To determine which of the given fractions is the smallest, let's consider that integers [tex]\( p, q, r, s \)[/tex] have specific values such that [tex]\( 0 < p < q < r < s \)[/tex].
Given the specific values:
- [tex]\( p = 1 \)[/tex]
- [tex]\( q = 2 \)[/tex]
- [tex]\( r = 3 \)[/tex]
- [tex]\( s = 4 \)[/tex]
We will calculate each of the four expressions provided:
### Option (a): [tex]\(\frac{p+q}{r+s}\)[/tex]
[tex]\[
\frac{p+q}{r+s} = \frac{1 + 2}{3 + 4} = \frac{3}{7} \approx 0.4286
\][/tex]
### Option (b): [tex]\(\frac{p+s}{q+r}\)[/tex]
[tex]\[
\frac{p+s}{q+r} = \frac{1 + 4}{2 + 3} = \frac{5}{5} = 1.0
\][/tex]
### Option (c): [tex]\(\frac{q+s}{p+r}\)[/tex]
[tex]\[
\frac{q+s}{p+r} = \frac{2 + 4}{1 + 3} = \frac{6}{4} = 1.5
\][/tex]
### Option (d): [tex]\(\frac{r+s}{p+q}\)[/tex]
[tex]\[
\frac{r+s}{p+q} = \frac{3 + 4}{1 + 2} = \frac{7}{3} \approx 2.3333
\][/tex]
Now, we compare the calculated values:
[tex]\[
\frac{p+q}{r+s} \approx 0.4286
\][/tex]
[tex]\[
\frac{p+s}{q+r} = 1.0
\][/tex]
[tex]\[
\frac{q+s}{p+r} = 1.5
\][/tex]
[tex]\[
\frac{r+s}{p+q} \approx 2.3333
\][/tex]
Among the values we have calculated, the smallest value is approximately [tex]\( 0.4286 \)[/tex], corresponding to option (a):
[tex]\[
\frac{p+q}{r+s}
\][/tex]
Hence, the smallest fraction is:
[tex]\[
\boxed{\frac{p+q}{r+s}}
\][/tex]
