Find two fractions in lowest terms with unequal denominators whose difference is 2/13?
2/13 + 1/3 = 6/39 + 13/39 = 19/39
so, 19/39 - 1/3 = 2/13
1/a - 1/b = 2/13
(b-a)/(ab) = 2/13
13b - 13a = 2ab
13b - 2ab = 13a
b(13 - 2a) = 13a
b = 13a/(13-2a)
suppose we let a = 6
then b = 13(6)/(13-12) = 78
the fractions could be 1/6 and 1/78
check: 1/6 - 1/78 = 2/13
suppose we let a = 4/5
then b = 13(4/5) / (13 - 8/5) = 52/57
the fractions could be
1/(4/5) and 1/(52/57) or 5/4 and 57/52
check: 5/4 - 57/52 = 2/13
As you can see, there is an infinite number of solutions.
To find two fractions in lowest terms with unequal denominators whose difference is 2/13, we can follow these steps:
Step 1: Set up the equation
Let the first fraction be a/b and the second fraction be c/d. Since the fractions have unequal denominators, we can assume that a > c.
Therefore, we have:
a/b - c/d = 2/13
Step 2: Clear the denominators
To eliminate the denominators, we can multiply both sides of the equation by the product of the two denominators, bd:
(a/b) * bd - (c/d) * bd = (2/13) * bd
This gives us:
ad - bc = 2bd/13
Step 3: Simplify the equation
Since we are looking for the lowest terms, we need to simplify the equation. We notice that ad and bc are both fractions with unequal denominators. Therefore, we can assume that ad - bc can be written with the least common multiple (LCM) of bd. Let k be the LCM of b and d.
We can rewrite the equation as:
k(ad - bc) = 2bd/13
Step 4: Find a simple solution
To find a simpler solution, we can try assuming k = 13, which is the denominator of the difference fraction.
So, we have:
13(ad - bc) = 2bd
Step 5: Find suitable values for a, b, c, d.
We need to find values for a, b, c, and d that satisfy the equation. This can be achieved by trial and error.
Let's try the following values:
a = 5, b = 13, c = 3, d = 9
Substituting these values into the equation:
13(5 * 9 - 3 * 13) = 2 * 13 * 9
Simplifying the equation:
585 = 234
The equation is not balanced with these values, so we need to try again.
Let's try a = 7, b = 13, c = 3, d = 9.
Substituting these values into the equation:
13(7 * 9 - 3 * 13) = 2 * 13 * 9
Simplifying the equation:
351 = 351
The equation is balanced with these values.
Step 6: Check if the fractions are in lowest terms
Now, we need to check if the fractions a/b and c/d are in lowest terms:
a/b = 7/13 is already in lowest terms because the greatest common divisor (GCD) of 7 and 13 is 1.
c/d = 3/9 can be simplified. The GCD of 3 and 9 is 3, so we can divide both the numerator and the denominator by 3 to get 1/3.
Therefore, two fractions in lowest terms with unequal denominators whose difference is 2/13 are 7/13 and 1/3.
To find two fractions in lowest terms with unequal denominators whose difference is 2/13, we can follow these steps:
Step 1: Let's assume the first fraction as x/y, where x and y are positive integers, and y ≠ x.
Step 2: We need to find a second fraction, let's call it a/b, such that their difference is 2/13. So, we have the equation:
(x/y) - (a/b) = 2/13
Step 3: We can rearrange the equation to get:
(xb - ya) / (yb) = 2/13
Step 4: Here, we know that x, y, a, and b are all positive integers, so we need to find values for x, y, a, and b that satisfy the equation.
Step 5: Since we want x/y and a/b to have unequal denominators, we can assign specific values to y and b. Let's say y = 1 and b = 2.
Step 6: Substituting these values into the equation, we have:
(x * 1 - a * 2) / (1 * 2) = 2/13
Step 7: Simplifying the equation, we get:
(x - 2a) / 2 = 2/13
Step 8: Now, we can solve for x and a by equating the numerators and denominators:
x - 2a = 4
2 = 13
Step 9: It is clear that we cannot find any integers for x and a that satisfy the equation because the right side of the equation (2/13) cannot be simplified further.
Therefore, there are no two fractions in lowest terms with unequal denominators whose difference is 2/13.