1.15 Practical problems with integers
Valentina and Maria want to meet at the cinema to watch a movie. Valentina lives 3\text{ km} west of the cinema, and Maria lives 5\text{ km} east of the cinema.
In the number lines below, the distance between each tick is 1\text{ km}. Which of the following number lines show the location of Valentina and Maria in relation to the cinema?
Draw a new number line that shows the location of Valentina and Maria using integers, where 0 is the cinema, east is the positive direction and west is the negative direction of the number plane.
Calculate how far Maria would have to travel to get to Valentina’s house.
At the start of the month, Laura owes \$60 to the bank. A week later, her employer deposits \$130 into her account.
Draw a number line that shows the change in Laura’s account balance.
Suppose that next month Laura now has \$130 in her account, and she spends \$60 on rent. Draw a number line that shows the change in Laura’s account balance for this month.
The following diagram shows how the location of a miner traveling up and down a mine shaft relates to an integer on the number line:
State the integer that represents 3\text{ m} above the surface.
State the integer that represents 4\text{ m} below the surface.
If Nadia is initially 2\text{ m} above the surface, and descends 6\text{ m} in the elevator, state the integer that represents her new location.
If Nadia is at a location represented by the integer - 4, and ascends 3\text{ m}, describe his new location.
The planet Anarres experiences drastic changes in temperature over the course of the day. At sunrise, the temperature is measured to be - 250\degree \text{F}, and it warms to a temperature of - 110\degree \text{F} by midday.
Calculate the difference of temperature from sunrise to midday.
A substance experiences rapid changes in temperature over the course of an experiment to test its electrical conductivity. At the start of the experiment, the substance is measured to be 70\degree \text{F} and by the end of the experiment, its temperature is measured as - 50\degree \text{F}.
Calculate the difference of temperature from the start to the end of the experiment.
Maximilian is going to cook a pot roast. He turns the oven on to warm it up. He puts the pot roast in the oven 15 minutes later when it is at the right temperature. The pot roast is ready 120 minutes after Maximilian first turned the oven on.
Calculate how long the pot roast baked for.
William arrives to a meeting 15 minutes early, and he leaves at the end, 75 minutes after he got there.
Find how long the meeting went for.
A submarine is initially 45\text{ ft} below sea level. It then descends 30\text{ ft} straight down to its final position.
If sea level is represented by the integer 0, and upwards is the positive direction, find the integer that represents the submarine's final position.
Describe the final position of the submarine.
A bird soaring at a height of 25\text{ ft} above sea level sees a school of fish and dives down 32\text{ ft}.
Calculate how far below sea level the bird reaches.
Tara is waiting for the next flight to Los Angeles, which was scheduled to be in 36 minutes, but there is a 48-minute delay. She takes a nap, and wakes up 24 minutes later.
Calculate how much longer Tara has to wait before the plane departs.
A bus departs the depot with 8 passengers on board. At the first stop, 4 people got off and 6 people got on. At the second stop, a further 3 people got off, and 7 more got on.
Calculate the number of people are on the bus when it leaves the second stop.
Luigi enters an elevator at the 7th floor (the ground floor being floor 0). The elevator goes down 3 floors, then up 9 floors and finally it goes down 2 floors, where Luigi gets out.
State which floor Luigi ends up.
When Luigi gets off the elevator, how many floors from his starting point is he?
At the start of the week, Quiana had \$131 in her bank account. On Monday, she withdrew \$43. On Wednesday, she deposited \$11, and on Thursday, she deposited a further \$28. Finally, she withdrew \$25 on Friday.
Find Quiana's final bank balance at the end of the week.
Valentina dives off a platform that is 8\text{ m} above the water. She descends 14\text{ m} before swimming up to the surface.
If Valentina's initial position on the diving platform is represented by the directed number 8, find the directed number that represents her lowest position throughout the dive.
Now, state the greatest depth that Valentina reaches during the dive.
One evening in Alaska, the temperature fell to - 2 \degree \text{F}. By midday the next day, the temperature was forecast to be 30 \degree \text{F}.
Find the expected rise in temperature.
Liquid nitrogen can be used to make ice cream. The ice cream mixture starts at 79 \degree \text{F}, then rapidly cools to - 301 \degree \text{F} after liquid nitrogen is added to it.
State the directed number that represents this change in temperature.
Three contestants are heading into the final round of a game show. Eileen has 39 points and Roxanne has - 3 points.
If Oprah has 14 more points than Eileen, calculate how many points separate Roxanne and Oprah.
The motion of a robot to the left or right along a straight line is determined by whether the robot is facing left or right, and whether it is stepping forwards or backwards.
Use a number line where 1 unit represents 1 step and the positive direction is to the right to answer the following questions:
| Facing | Stepping | Moving |
|---|---|---|
| Right | Forward | Right |
| Right | Backward | Left |
| Left | Forward | Left |
| Left | Backward | Right |
For each of the following, find the integer that represents the final position:
The robot starts at 0, facing to the right, and takes 5 steps forward.
The robot starts at 0, facing to the left, and takes 13 steps forward.
The robot starts at 0, facing to the left, and takes 11 steps backwards.
Suppose the robot performs the following instructions in order:
Start at -5, facing right.
Take 7 steps forward.
Take 21 steps backward.
Turn to face left.
Take 22 steps backward.
State the integer that represents the robot's final position.