Integers at work

Now put the rules to work on real situations, where the signs carry real meaning.

An exam has 50 multiple choice questions. Five marks are given for every correct answer and two negative marks for every wrong one. Mala had 30 correct and 20 wrong. What are her total marks?

Use positive and negative integers. The mark for each correct answer is +5 and for each wrong answer is −2. So marks for 30 correct answers = 30 × 5, and marks for 20 wrong answers = 20 × (−2).

The expression is 30 × 5 + 20 × (−2) = 150 + (−40) = 110. Mala got 110 marks. (Ask yourself: what are the maximum and minimum possible marks in this exam?)

Next, an elevator in a mining shaft. Positions ABOVE the ground are positive integers and positions BELOW the ground are negative. The elevator moves 3 metres per minute. If it descends from ground level, where is it after one hour?

There are two ways, and both are worth having. METHOD 1 uses subtraction: in one hour it moves 60 × 3 = 180 metres, and since it descends from 0 we subtract: 0 − 180 = −180. It is 180 metres below ground.

METHOD 2 puts the direction INTO the number: let the speed and direction together be −3 metres per minute while descending. Then in 60 minutes it goes 60 × (−3) = −180. Same answer, and now the method scales. Starting 15 m above ground and descending for 45 minutes: Ending position = 15 + (45 × (−3)) = 15 + (−135) = −120. The elevator is 120 metres below the ground.