Problem solving ratios

To solve problems involving ratios and proportions, you can use the method of cross-multiplication. 

The procedure for cross-multiplication is straightforward and relatively easy. The challenge in solving word problems using ratios is in the set-up of the proportion. Take care to keep all terms in order. Remember that two ratios are being compared and that the order of the ratio set-up has meaning. 

When approaching a word problem involving ratios, in addition to a proper set-up, be clear on what the problem is asking for you to solve. Study the next example. 

You can use proportions to solve ratio problems that ask you to determine how much of something is needed based on how much you have of something else. 

Ability to solve problems involving ratio and direct and inverse proportions, exponents, and scientific notation... 

The radiation detector is the essential part of every spectrometer. It employs different elementary processes to transform an input, e.g., radiant power (W), irradiance (W cm ), or radiant energy (W s) into an output, e.g., an electric charge, a current, or a potential. This is either recorded directly as a. spectrum or processed after being digitized in order to solve analytical problems. The ratio between detector output and detector input is defined as the responsivity of the detector (lUPAC, 1992). Related to the responsivity is the quantum efficiency, which describes the number of elementary events produced by one incident photon. 

The ideal innovation concept is borrowed from the Theory of Inventive Problem Solving (TRIZ), which calls this perfect state the ideal final result. As a ratio, the value quotient approaches infinity, or a state where all benefits of a solution are achieved at zero cost and zero harm. In TRIZ terminology, this is called working backward from perfect, which forces the innovator to break through his or her psychological inertia into new, less limiting domains of thinking. 

We have set up the dimensional equation with vertical lines to separate each ratio, and these lines retain the same meaning as an X or multiplication sign placed between each ratio. The dimensional equation will be retained in this form throughout most of this text to enable you to keep clearly in mind the significance of units in problem solving. It is recommended that you always write down the units next to the associated numerical value (unless the calculation is very simple) until you become quite familiar with the use of units and dimensions and can carry them in your head. 

Mole-mole problems are sort of like introductory, or skill-building, problems that will help you practice using the molar ratios given by balanced chemical reactions. The harder stoichiometry problems, which we will begin in the next lesson, all make use of mole-mole problems as a step in the problem-solving process. This lesson will give you an opportunity to become comfortable with the molar ratio without worrying about more complex problems at the same time. 

Problem-Solving Tip Use the Mole Ratio in Calculations with... 

To convert masses to moles or vice versa, we use the molar mass of the substance. Molar mass has the same numeric value as the number of atomic mass units in a formula unit, but it is expressed in units of grams per mole. For example, the molar mass of water is 18.0 g/mol because the formula mass of water is 18.0 amu/molecule. Because molar mass is a ratio, it can be used as a factor in problem solving. 

In Chapters 3 and 4, we encountered many reactions that involved gases as reactants (e.g., combustion with O2) or as products (e.g., a metal displacing H2 from acid). From the balanced equation, we used stoichiometrically equivalent molar ratios to calculate the amounts (moles) of reactants and products and converted these quantities into masses, numbers of molecules, or solution volumes (see Figure 3.10). Figure 5.11 shows how you can expand your problem-solving repertoire by using the ideal gas law to convert between gas variables (F, T, and V) and amounts (moles) of gaseous reactants and products. In effect, you combine a gas law problem with a stoichiometry problem it is more realistic to measure the volume, pressure, and temperature of a gas than its mass. 

Some authors have sought to circumvent the problems of ratio correlation by txansforming their data into logarithmic form. Unfortunately this also does not solve any problems for the problems are preserved even as log-ratios (see Kenny, 1982 Rollinson and Roberts, 1986). 

We have 4 equations and 2 unknowns at this point. This is where the problem solving requires some ingenuity. First, lets see what happens when we combine this information with the splitting ratio and constant concentration at the splitter ... 

Normally, the observations of nature that are used to develop theories and models need to be quantitative—that is, they assess what is being observed with some level of numerical detail. The need for numerical observations throughout the development of chemistry (and other sciences) has given rise to systematic ways to communicate this information. A number alone is not sufficient to impart all the meaning of a measurement experimental observations are expected to include units for the observations. An important skill in the study of chemistry or other quantitative sciences is the ability to manipulate numerical information—including the units attached to it. The use of ratios to convert between a measurement in one unit and desired information in another related unit represents a core skill for problem solving in chemistry. The method of dimensional analysis, sometimes called the factor-label method, provides one common way to carry out these transformations in chemistry problems. 

The organization maintains safe staffing through such activities as crosstraining, adequate volume ratios, appropriate skill mix, and limited work hours. Education and career development plans foster core competencies of continuous performance improvement, direct and open communication, innovation, and problem solving. 

To construct a pH problem-solving method we proceed just as with the ratios and find the material balance expression for the fractions just derived from the equilibrium condition equation. From the derivation of equation (3-2) we have (C is the total of HX and X )... 

Species fractions, a s, and ratios between the members of any pair of species in equilibrium in mononuclear polyprotic systems have been shown to depend only upon H and the K values. Uses of plots of these functions have been illustrated in problem solving. The n function related to a s is useful because it can be determined experimentally from analytical concentrations and a pH measurement. It also provides a fundamentally complete and rigorous approach to pH calculations. 

Dimensional analysis Dimensional analysis is a method of problem solving that focuses on the units that are used to describe matter. Dimensional analysis often uses conversion factors. A conversion factor is a ratio of equivalent values used to express the same quantity in different units. A conversion factor is always equal to 1. Multiplying a quantity by a conversion factor does not change its value—because it is the same as multiplying by 1— but the units of the quantity can change. 

This is an NLR problem. Solve this by minimizing the sum of squares of residuals using Solver. After you have determined the optimal values for Kj and Ajj, calculate numerically (using second-order correct formulas) the derivatives Zj and Z2 at each data point, form the G matrix, calculate the parameter standard deviations, and calculate the t-ratios for each parameter. 

Density and Percent Composition Their Use in Problem Solving—Mass and volume are extensive properties they depend on the amount of matter in a sample. Density, the ratio of the mass of a sample to its volume, is an intensive property, a property independent of the amount of matter sampled. Density is used as a conversion factor in a variety of calculations. 

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