  In this paper, an attempt is made to construct a
Friedmann-Lemaitre-Robertson-Walker model in $f(R,T)$ gravity with a perfect
fluid that yields acceleration at late times. We take $f(R,T)$ as $R$ + $8\pi
\mu T$. As in the $\Lambda$CDM model, we take the matter to consist of two
components, viz., $\Omega_m$ and $\Omega_{\mu}$ such that $\Omega_m$ +
$\Omega_{\mu}$=1. The parameter $\Omega_m$ is the matter density (baryons +
dark matter), and $\Omega_{\mu}$ is the density associated with the Ricci
scalar $R$ and the trace $T$ of the energy momentum tensor, which we shall call
dominant matter. We find that at present $\Omega_{\mu}$ is dominant over
$\Omega_m$, and that the two are in the ratio 3:1 to 3:2 according to the three
data sets: (i) 77 Hubble OHD data set (ii) 580 SNIa supernova distance modulus
data set and (iii) 66 pantheon SNIa data which include high red shift data in
the range $0\leq z\leq 2.36$. We have also calculated the pressures and
densities associated with the two matter densities, viz., $p_{\mu}$,
$\rho_{\mu}$, $p_m$ and $\rho_m$, respectively. It is also found that at
present, $\rho_{\mu}$ is greater than $\rho_m$. The negative dominant matter
pressure $p_{\mu}$ creates acceleration in the universe. Our deceleration and
snap parameters show a change from negative to positive, whereas the jerk
parameter is always positive. This means that the universe is at present
accelerating and in the past it was decelerating. State finder diagnostics
indicate that our model is at present a dark energy quintessence model. The
various other physical and geometric properties of the model are also
discussed.
