Before it started to rain, Amnar cut 16

To determine which statement is true, we need to analyze how much grass Amnar and Aayush cut.

Amnar cut \(\frac{16}{x}\) of the yard, and Aayush cut \(\frac{13}{x}\) of the yard, where \(x\) represents the total area of the yard.

To find out what they cut together, we can add their portions:

\[
\frac{16}{x} + \frac{13}{x} = \frac{29}{x}
\]

Now let's evaluate the options:

1. **The boys cut almost the whole yard.** (This could be true if the total yard is less than 29, but not definitive without specific values.)
2. **Together the boys cut three-fourths of the yard.** (This means they would have to cut \(\frac{3}{4}\) of the yard, or \(0.75\) of it. \(\frac{29}{x} = 0.75\) could be true for certain values of \(x\).)
3. **The boys cut exactly half of the yard.** (This means \(\frac{29}{x} = 0.5\), which is not possible since \(29\) is greater than half of any reasonable yard size.)
4. **Together the boys did not cut half of the yard.** (This statement implies that \(\frac{29}{x} < 0.5\), which is also not possible as \(29\) is a sizeable portion.)

Without specific values for the yard size \(x\), we can't definitively determine the truth of the first three options. However, the last statement indicates they did cut more than half of the yard if \(x\) is reasonably sized.

Based on this reasoning, **the statement "Together the boys did not cut half of the yard" is true.**