### Learning Outcomes

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

In this section we will bring together the mathematical tools we’ve reviewed, and use them to approach more complex problems. In many problems, it is tempting to take the given information, plug it into whatever formulas you have handy, and hope that the result is what you were supposed to find. Chances are, this approach has served you well in other math classes.

This approach does not work well with real life problems, however. Read on to learn how to use a generalized problem solving approach to solve a wide variety of quantitative problems, including how taxes are calculated.

## Problem Solving and Estimating

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

- Identify the question you’re trying to answer.
- Work backwards, identifying the information you will need and the relationships you will use to answer that question.
- Continue working backwards, creating a solution pathway.
- If you are missing necessary information, look it up or estimate it. If you have unnecessary information, ignore it.
- 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.

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. For the last

### 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.

### Try It

## Taxes

Governments collect taxes to pay for the services they provide. In the United States, federal income taxes help fund the military, the environmental protection agency, and thousands of other programs. Property taxes help fund schools. Gasoline taxes help pay for road improvements. While very few people enjoy paying taxes, they are necessary to pay for the services we all depend upon.

Taxes can be computed in a variety of ways, but are typically computed as a percentage of a sale, of one’s income, or of one’s assets.

### Example: Sales Tax

The sales tax rate in a city is 9.3%. How much sales tax will you pay on a $140 purchase?

When taxes are not given as a fixed percentage rate, sometimes it is necessary to calculate the **effective tax rate**:** **the equivalent percent rate of the tax paid out of the dollar amount the tax is based on.

### Example: Property Tax

Jaquim paid $3,200 in property taxes on his house valued at $215,000 last year. What is the effective tax rate?

Taxes are often referred to as progressive, regressive, or flat.

- A
**flat tax**, or proportional tax, charges a constant percentage rate. - A
**progressive tax**increases the percent rate as the base amount increases. - A
**regressive tax**decreases the percent rate as the base amount increases.

### Example: Federal Income Tax

The United States federal income tax on earned wages is an example of a progressive tax. People with a higher wage income pay a higher percent tax on their income.

For a single person in 2011, adjusted gross income (income after deductions) under $8,500 was taxed at 10%. Income over $8,500 but under $34,500 was taxed at 15%.

#### Earning $10,000

Stephen earned $10,000 in 2011. He would pay 10% on the portion of his income under $8,500, and 15% on the income over $8,500.

8500(0.10) = 850 10% of $8500

1500(0.15) = 225 15% of the remaining $1500 of income

Total tax: = $1075

What was Stephen’s effective tax rate?

#### Earning $30,000

D’Andrea earned $30,000 in 2011. She would also pay 10% on the portion of her income under $8,500, and 15% on the income over $8,500.

8500(0.10) = 850 10% of $8500

21500(0.15) = 3225 15% of the remaining $21500 of income

Total tax: = $4075

What was D’Andrea’s effective tax rate?

Notice that the effective rate has increased with income, showing this is a progressive tax.

### Example: Gasoline Tax

A gasoline tax is a flat tax when considered in terms of consumption. A tax of, say, $0.30 per gallon is proportional to the amount of gasoline purchased. Someone buying 10 gallons of gas at $4 a gallon would pay $3 in tax, which is $3/$40 = 7.5%. Someone buying 30 gallons of gas at $4 a gallon would pay $9 in tax, which is $9/$120 = 7.5%, the same effective rate.

However, in terms of income, a gasoline tax is often considered a regressive tax. It is likely that someone earning $30,000 a year and someone earning $60,000 a year will drive about the same amount. If both pay $60 in gasoline taxes over a year, the person earning $30,000 has paid 0.2% of their income, while the person earning $60,000 has paid 0.1% of their income in gas taxes.

### Try It

A sales tax is a fixed percentage tax on a person’s purchases. Is this a flat, progressive, or regressive tax?