Percent Composition. Amount of A Total Quantity of Mixture

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1 The Mathematics 11 Competency Test Percent Composition When two or more types of things (ingredients) are mixed together, it is common to speak of the individual ingredients as making up a certain percentage of the whole. The formal relationship can be written percent concentration of ingredient A Amount of A Total Quantity of Mixture 100 We have shown the multiplication by 100 necessary to get the actual percent value. When you do calculations involving this percent, you ignore the 100 and work directly with the decimal or fractional equivalent of the percent. Note particularly that the quantity forming the denominator on the right-hand side is the entire quantity of material present everything including the amount of ingredient A. Problems involving percent composition can be quite complicated, and a few of the more involved types of percent problems are illustrated in the section on mixtures word problems later in these notes. In this section we ll focus more on directly applying the basic formula given above. Example 1: A storage tank contains 750 kg of a solution which is 8.5% salt by weight. If 380 kg of pure water is added to the tank, what will be the percent salt of the resulting solution? We are asked to calculate the percent salt in the. Applying the property shown above, we can write The amounts and quantities here will have units of kilograms. Now, the total number of kilograms of will be just 750 kg kg 1130 kg since 380 kg of water was added to the existing 750 kg of salt solution to form the. The kilograms of salt in the is just the kilograms of salt in the initial kg salt kg solution x percent salt in solution (750 kg) (0.085) kg. The 380 kilograms of pure water added to the original solution contains no salt, and so the only salt in the must come from the initial solution. Thus David W. Sabo (2003) Percent Composition Page 1 of 5

2 percent concentration of salt in the kg salt 1130 kg solution % rounded to four decimal places. (This is a plausible answer, since we expect the percent salt to decrease when water is added to dilute the original solution, so we are expecting an answer somewhat smaller than 8.5%.) Example 2: Suppose a tank contains 1200 kg of a solution which is 6.39% salt by weight. If another 50 kg of salt are dissolved in this tank, what will be the percent salt of the resulting solution? This problem is very similar to the previous one. Again In this situation, 50 kg of salt are added to 1200 kg of an initial salt solution, so that the total quantity of is 1200 kg + 50 kg 1250 kg Now, the salt in the is the 50 kg of salt which was added, plus the number of kilograms of salt already in the original solution. Since the original solution has a mass of 1200 kg and is 6.3% salt, then kg of salt in the percent salt in the kg solution original solution original soltuion ( kg )( ) So, the total kilograms of salt in the is So, finally, kg + 50 kg kg kg percent concentration of kg salt in the 1250 kg % David W. Sabo (2003) Percent Composition Page 2 of 5

3 (unrounded). After addition of the 50 kg of salt to the initial solution, the concentration will have increased to % by weight. Example 3: A tank contains 2500 kg of a solution which is 7.6% salt. Over a period of time, 268 kilograms of water evaporate from the tank. What is the percent salt of the solution in the tank now? We begin with the same general formula as in the previous two examples: In this case, the final quantity of solution is 2500 kg 268 kg 2232 kg because the evaporation of the 268 kg of water actually reduces the mass of the remaining solution by 268 kg. However, only water evaporates, so the kilograms of salt remaining in the 2232 kg of solution will be the same as the kilograms of salt in the original kilograms of salt remaining kilograms of salt originally present percent salt in ( kilograms of solution) original solution So ( kg ) ( ) kg percent concentration of 190 kg salt in the 2232 kg rounded to four decimal places % Thus, due to the evaporation of the 268 kg of water, the concentration of salt in this solution increases to %. David W. Sabo (2003) Percent Composition Page 3 of 5

4 Example 4: A tank contains 4300 kg of a solution which is 5.8% salt. To this tank is added 2200 kg of a solution which is 9.3% salt. What is the percent salt in the resulting solution? Once again, we use the general principle: In this case, the kilograms of solution of the final mixture is just the sum of the number of kilograms of each of the two solutions mixed: 4300 kg kg 6500 kg However, in this example, salt is contributed to the from both of the solutions which are mixed together. The first component is 4300 kg of a 5.8% salt solution, so it contributes (4300 kg) x (0.058) kg of salt. The second component is 2200 kg of a 9.3% salt solution, so it contributes (2200 kg) x (0.093) kg of salt. Thus, the total number of kilograms of salt in the is So kg kg 454 kg percent concentration of 454 kg salt in the 6500 kg % rounded to four decimal places. Combining these two solutions in the amounts stated results in a which is % salt. Note that in Example 4, we could not get the percent salt for a combination of two solutions by adding the original percents (5.8% + 9.3% 15.1% %) or even by averaging ( 5.8% + 9.3% 7.55% % ) the percent concentrations of the original two solutions. 2 This is because it is quantities of material (the kilograms of salt, and the kilograms of solution) which can be added. You can t simply combine percent concentrations by themselves, since percent numbers have no real meaning until they are related to actual amounts of mixture or solutions. David W. Sabo (2003) Percent Composition Page 4 of 5

5 Example 5: In a survey, it was found that 512 of the people contacted wore eyeglasses, and the remaining 428 of the people questioned did not wear eyeglasses. What percentage of these people questioned wore eyeglasses? This is a very simple problem. The general principle here will be percent of people who wear eyeglasses number of people who wear eyeglasses total number of people % rounded to two decimal places. David W. Sabo (2003) Percent Composition Page 5 of 5