Probably the easiest approach (and most likely the approach that the source of this question recommends) is the empirical rule for the normal distribution. It has several variations, but one succinct way to summarize it is in the following statements:
The normal distribution is symmetrical around its mean, so 50% is above the mean, and 50% is below it.
68% of the data in a normal distribution is within 1 standard deviation of the mean.
95% of the data in a normal distribution is within 2 standard deviations of the mean.
More could be said, but this is enough to get us started on our problem. Consider what we're given. The mean is 80, and the standard deviation is 5. This means that the standard deviations go above and below the mean in 5 point increments. So already we can see that 75 is 1 standard deviation below the mean.
So let's put some of the above information together. 68% is within 1 standard deviation of the mean, so 34% is between the mean and 1 standard deviation below, or between 75 and 80.
We're wanting the percentage of scores below 75. We know that 50% are below 80, and 34% are between 75 and 80. So the proportion below 75 must be the remaining 16% (50 - 34). So we can conclude that 16% scored below a 75.
