Question

Sum of three fractions is 2 11/24. If the greatest fraction is divided by the smallest fraction, the result is 7/6, which is greater than the middle fraction by 1/3. find all the three fractions ?​

Answer 1

The three fractions are:

`\[ A = (35)/(6) \]`

`\[ B = (5)/(6) \]`

`\[ C = (66)/(13) \]`

Let's denote the three fractions as A, B, and C, with A being the greatest, B being the middle, and C being the smallest fraction.

Given that the sum of the three fractions is
`\(2 (11)/(24)\)`, we can set up the equation:

`\[A + B + C = 2 (11)/(24)\]`

Now, we are told that if the greatest fraction (A) is divided by the smallest fraction (C), the result is
`\( (7)/(6) \)`, and this result is greater than the middle fraction (B) by
`\( (1)/(3) \)`. This can be expressed as two equations:

`\[ (A)/(C) = (7)/(6) \]`

`\[ (A)/(C) - B = (1)/(3) \]`

Now, let's solve these equations simultaneously.

Equation 1:

`\[ A + B + C = 2 (11)/(24) \]`

Equation 2:

`\[ (A)/(C) = (7)/(6) \]`

Equation 3:

`\[ (A)/(C) - B = (1)/(3) \]`

First, let's deal with Equation 2:

`\[ (A)/(C) = (7)/(6) \]`

Cross-multiply to get:

`\[ 6A = 7C \]`

Now, substitute this into Equation 1:

`\[ 6A + 6B + 6C = 2 \cdot 35 + 11 \]`

Simplify:

`\[ 6A + 6B + 6C = 71 \]`

Now, substitute
`\(6A = 7C\)` into the equation:

`\[ 7C + 6B + 6C = 71 \]`

Combine like terms:

`\[ 13C + 6B = 71 \]`

Now, consider Equation 3:

`\[ (A)/(C) - B = (1)/(3) \]`

Substitute
`\( (A)/(C) = (7)/(6) \)` into this equation:

`\[ (7)/(6) - B = (1)/(3) \]`

Multiply through by 6 to get rid of the fraction:

`\[ 7 - 6B = 2 \]`

Now, solve for B:

`\[ -6B = -5 \]`

`\[ B = (5)/(6) \]`

Now that we have B, substitute it back into the equation we derived earlier:

`\[ 13C + 6B = 71 \]`

`\[ 13C + 6 \cdot (5)/(6) = 71 \]`

`\[ 13C + 5 = 71 \]`

`\[ 13C = 66 \]`

`\[ C = (66)/(13) = 5 (1)/(13) \]`

Now that we have B and C, we can find A using the relationship
`\(6A = 7C\)`:

`\[ 6A = 7 \cdot (66)/(13) \]`

`\[ 6A = 7 \cdot 5 \]`

`\[ 6A = 35 \]`

`\[ A = (35)/(6) \]`

So, the three fractions are:

`\[ A = (35)/(6) \]`

`\[ B = (5)/(6) \]`

`\[ C = (66)/(13) \]`