AIgebra is the language of quantitative science. As such, its methods and techniques can be found in most of the examples in this volume. The problems selected for this chapter are those that do not also draw heavily on other mathematical areas. Several use the distance, time, and rate relationship, and others use direct and inverse variation. Some approximation techniques that are frequently used to solve otherwise intractable problems are also included.

During 1982, the planets Jupiter and Saturn were in conjunction, so that they appeared very close to each other in the night sky. In the problem that follows, we see how frequently such an event happens.

**PROBLEM 1.** The planets Earth, Jupiter, Saturn, and Uranus revolve around the Sun approximately once every 1,12, 30, and 84 years respectively.

**a.** How often will Jupiter and Saturn appear close to each other in the night sky as seen from Earth?

**Solution: **The time required must be a multiple of both 12 years and 30 years. This event will recur at intervals of the least common multiple of 12 and 30, or 60 years.

**b. **How often will Jupiter, Saturn, and Uranus all appear in the same area in the night sky as seen from Earth?

**Solution: **Now we need the least common multiple of 12, 30, and 84, which is 420 years.

In addition to the electromagnetic radiation that we know as heat and light, the Sun continuously sends out charged particles known as the solar plasma (see also Chapter 10, Problem 1). From time to time, there is a strong burst of highly energetic particles called a solar flare from a small source in the Sun's atmosphere.