# I-2. Problem-Solving Skills

### Introduction

There are no magic recipes that work for solving all problems. The hints and strategies described below provide a starting point to improve your problem-solving skills. The best way to develop and refine your problem-solving skills is simply to practice solving a variety of problems.

### General Strategies

1. Identify what the problem is asking.

2. Identify what information is given. Not everything will be spelled-out in a problem. You might have to draw on your knowledge or reference sources, such as textbooks, for equations or conversion factors. There can also be extra information given that is not needed to solve the problem.

3. Look for paths to get from what is given to what is asked. If a problem-solving strategy is not obvious, and it usually isn't, start working forward from the given information, and work backwards from what the problem asks. If possible, break the problem down into smaller parts.

4. Evaluate the reasonableness of a result. If the result is not reasonable, reevaluate your approach and look for possible errors.

### Other Hints

Read the problem carefully. You might have to read the problem several times to understand what the problem is asking, and what information it provides. Dissect the problem as much as possible.

Use dimensional analysis. Check the units as you are working through a problem, and be sure that the units of your result matches the units of the requested...

### Example 1

What profit can you make if you sell 300 0.5-kg fruit shakes for \$1.00 each? Each shake is one-third bananas, strawberries, and ice. 50 kg of bananas cost \$44.00, 50 kg of strawberries cost \$150.00, and 50 kg of ice costs \$10.00.

1. What is the problem asking? The problem is asking what profit you can make. What is profit? Profit is the your gross income minus your cost. To solve this problem find your gross income and your costs.

2. What are you given? You are given how many fruit shakes you can sell, their price, their composition, and the cost of the raw materials.

3. To solve this problem you must calculate your gross income and your costs. What information in 2. gives you this information? Your income is the number of shakes you can sell x the price: 300 x \$1.00 = \$300.00. Your costs are \$44.00 + \$150.00 + \$10.00 = \$204.00. Your profit is gross income - costs: \$300.00 - \$204.00 = \$96.00.

4. Is this a reasonable result? For this simple problem we can expect the profit to be less than our income and greater than our costs, and the units should be in dollars. So yes this result is reasonable.

### Example 2

What is the pH of the resulting solution when 0.100 L of 0.500 M NaOH is added to 0.150 L of 0.400 M HCl?

1. What is the problem asking? The problem is asking for the pH of a solution. What is pH? If you don't know the first thing is to look up the definition of pH. pH is the -log[H+]. To solve this problem you must find [H+], the hydrogen ion concentration in moles/liter, after the two solutions are mixed.

2. What are you given? You are given the amount and composition of the two solutions that will be mixed. What is in each solution? What will happen when these solutions are mixed? At this point you must rely on your knowledge of chemistry (or a reference source) to predict what will happen when the two solutions are mixed. In this case, HCl is an acid and NaOH is a base, which will undergo a neutralization reaction when they are mixed.

3. To solve this problem you must determine [H+] in moles/liter. You are given starting information and you know that a reaction will take place. You can break this problem into several parts. Determine what you will have after the reaction takes place, find the number of moles of H+ in the resulting solution, find the volume of the resulting solution, determine [H+], then take the negative log to find pH.

moles of OH- = (0.100 L)(0.500 M) = 0.0500 moles
moles of H+ = (0.150 L)(0.400 M) = 0.0600 moles
The OH- and H+ will react to leave 0.0100 moles of H+.

The volume of the resulting solution is the volume of the two original solutions:
0.100 L + 0.150 L = 0.250 L

Now find [H+]: [H+] = (0.0100 moles)/(0.250 L) = 0.0400 M

The pH is therefore -log(0.0400) = 1.398

4. Is this a reasonable result? The pH range of most solutions is from 0 to 14, and acidic solutions will have a pH less than 7. So yes, this result is reasonable.

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