2.4: Word problems for adding and subtracting with negative integers

Notes

Real-world interpretations of negative integers

We can work with negative integers more easily if we can interpret them using meaningful real-world phenomena, such as temperatures below zero, building floors below ground, or amounts of money owed. For example, –4 could represent a fridge temperature of –4ºC, a parking level 4 floors below ground, or a \$4 debt. Some other ideas for negative amounts that lend themselves to story problems are:

• Hot-air balloons with puffs of gas (positive) and sand bags (negative). Adding a puff of gas or taking away a sand bag makes the balloon go up by 1 metre, while taking away a puff of gas or adding a sand bag makes the balloon go down by 1 metre.
• Magic soup with hot cubes (positive) and cold cubes (negative). Adding a hot cube or taking away a cold cube makes the soup temperature go up by 1 degree, while taking away a hot cube or adding a cold cube makes the soup temperature go down by 1 degree.
• People with happy thoughts (positive) and sad thoughts (negative). Adding a happy thought or taking away a sad thought makes the person’s happiness go up by 1 level, while taking away a happy thought or adding a sad thought makes the person’s happiness go down by 1 level.
Adding a negative integer to a positive integer

Add To Problem: Add a negative amount of an object to a positive amount of that same object. For example, add a \$4 debt to a \$7 credit. Recall from 3b: Adding and subtracting with negative integers that we use the subtraction algorithm to calculate A + (–B) by rewriting it as A – B (if A > B) or –(B – A) (if A < B).

Here’s an example of an “add to” problem involving a negative integer:

• Arithmetic problem: Solve 7 + (–4).
• Word problem: Keenan has \$7 in his pocket but he owes his friend \$4. How much money does Keenan have to spend?
• 7 + (–4) = 3, so Keenan has \$3 to spend.
Subtracting a negative integer from a positive integer

Take From Problem: Taking away something negative has the same effect as adding something positive. For example, taking away a \$4 debt has the same effect as adding a \$4 credit. Recall from 3b: Adding and subtracting with negative integers that we use the addition algorithm to calculate A – (–B) by rewriting it as A + B.

Here’s an example of a “take from” problem involving a negative integer:

• Arithmetic problem: Solve 7 – (–4).
• Word problem: A hot-air balloon with 4 puffs of gas and 4 sand bags is balanced (neither going up nor going down). What happens to the balloon if we add 7 more puffs of gas and take away the 4 sand bags?
• 7 – (–4) = 11, so the ballon goes up by 11 metres.

Compare Problem: By comparing a positive integer with a negative integer, we can see that the difference between A and –B is the same as the sum A + B. For example:

• Arithmetic problem: Solve 7 – (–4).
• Word problem: At 12 noon it is 7ºC in Vancouver and –4ºC in Whistler. How much warmer is it in Vancouver than in Whistler?
• 7 – (–4) = 11, so it is 11ºC warmer in Vancouver than in Whistler.

The video below works through some examples of word problems for adding and subtracting with negative integers.

Practice Exercises

Do the following exercises to practice matching negative integer addition/subtraction problems and word problems.