  Given $n\leq d<\infty$, we investigate the existence of algebras of global
dimension $d$ which admit an $n$-cluster tilting subcategory. We construct many
such examples using representation-directed algebras. First, given two
representation-directed algebras $A$ and $B$, a projective $A$-module $P$ and
an injective $B$-module $I$ satisfying certain conditions, we show how we can
construct a new representation-directed algebra $\Lambda$ in such a way that
the representation theory of $\Lambda$ is completely described by the
representation theories of $A$ and $B$. Next we introduce $n$-fractured
subcategories which generalize $n$-cluster tilting subcategories for
representation-directed algebras. We then show how one can construct an
$n$-cluster tilting subcategory for $\Lambda$ by using $n$-fractured
subcategories of $A$ and $B$. As an application of our construction, we show
that if $n$ is odd and $d\geq n$ then there exists an algebra admitting an
$n$-cluster tilting subcategory and having global dimension $d$. We show the
same result if $n$ is even and $d$ is odd or $d\geq 2n$.
