Estimating answers

When we work problems, we assume that the calculator is working properly the numbers were all put into the calculator and that we keyed them in correctly. Suppose that one or more of these suppositions is incorrect will the incorrect answer be accepted A very important skill is to determine, by visual inspection, an approximate answer. Especially important is the correct order of magnitude, represented by the location of the decimal point (or the power of 10). Sometimes the answer may contain the correct digits, but the decimal point is in the wrong location. A little practice to learn how to estimate answers and a few seconds used to do so when working problems can boost accuracy (and your grades) significantly. 

EXAMPLE 7 Consider the multiplication 122g x 0.0518 = 6.32g. Visual inspection shows that 0.0518 is a little more than 1 /20th (0.05) the value of l/20th of 122 is a little more than 6. This relationship tells us that the answer should be a little more than 6 g, which it is. Suppose that the answer were given as 63.2 g this answer is not logical because it is much larger than the estimated answer of somewhere around 6 g. 

There is considerable empirical evidence to suggest that due to a variety of heuristics employed in human thought processes cognitive biases may result in "best estimates" that are not actually very good. Even if all that is needed is a "best estimate" answer the quality of that answer may be improved by an analysis that incorporates and deals with the full uncertainty. 

A friend once remarked cynically that calculators let you get the wrong answer more quickly. He was implying that unless you have the correct strategy for solving a problem and have punched in the correct numbers, the answer will be incorrect. If you learn to estimate answers, however, you wiU be able to check whether the answers to your calculations are reasonable. 

Estimating Answers 28 The Importance of Practice 31 The Features of This Book 32 How to Take a Test 71 Problem Solving 92 Design an Experiment 110 Analyzing Chemical Reactions 146 Using Enthalpy as a Guide 181 Calculations Involving Many Variables 410 What Now 1081... 

The number 72 is larger than the number 22, so they meet that standard. Beyond that, checking the answer for reasonableness involves some mental arithmetic. The numerator is 72 — 32, which is 40. The denominator is about 2. Dividing 40 by 2 gives 20, which is close to the calculated answer, 22. You will find more about estimating answers in Appendix I, Part D. 

The six-step strategy for solving problems is applied in every example in this chapter to which it can be applied. We will use it consistently throughout the book. We omit the last step later, since the check is largely a mental operation. You should never omit the last step. Always check your answers, both numbers and units, to be sure they are reasonable. We strongly recommend that you read the section in Appendix I.D on estimating answers for this purpose. 

The units cancel properly to give the desired answer in °C. The answer is correctly given to three significant digits. The mass of the solute is approximately one—fourth its molar mass and would give 0.75 mol of ions in the 1 kg of solvent, so the estimated answer of 0.75 x (—1.86°C) = 1.4°C supports our computation. 

---