1.05 Problem solving with integers

We can use our knowledge of addition and subtraction on the number line to describe how the real world quantities change.

We can also talk about changes in the quantity we are representing using integer arithmetic. Given a starting temperature and some change in a certain direction, what is the final temperature? Given a starting balance and an ending balance of money in an account, what has been the amount and sign of the change?

Answers that are integers can be positive or negative:

Question: What is the balance of your account?

Answer: Balance =-\$31

When describing this situation in words, it is more natural to combine a positive number with a directional word:

Question: How much do you owe the bank?

Answer: I owe the bank \$31.

Let's look at the following examples, how we can apply the number line and arithmetic in solving problems with integers.

Examples

Example 1

The image below shows how the location of a miner traveling up and down a mine shaft relates to an integer on the number line:

a

If Nadia is initially 2\text{ m} above the surface, and descends 6\text{ m} in the elevator, what integer represents her end point?

Worked Solution

Create a strategy

The height of 2\text{ m} above the surface is represented by the integer 2 on the number line. Descending 6\text{ m} means we move in the negative direction on the number line by 6 units.

Apply the idea

\displaystyle \text{Location}	\displaystyle =	\displaystyle 2 - 6	Set up the equation
\displaystyle \text{}	\displaystyle =	\displaystyle -4	Evaluate

Nadia ends up below the surface, so the integer representing the end point is -4.

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b

If Nadia is at a location represented by the integer -4, and ascends 3\text{ m} , which option describes her new location?

A

7\text{ m} below the surface

B

1\text{ m} below the surface

C

3\text{ m} above the surface

D

7\text{ m} above the surface

Worked Solution

Create a strategy

Ascending 3\text{ m} in the elevator corresponds to moving in the positive direction on the number line by 3 units.

Apply the idea

\displaystyle \text{Location}	\displaystyle =	\displaystyle -4 + 3	Set up the equation
\displaystyle \text{ }	\displaystyle =	\displaystyle -1	Evaluate

Nadia's new location is 1\text{ m} below the surface. So, the correct answer is B.

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Example 2

Tara is waiting for the next flight to Los Angeles, which was scheduled to be in 64 minutes, but there is a 34-minute delay. She takes a nap and wakes up 23 minutes later. How much longer does Tara have to wait before the plane departs?

Worked Solution

Create a strategy

Add the delay and subtract her sleep time from her wait time.

Apply the idea

\displaystyle \text{Waiting time}	\displaystyle =	\displaystyle 64 + 34 - 23	Set up the equation
\displaystyle \text{ }	\displaystyle =	\displaystyle 98 - 23	Perform 64 + 34
\displaystyle \text{ }	\displaystyle =	\displaystyle 75 \text{ minutes}	Evaluate

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Example 3

A submarine dives 22 \text{ m} each minute for 16 minutes. What integer represents the total depth of the dive after 16 minutes?

Worked Solution

Create a strategy

Diving 22 meters is represented by the integer -22. To find the total depth of the dive, multiply the rate of descent by the time passed.

Apply the idea

\displaystyle \text{Depth of the dive}	\displaystyle =	\displaystyle -22 \times 16	Set up the equation
	\displaystyle =	\displaystyle -352 \text{ m}	Evaluate