# The ratio of the lengths of two pieces of ribbon is 1:3. If 4 ft were cut from each piece, the sum of the new lengths would be 4 ft. How long would each piece be?

Jan 19, 2017

One piece has length $3$ feet, the other has length $9$ feet.

#### Explanation:

If the ratio of the length of the two pieces is $\frac{1}{3}$, then if $a$ is the length of the small piece, the big piece will have length $3 a$. If we cut $4$ feet from each piece, their lengths are now

$a - 4$ and $3 a - 4$.

So, we know that their new lengths' sum is $4$ feet, or

$\left(a - 4\right) + \left(3 a - 4\right) = 4 \implies 4 a - 8 = 4 \implies 4 a = 12 \implies a = 3$

So one piece would have length $3$ feet, and the other, $9$ feet.

However, this problem seems a little weird, since we can't really cut $4$ feet from a piece of length $3$ feet. Nonetheless, a first degree equation, without any involvement of absolute values, can only have one root, and since the root is $a = 3$ and the length of the other piece depends directly on this value, there are no other possible solutions to the problem.