General Problem Solving: Learn It 2

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General Problem Solving: Learn It 2

Strategy Makes The Difference

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. Instead, problem-solving is best approached by first starting at the end: identifying exactly what you are looking for. From there, you then work backward, asking “what information and procedures will I need to find this?”

How To: Problem-Solving Process

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

Now that we have a process for problem-solving, let’s talk about the different approaches we can take to solve a problem.

Break It Down: Complex problems can often be daunting. But here’s a secret – they’re just a bunch of simple problems grouped together. Our first approach should always be to break down complex problems into smaller, more manageable parts.
- For example, if you are asked “”A farmer has chickens and cows in his farm. He counts [latex]50[/latex] heads and [latex]140[/latex] legs. How many chickens and cows does he have?” you may think this sounds super complex but if we break it down we see it isn’t so bad
  - All animals have [latex]1[/latex] head. So, the [latex]50[/latex] heads mean we have [latex]50[/latex] animals.
  - Chickens have [latex]2[/latex] legs, cows have [latex]4[/latex]. So if all [latex]50[/latex] animals were chickens, we would have [latex]100[/latex] legs.
  - But we have [latex]140[/latex] legs, which is [latex]40[/latex] more than [latex]100[/latex]. Since each cow has [latex]2[/latex] extra legs compared to a chicken, the [latex]40[/latex] extra legs mean we have [latex]20[/latex] cows ( [latex]40 ÷ 2 = 20[/latex]).
  - Finally, since we have [latex]50[/latex] animals in total, the remaining [latex]30[/latex] must be chickens.
Trial and Error: Sometimes, problems don’t have a clear path to the solution. In these cases, good old trial and error can come to our rescue.
- For example, if you are asked “What is the value of [latex]x[/latex] in the equation [latex]2^x = 32?[/latex]” it may be tempting to jump right into logarithms, but if you try a few values for [latex]x[/latex] first it may help to find the answer.
  - If [latex]x = 4[/latex], then [latex]2^x = 2^4 = 16[/latex]. Too small.
  - If [latex]x = 5[/latex], then [latex]2^x = 2^5 = 32[/latex]. Bingo!
Pattern Recognition: In mathematics, patterns are everywhere! Recognizing these patterns can make problem-solving a breeze.
- For example, if you are asked “What is the [latex]5[/latex]th term in the sequence: [latex]2, 4, 8, 16,[/latex]…?” looking for a pattern will help solve the problem.
  - Here, we can see that each term is twice the previous one. So, the [latex]5[/latex]th term is [latex]16*2 = 32[/latex].
Logical Reasoning: using logical reasoning can be a powerful problem-solving strategy. This involves creating a logical sequence of steps to solve the problem.
- For example, if you are asked “If all squares are rectangles, and all rectangles have four sides, do all squares have four sides?” you can use logic to find a solution
  - We can logically reason that since every square is a rectangle, and every rectangle has four sides, it must be that every square has four sides too.

Let’s try using our problem-solving process and the approaches we just learned to solve a few examples.

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.

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.
[latex]\dfrac{a}{b}\cdot\dfrac{c}{d} = \dfrac {ac}{bd}[/latex]
To divide fractions, multiply the first by the reciprocal of the second.
[latex]\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}[/latex]
To simplify fractions, find common factors in the numerator and denominator that cancel.
[latex]\dfrac{24}{32}=\dfrac{2\cdot2\cdot2\cdot3}{2\cdot2\cdot2\cdot2\cdot2}=\dfrac{3}{2\cdot2}=\dfrac{3}{4}[/latex]
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.
[latex]\dfrac{a}{b}\pm\dfrac{c}{d} = \dfrac{ad \pm bc}{bd}[/latex]

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.

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

We can infer a measurement by using scaling. If [latex]500[/latex] 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.

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

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 we have to make comparisons. Making comparisons mathematically often involves using ratios and proportions. In the next examples we will will use proportions and rates to make a comparison.

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

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

To make this decision, we must first decide what our basis for comparison will be. For the purposes of this example, we’ll focus on fuel and purchase costs, but environmental impacts and maintenance costs are other factors a buyer might consider.

It might be interesting to compare the cost of gas to run both cars for a year. To determine this, we will need to know the miles per gallon both cars get, as well as the number of miles we expect to drive in a year. From that information, we can find the number of gallons required from a year. Using the price of gas per gallon, we can find the running cost.

From Hyundai’s website, the 2013 Sonata will get [latex]24[/latex] miles per gallon (mpg) in the city, and [latex]35[/latex] mpg on the highway. The hybrid will get [latex]35[/latex] mpg in the city, and [latex]40[/latex] mpg on the highway. An average driver drives about [latex]12,000[/latex] miles a year.

Suppose that you expect to drive about [latex]75\%[/latex] of that in the city, so [latex]9,000[/latex] city miles a year, and [latex]3,000[/latex] highway miles a year. We can then find the number of gallons each car would require for the year.

Sonata: [latex]\displaystyle{9000}\text{ city miles}\cdot\frac{1\text{ gallon}}{24\text{ city miles}}+3000\text{ highway miles}\cdot\frac{1\text{ gallon}}{35\text{ highway miles}}=460.7\text{ gallons}[/latex]

Hybrid: [latex]\displaystyle{9000}\text{ city miles}\cdot\frac{1\text{ gallon}}{35\text{ city miles}}+3000\text{ highway miles}\cdot\frac{1\text{ gallon}}{40\text{ highway miles}}=332.1\text{ gallons}[/latex]

If gas in your area averages about [latex]$3.50[/latex] per gallon, we can use that to find the running cost:

Sonata: [latex]\displaystyle{460.7}\text{ gallons}\cdot\frac{\$3.50}{\text{gallon}}=\$1612.45[/latex]

Hybrid: [latex]\displaystyle{332.1}\text{ gallons}\cdot\frac{\$3.50}{\text{gallon}}=\$1162.35[/latex]

The hybrid will save [latex]$450.10[/latex] a year. The gas costs for the hybrid are about [latex]\displaystyle\frac{\$450.10}{\$1612.45} = 0.279 = 27.9\%[/latex] lower than the costs for the standard Sonata.

While both the absolute and relative comparisons are useful here, they still make it hard to answer the original question, since “is it worth it” implies there is some tradeoff for the gas savings. Indeed, the hybrid Sonata costs about [latex]$25,850[/latex], compared to the base model for the regular Sonata, at [latex]$20,895[/latex].

To better answer the “is it worth it” question, we might explore how long it will take the gas savings to make up for the additional initial cost. The hybrid costs [latex]$4,965[/latex] more. With gas savings of [latex]$451.10[/latex] a year, it will take about [latex]11[/latex] years for the gas savings to make up for the higher initial costs.

We can conclude that if you expect to own the car [latex]11[/latex] years, the hybrid is indeed worth it. If you plan to own the car for less than [latex]11[/latex] years, it may still be worth it, since the resale value of the hybrid may be higher, or for other non-monetary reasons. This is a case where math can help guide your decision, but it can’t make it for you.

To see this example worked out in a video, watch the following:

You can view the transcript for “Guiding decision using math: Sonata vs Hybrid” here (opens in new window).

Try using the problem-solving process and the approaches you learned to solve some questions on your own.