Problem on fractions > John's front lawn is $\dfrac13$ the size of his back lawn. If John mows $\dfrac12$ of his front lawn and $\dfrac23$ of his back lawn, what fraction of his lawn is left unmoved? How to solve it without picking numbers? The way to solve with number picking: > (size back lawn) = $6$ units > > (size front lawn) = ($\dfrac13$)(size back lawn) = $2$ units > > (size total lawn) = (size back lawn) + (size front lawn) = $8$ units Now we can use these numbers to calculate how much of each lawn has been mowed: > Front lawn: $(\frac 12)(2) = 1$ unit > > Back lawn: $(\frac 23)(6) = 4$ units So, in total, $5$ units of lawn have been mowed. This represents $\dfrac58$ of the total, meaning $\dfrac38$ of the lawn is left unmowed.

> Let John's lawn be $J$. Then, John's front lawn would be $\dfrac 14J$ and his back lawn would be $\dfrac 34J$. If he is mowing $\dfrac 12$ of his front lawn and $\dfrac 23$ of his back lawn, then he is mowing $$\frac 14J \cdot \frac 12+\frac 34J \cdot \frac 23=\frac{5}{8}J$$ If $\dfrac 58J$ was mowed, you can see that $\dfrac 38J$ was not mowed.

Although I wouldn't know why you would want to do it this way; picking numbers is a lot easier.