Question

Solve each proportion.
6/n-6 = 8/n

* please show the work too!

Answer 1

9514 1404 393

Answer:

n = 24

Explanation:

Multiply the given equation by the product of denominators. Then subtract 6n and go from there.

`(6)/(n-6)=(8)/(n)\\\\6n=8(n-6)\qquad\text{rid of fractions}\\\\0=2n-48\qquad\text{subtract 6n, simplify}\\\\0=n-24\qquad\text{divide by 2}\\\\\boxed{24=n}\qquad\text{add 24}`

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Alternate solution

Any proportion can be written 4 ways. If we write this one "sideways," we have ...

6/8 = (n -6)/n

1 -1/4 = 1 -6/n . . . . . expand in parallel fashion

6/n = 1/4 . . . . . . . . subtract 1, multiply by -1

n = 24 . . . . . . . . . multiply by 4n

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Another alternate solution (can be done mentally)

The above "sideways" proportion can be reduced to ...

3/4 = (n -6)/n

The difference between values on the left is 1 unit. On the right, the difference between values is 6. (The numerator is 6 less than the denominator.) The denominator on the left (4) is 4 times the difference, so the denominator on the right (n) will likewise be 4 times the difference: n = 4·6 = 24. Check: 3/4 = 18/24.