6 = 1/7 + 1/42
1/6 = 1/4 + 1/12
1/6 = 1/3 + 1/4
1/6 = 1/2 + 1/14