Lesson 14Solving Problems with Rational Numbers

Learning Goal

Let’s use all four operations with signed numbers to solve problems.

Learning Targets

I can represent situations with expressions that include rational numbers.
I can solve problems using the four operations with rational numbers.

Warm Up: Which One Doesn’t Belong: Equations

Which equation doesn’t belong?

$\frac{1}{2} x = - 50$
$x + 90 = - 100$
$- 60 t = 30$
$- 0.01 = - 0.001 x$

Activity 1: Draining and Filling a Tank

A tank of water is being drained. Due to a problem, the sensor does not start working until some time into the draining process. The sensor starts its recording at time zero when there are 770 liters in the tank.

Given that the drain empties the tank at a constant rate of 14 liters per minute, complete the table:
time after
sensor starts
(minutes)
change
in water
(liters)
expression
water in
the tank
(liters)
$0$
$0$
$770 + (0) (- 14)$
$770$
$1$
$- 14$
$770 + (1) (- 14)$
$756$
$5$
$- 70$
$10$
Later, someone wants to use the data to find out how long the tank had been draining before the sensor started. Complete this table:
time after
sensor starts
(minutes)
change
in water
(liters)
expression
water in
the tank
(liters)
$1$
$- 14$
$770 + (1) (- 14)$
$756$
$0$
$0$
$770 + (0) (- 14)$
$770$
$- 1$
$14$
$770 + (- 1) (- 14)$
$784$
$- 2$
$28$
$- 3$
$- 4$
$- 5$
If the sensor started working 15 minutes into the tank draining, how much was in the tank to begin with?

time after sensor starts (minutes)	change in water (liters)	expression	water in the tank (liters)
$0$	$0$	$770 + (0) (- 14)$	$770$
$1$	$- 14$	$770 + (1) (- 14)$	$756$
$5$	$- 70$
$10$

time after sensor starts (minutes)	change in water (liters)	expression	water in the tank (liters)
$1$	$- 14$	$770 + (1) (- 14)$	$756$
$0$	$0$	$770 + (0) (- 14)$	$770$
$- 1$	$14$	$770 + (- 1) (- 14)$	$784$
$- 2$	$28$
$- 3$
$- 4$
$- 5$

Activity 2: Buying and Selling Power

A utility company charges $0.12 per kilowatt-hour for energy a customer uses. They give a credit of $0.025 for every kilowatt-hour of electricity a customer with a solar panel generates that they don’t use themselves.

A customer has a charge of $82.04 and a credit of -$4.10 on this month’s bill.

What is the amount due this month?
How many kilowatt-hours did they use?
How many kilowatt-hours did they generate that they didn’t use themselves?

Are you ready for more?

Problem 1

Find the value of the expression without a calculator.

$(2) (- 30) + (- 3) (- 20) + (- 6) (- 10) - (2) (3) (10)$

Problem 2

Write an expression that uses addition, subtraction, multiplication, and division and only negative numbers that has the same value.

Lesson Summary

We can apply the rules for arithmetic with rational numbers to solve problems

In general:

$a - b = a + (- b)$

If $a - b = x$ , then $x + b = a$ . We can add $- b$ to both sides of this second equation to get that $x = a + (- b)$

Remember: the distance between two numbers is independent of the order, but the difference depends on the order.

And when multiplying or dividing:

The sign of a positive number multiplied or divided by a negative number is always negative.
The sign of a negative number multiplied or divided by a positive number is always negative.
The sign of a negative number multiplied or divided by a negative number is always positive.