# Lesson 14

Solving Problems with Rational Numbers

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

### 14.1: Which One Doesn’t Belong: Equations

Which equation doesn’t belong?

$$\frac12 x = \text-50$$

$$\text-60t = 30$$

$$x + 90 = \text-100$$

$$\text-0.01 = \text-0.001x$$

### 14.2: 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.

1. 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   $$770 + (0) (\text-14)$$   770
1 -14 $$770 + (1) (\text-14)$$ 756
5 -70
10
2. 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   $$770 + (1) (\text-14)$$   756
0 0 $$770 + (0) (\text-14)$$ 770
-1 14 $$770 + (\text-1) (\text-14)$$ 784
-2 28
-3
-4
-5
3. If the sensor started working 15 minutes into the tank draining, how much was in the tank to begin with?

### 14.3: 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.

1. What is the amount due this month?
2. How many kilowatt-hours did they use?
3. How many kilowatt-hours did they generate that they didn't use themselves?

1. Find the value of the expression without a calculator.

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

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

### Summary

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

In general: $$a - b = a + \text- b$$

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

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.

### Glossary Entries

• rational number

A rational number is a fraction or the opposite of a fraction.

For example, 8 and -8 are rational numbers because they can be written as $$\frac81$$ and $$\text-\frac81$$.

Also, 0.75 and -0.75 are rational numbers because they can be written as $$\frac{75}{100}$$ and $$\text-\frac{75}{100}$$.