## Problem Solving and Estimating

### Learning Outcomes

• Identify and apply a solution pathway for multi-step problems

Problem solving is best approached by first starting at the end: identifying exactly what you are looking for. From there, you then work backwards, asking “what information and procedures will I need to find this?” Very few interesting questions can be answered in one mathematical step; often times you will need to chain together a solution pathway, a series of steps that will allow you to answer the question.

## Problem Solving Process

1. Identify the question you’re trying to answer.
2. Work backwards, identifying the information you will need and the relationships you will use to answer that question.
3. Continue working backwards, creating a solution pathway.
4. If you are missing necessary information, look it up or estimate it. If you have unnecessary information, ignore it.
5. Solve the problem, following your solution pathway.

In most problems we work, we will be approximating a solution, because we will not have perfect information. We will begin with a few examples where we will be able to approximate the solution using basic knowledge from our lives.

### Recall: operations on Fractions

When simplifying algebraic expressions, we may sometimes need to add, subtract, simplify, multiply, or divide fractions. It is important to be able to do these operations on the fractions without converting them to decimals.

• To multiply fractions, multiply the numerators and place them over the product of the denominators.
•  $\dfrac{a}{b}\cdot\dfrac{c}{d} = \dfrac {ac}{bd}$
• To divide fractions, multiply the first by the reciprocal of the second.
•  $\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}$
• To simplify fractions, find common factors in the numerator and denominator that cancel.
•  $\dfrac{24}{32}=\dfrac{2\cdot2\cdot2\cdot3}{2\cdot2\cdot2\cdot2\cdot2}=\dfrac{3}{2\cdot2}=\dfrac{3}{4}$
• To add or subtract fractions, first rewrite each fraction as an equivalent fraction such that each has a common denominator, then add or subtract the numerators and place the result over the common denominator.
•  $\dfrac{a}{b}\pm\dfrac{c}{d} = \dfrac{ad \pm bc}{bd}$

In the first example, we will need to think about time scales, we are asked to find how many times a heart beats in a year, but usually we measure heart rate in beats per minute.

### Examples

How many times does your heart beat in a year?

The technique that helped us solve the last problem was to get the number of heartbeats in a minute translated into the number of heartbeats in a year. Converting units from one to another, like minutes to years is a common tool for solving problems.

In the next example, we show how to infer the thickness of something too small to measure with every-day tools. Before precision instruments were widely available, scientists and engineers had to get creative with ways to measure either very small or very large things. Imagine how early astronomers inferred the distance to stars, or the circumference of the earth.

### Example

How thick is a single sheet of paper? How much does it weigh?

The first two example questions in this set are examined in more detail here.

We can infer a measurement by using scaling.  If 500 sheets of paper is two inches thick, then we could use proportional reasoning to infer the thickness of one sheet of paper.

In the next example, we use proportional reasoning to determine how many calories are in a mini muffin when you are given the amount of calories for a regular sized muffin.

### Example

A recipe for zucchini muffins states that it yields 12 muffins, with 250 calories per muffin. You instead decide to make mini-muffins, and the recipe yields 20 muffins. If you eat 4, how many calories will you consume?

View the following video for more about the zucchini muffin problem.

We have found that ratios are very helpful when we know some information but it is not in the right units, or parts to answer our question. Making comparisons mathematically often involves using ratios and proportions. In the next examples we will

### Example

You need to replace the boards on your deck. About how much will the materials cost?

This example is worked through in the following video.

### Example

Is it worth buying a Hyundai Sonata hybrid instead the regular Hyundai Sonata?

This question pulls together all the skills discussed previously on this page, as the video demonstration illustrates.