Question

Treat the percents given in this exercise as exact numbers, and work to three significant digits. A certain paint mixture weighing 315 lb contains \(20 \%\) solids suspended in water. How many pounds of water must be allowed to evaporate to raise the concentration of solids to \(25 \% ?\)

Short answer

63.0 lb of water must be allowed to evaporate.

Step-by-step solution

Determine the original amount of solids

To find the original amount of solids in the paint mixture, calculate 20% of the total weight. Multiply 315 lb by 20% (or 0.20) to get the weight of the solids.

Calculate the new total weight

When the concentration of solids increases to 25% without adding more solids, only the amount of water changes. Let the new total weight be x lb. Since the amount of solids remains constant, 25% of x lb will be equal to the original amount of solids found in Step 1.

Set up and solve the equation

Set up the equation based on the fact that the weight of solids is constant: 0.25x = amount of solids (from Step 1). Solve for x to find the new total weight of the mixture.

Calculate the weight of evaporated water

Subtract the new total weight found in Step 3 from the original total weight (315 lb) to get the weight of water that must evaporate.

Key concepts

Significant Digits

Understanding significant digits is crucial when measuring the precision of your calculations. When the exercise mentions that you should work to three significant digits, it means that your answer should be precise to three numbers that carry meaning contributing to its measurement accuracy. For instance, in the number 0.0523, the significant digits are 523 because they are the first non-zero figures.

To adhere to the rule of significant digits, start by identifying them in your given values. In our example, the number 315 is to be treated as an exact number, but when we perform calculations, we should ensure our answer is rounded to three significant digits. This is important in scientific work where precision is more critical than in everyday calculations.

Percentage Calculation

Percentage calculations are a fundamental part of many math problems, including those involving percent concentration. To calculate a percentage of a quantity, you multiply the quantity by the percentage expressed as a decimal. For instance, calculating 20% of 315 lb involves converting 20% to a decimal (0.20) and multiplying it by 315, resulting in 63 lb.

Understanding how to manipulate percentages allows you to solve problems relating to changes in concentration, such as determining how much water must evaporate from a mixture to achieve a desired percentage of a substance.

Solving Equations

Equations are mathematical statements that assert the equality of two expressions. Solving an equation entails finding the value(s) of the variable(s) that make the equation true. In the context of percentage concentration calculations, setting up the correct equation is a critical step. In the provided solution, we express the desired concentration as an equation (0.25x = amount of solids). We solve for x, which represents the new total weight of the mixture.

Mastering equation solving enables you to determine unknown quantities in various scientific and mathematical contexts. By isolating the variable and simplifying the equation, you can arrive at the solution.

Weight of Mixtures

The weight of mixtures is a concept that comes into play when dealing with combined substances, such as the given paint mixture. In such scenarios, we often need to manipulate one component (like water in the example) while keeping the other constant (like the solids). The weight of the mixture is the sum of the weights of its components. It can change if one of the components (water) evaporates or is separated.

To adjust the percent concentration, you need to determine the weight of the evaporated water by subtracting the new total weight of the mixture (after evaporation) from the original weight. Mastery of this can be applied to various fields, such as chemistry, cooking, or even in commercial product formulation.