# There are three pumpkins.Each two of them are weighed in pairs and the final results are: 12 "kg", 13 "kg", 15 "kg", What is the weight of the lightest pumpkin?

Nov 2, 2016

The weight of the lightest pumpkin is $5 k g$

#### Explanation:

If we weigh pumpkin 1 (let's call it $x$) and pumpkin 2 (let's call it $y$) we know these two added together are $12 k g$ so:

$x + y = 12 k g$

Then solve for $y$
$y = 12 k g - x$

Next, if we weigh pumpkin 1 (still calling it $x$) and pumpkin 3 (let's call it $z$) we know these two added together are $13 k g$ so:

$x + z = 13 k g$

Then solve for $z$
$z = 13 k g - x$

Next, if we weigh pumpkin 2 (still calling it $y$) and pumpkin 3 (still calling it $z$) we know these two added together are $15 k g$ so:

$y + z = 15 k g$

But from above we know what $y$ is in terms of $x$ and we know what $z$ is in terms of $x$ so we can substitute this for $y$ and $z$ in this formula and solve for $x$:
$12 k g - x + 13 k g - x = 15 k g$
$25 k g - 2 x = 15 k g$
$25 k g - 15 k g = 2 x$
$2 x = 10 k g$
$x = 5 k g$

Substituting the value of $x$ back into the first formula and calculating $y$ gives:
$y = 12 k g - 5 k g$
$y = 7 k g$

And substituting the value of $x$ back into the second formula and calculating $z$ gives:
$z = 13 k g - 5 k g$
$z = 8 k g$