19. The letters $p, q, r, s$ and $t$ represent different positive single-digit numbers such that $p-q=r$ and $r-s=t$.
How many different values could $t$ have?
A 6
B 5
C 4
D 3
E 2
Solution
A
Because $r=p-q$, we have $t=r-s=(p-q)-s=p-(q+s)$.
It follows that the maximum value of $t$ occurs when $p$ takes its maximum value, and $q+s$ takes its minimum value.

The maximum value of $p$ is 9 .
Because $q$ and $s$ have different values, the minimum value of $q+s$ is $1+2=3$.
Therefore the maximum possible value of $t$ is $9-3=6$.
The following table shows that $t$ can take all the positive integer values from 1 to 6 .
\begin{tabular}{ccccc}
\hline$p$ & $q$ & $r$ & $s$ & $t$ \\
\hline 9 & 1 & 8 & 2 & 6 \\
9 & 1 & 8 & 3 & 5 \\
9 & 2 & 7 & 3 & 4 \\
9 & 2 & 7 & 4 & 3 \\
9 & 3 & 6 & 4 & 2 \\
9 & 3 & 6 & 5 & 1 \\
\hline
\end{tabular}

Therefore the number of values that $t$ could take is 6 .
For INVESTIGATION
19.1 In how many different ways is it possible to choose different single-digit values for $p, q, r$ and $s$ so that $t=1$ ?
19.2 The letters $p, q, r$ and $s$ represent different positive single-digit numbers such that $p-q=r$ and $q-r=s$. How many different values could $s$ have?
19.3 The letters $p, q, r$ and $s$ represent different positive single-digit numbers such that $p-q=r$ and $p-r=s$. How many different values could $s$ have?