Lesson 13: Solutions

ave you ever bought a container of juice and somewhere on the container it said 100% juice from concentrate? What does that mean? When something is concentrated, all of the water is removed. A juice concentrate is much stronger than regular juice. It would be like drinking a glass of lemonade that was made without water. There would be only lemon juice and sugar. To make a juice concentrate, the water is removed, leaving only a syrup that consists of sugar and other ingredients. For preservation and easier shipping, juices are sold in their concentrated form. Once purchased by beverage bottling companies, water is added to the concentrate, and the juice is repackaged for sale to stores and consumers. In this lesson, you will learn about solutions. You will learn about different types of solutions and how they mix. You will also learn about different types of mixtures, dissolving, and solubility.
What Is a Mixture?
A mixture is a substance made by mixing materials physically. Mixtures are not chemically combined and do not undergo any sort of chemical reaction. They can usually be separated into their original parts and retain many of their own properties. Anything you can combine is a mixture. Mixtures are categorized as heterogeneous or homogeneous based on how they mix. The prefix hetero- means other or different. Aheterogeneous mixture is a mixture composed of substances that are different or not evenly mixed. In a heterogeneous mixture, you can still identify its parts. A salad is one example of a heterogeneous mixture. Even when mixed, you can distinguish all of its ingredients (the cucumbers from lettuce, cheese from tomatoes, and so on) just by looking at it. Oftentimes, a heterogeneous mixture consists of substances in different phases. For example, a glass of soda with ice is a heterogeneous mixture of ice (a solid) and soda (a liquid). heterogeneous mixture homogeneous mixture The prefix homo- means same or one. A homogeneous mixture is a mixture composed of substances that are the same throughout. The individual parts lose their identities and cannot be distinguished from each other by sight. Homogeneous mixtures have a uniform composition, appearance, and definite properties. Air is a homogeneous mixture. It is composed of different gases: hydrogen, oxygen, nitrogen, argon, water vapor, carbon dioxide, and others. Tap water is also a homogenous mixture. It is composed of water and several minerals, yet you cannot see each of the individual ingredients.
What Is a Solution?
Scuba diving is swimming underwater using special equipment. Scuba gear consists of fins for the feet, large goggles to protect the eyes, and self-contained breathing equipment called scuba diving tanks. The tanks are filled with gas and allow divers to breathe while underwater. Scuba divers wear tanks filled with a gas solution. What does scuba equipment look like? The tanks are large metal containers strapped to the diver's back. A hose extends from the top of the tank to the face. The hose connects to a mouthpiece and transports the gas from the tank to the diver, allowing him or her to breathe. The breathing gas is composed of several different gases, including air. A scuba diver would not be able to tell which gases are present in the scuba tank just by looking. The gases form a type of mixture called asolution. A solution is made up of two or more different substances combined to form a homogeneous mixture. The substances are evenly distributed throughout the mixture and cannot be separated. They are the same color, composition, density, and taste all the way through. A substance dissolves when it is mixed with another substance by breaking up into particles too small for the eyes to detect. A solution is composed of two parts: a solvent and a solute. The solvent is the substance in which another substance is dissolved. The solute is the substance that gets dissolved. In a solution, the solute cannot be separated from the solvent, not even with a filter. The substance used in the largest amount is usually the solvent. Fruit juices from concentrate are solutions in which fruit juice concentrate is the solute and water is the solvent.
Dissolving Salt dissolves by ionizing or dissociating in water. Dissolving causes a solute to mix with a solvent to form a solution. The solute particles are attracted to the solvent particles. In order for a substance to dissolve, the forces that hold the two substances together have to be conquered. There are three different ways to dissolve: dissociation, ionization, and dispersion. Dissociation is the process that occurs when an ionic compound breaks apart into its ions as it dissolves to form a solution. Sodium chloride (table salt) is an example of one such ionic compound. When mixed with water, sodium chloride separates into its ions, sodium (Na+) and chloride (Cl-). Dispersion takes place when a substance breaks apart into tiny particles and spreads throughout a liquid. When sugar is added to water, the sugar particles disperse forming a solution. Dispersion is a physical process. The particles do not react or separate into ions. They just simply spread apart. Ionization occurs when a neutral molecule gains or loses electrons while forming a solution. The compound separates into ions. The resulting solution is not neutral; it carries an electrical charge. These solutions are able to conduct electricity. Dissolving by ionization is a chemical change unlike dissociation and dispersion, which are physical changes.
Properties of Solutions There are several different types of solutions that can exist as solids, liquids, and gases in any combination. For example, water vapor is a liquid but forms a gas when in a solution with air. Salt is a solid that forms a liquid solution when combined with water. Air is a gaseous solution formed from nitrogen, oxygen, hydrogen, and other gases. Brass is a solid solution of copper and zinc. The most common solutions have liquids as the solvent. An aqueous solution is a solution in which water is the solvent. Most beverages are aqueous solutions. Soda is an aqueous solution that uses water as the solvent with carbon dioxide and sugar as the solutes, among other ingredients. Carbon dioxide and sugar dissolve in water to form soda. Many substances, when dissolved in water, form solutions that are able to conduct electricity. These substances alone are poor conductors. However, once dissolved in water they are able to dissociate or ionize. This allows the ions to move freely throughout the solution and give it an electric charge. Substances forming aqueous solutions that can conduct electricity are called electrolytes. Electrolytes dissociate, or ionize, in water. If an aqueous solution conducts electricity, the solute must be an electrolyte. Sodium chloride and hydrogen chloride are both electrolytes.
What Factors Affect Dissolving?
Remember that during a chemical reaction the particles of the reactants collide with one another. When particles collide with enough energy to overcome the activation energy, a reaction will occur. The same principle applies to solutions. When a solute and solvent come together, the particles collide and the solute dissolves. The speed at which the solute dissolves depends on the collisions. Three factors that affect the collisions and the rate at which they dissolve are surface area, temperature, and stirring.
Surface Area Dissolving takes place at the surface of a solid. Because dissolving takes place on the surface, increasing the surface area increases the rate at which a substance dissolves. When a medicine tablet dissolves in a cup of water, it dissolves from the outside first. The larger the outside of the tablet (the surface area), the more collisions are possible, and the faster the tablet will dissolve. More collisions means a faster rate of dissolving. If you break up the tablet of medicine, it dissolves even faster. Breaking the tablet increases the surface area by allowing the solvent to come in contact with more of the tablet.
Stirring If you need a medicine tablet to dissolve even faster, you could stir the mixture. Stirring speeds up the rate of dissolving. The dissolved particles are pushed away from the surface of the solid (in this case the medicine tablet) and make more room for collisions. Stirring the mixture increases the collisions between the solute and solvent. This, in turn, increases the rate at which it dissolves.
Temperature Increasing the temperature of the solvent also increases the rate of dissolving. Raising the temperature causes the solvent particles to move faster, increasing its energy. This results in even more collisions between the solvent and the solute, and the solute dissolves faster.
What Is Solubility?
Have you ever wondered why substances such as salt and sugar dissolve in water but fats such as oil do not? It has to do withsolubility. Solubility is the ability of one substance to dissolve in another. It is a physical property that measures the maximum amount of solute that can be dissolved in a specific amount of solvent. But not all solutes dissolve in all solvents. A substance capable of being dissolved is soluble. A substance that cannot be dissolved isinsoluble. Salt is soluble in water. It has a high solubility when it comes to water. Oil has a low solubility and is insoluble in water. Liquids capable of being mixed or dissolving in one another are miscible. Liquids, such as oil and water, that are incapable of being mixed with one another are immiscible. Solubility is often expressed in grams of solute per 100 grams of solvent. The more grams of solute per 100g of solvent, the more soluble the substance. The table below shows the solubility of some substances in water at room temperature (20° C). Substance   Solubility (g/100g water) at 20° C  baking soda   9.6 salt   36.0 sugar   203.9 Of the items on the chart, sugar is the most soluble, and baking soda is the least soluble.
Temperature and Pressure Tea and sugar dissolve faster in hot water. Temperature and pressure are two factors that affect solubility. Substances are likely to be more soluble at higher temperatures. The solubility of a substance generally increases as the temperature of the solvent increases. This is the reason drinks such as iced tea are made using hot water. Once the sugar has been dissolved, the tea can then be cooled or put on ice. Pressure has a similar affect on solubility. Pressure can be used to force one substance to dissolve in another. Increasing the pressure on a gas increases its solubility. Carbonated beverages consist of a gas, carbon dioxide, dissolved in a liquid. Carbon dioxide under pressure is able to dissolve in water. The pressure inside a can full of a carbonated beverage is very high. This is the reason that shaking a can of soda and opening it causes the liquid to squirt out everywhere. Polarity Water is the most common solvent and is often called the universal solvent. Why is water such a great solvent? Water is a compound with very unique properties and is able to dissolve many different substances. Water has oppositely charged poles. Water is a covalent compound that is composed of two hydrogen atoms and one oxygen atom. Its structure consists of oppositely charged poles which result in an unbalanced charge. The oxygen carries a slightly negative charge and the hydrogen atoms have a slightly positive charge. Substances that have an unbalanced charge are polar. Polarity affects solubility. Because water carries both a positive and negative charge, it is able to attract solutes with a positive or negative charge. The negatively charged poles of water are attracted to the positive parts of the solute and the positively charged poles of the water are attracted to the negative parts of the solute. Substances such as oil are immiscible in water. They do not dissolve. Oil does not dissolve in water because it is non-polar. How do you know whether two substances are soluble? Scientists use a rule, "like dissolves like", to predict solubility. Similar substances are soluble in each other. Polar substances are soluble in polar substances. Non-polar substances are soluble in other non-polar substances.
What Are the Different Types of Solutions?
Lemonade is a popular beverage during the spring and summer. The solvent is water, and the solutes are lemon juice and sugar. The type of solution that forms (for example, how sweet the lemonade is) depends on the concentration. The concentration of a solution tells how much solute has been dissolved into the solvent. The more sugar is added to the lemonade, the greater the concentration of sugar. A solution in which a lot of solute has been dissolved into the solvent is called a concentrated solution. The opposite of concentrate is dilute. A solution is diluted if it contains a small quantity of solute compared to the amount of solvent. For example, if you make a pitcher of lemonade and it is too sweet; you added too much sugar. You can dilute the lemonade by adding more of the solvent, or water. This reduces the concentration of the solution. The lemonade is not as sweet anymore because adding more solvent dilutes (weakens) or reduces the sugar concentration. There are specific terms for the concentration of a solution. Solutions can be saturated, unsaturated, or supersaturated depending on the amount of the solute in the solution and its solubility.
Saturated Solutions A saturated solution is a solution that contains as much dissolved solute as possible at a specific temperature. A solution becomes saturated when no more solute can be dissolved at that specific temperature. Any more solute added does not dissolve; it settles at the bottom of the container. It cannot dissolve because the solvent is filled with solute.
Unsaturated Solutions An unsaturated solution is a solution that has less than the maximum amount of solute. It can dissolve more solute if necessary. Most of the beverages you drink are unsaturated solutions. When making tea or lemonade, you might add sugar to make it sweeter. If all of the sugar dissolves, it is an unsaturated solution.
Supersaturated Solutions A supersaturated solution is a solution that contains more than the maximum amount of solute that can normally be dissolved in the solvent at a specific temperature. These solutions are unstable. If even the smallest amount of solute is added to a supersaturated solution, the solution will crystallize. Crystals are solids made up of particles that are neatly arranged in regular and specific repeating patterns. Rock candy is made from a supersaturated solution that turns into crystallized sugar. Remember that temperature affects solubility. The higher the temperature of the solvent, the more solute can be added and dissolved. Increasing the temperature can cause a supersaturated solution to become saturated or even unsaturated.
Think About It What happens to a saturated solution if it is heated?
What Is a Suspension? What Is a Colloid?
You have already learned about the two types of mixtures: heterogeneous and homogeneous. You have also learned that a solution is one type of mixture. This is another way to classify mixtures. Depending on the size of a mixture's particles, it might be classified as a solution, a suspension, or a colloid.
Suspensions What does a bottle of chocolate milk have in common with a bottle of Italian salad dressing? Both bottles read "shake well before using". They are both suspensions. A suspension is a heterogeneous mixture that separates into its parts over time. In a suspension, the solute particles are temporarily and unevenly mixed, forming a cloudy mixture. Particles in suspension are large and often can be seen floating around in the solvent. For example, imagine you have a clear plastic bottle filled with dirt and water. If you shake it up, the dirt mixes with the water and forms a suspension of mud. But after a while, the dirt particles settle to the bottom of the container and you are again left with dirt and water. Some suspensions last longer than others, but all will eventually settle. A very common suspension is salad dressing. Italian dressing is a suspension of oil, vinegar, and other ingredients. The ingredients temporarily mix when shaken, but after a while they settle and separate into layers. Other examples of suspensions include dust particles suspended in air, sand in water, oil in water, and peanut butter. Suspensions are responsible for the formation of substances called precipitates. Aprecipitate is a solid that forms in a solution as a result of a chemical reaction. When two liquids are mixed and a precipitate forms at the bottom of the container, the two substances have undergone a chemical reaction. The solution remaining above the precipitate is called the supernatant. Precipitates also form when a solution has been supersaturated.
Colloids Some substances do not form solutions or suspensions. They form colloids, substances that have properties of both suspensions and solutions. A colloid is made up of small particles of matter dispersed throughout a solid, liquid, or gas. Colloids are like solutions because the particles do not settle when left standing. The solute particles in a colloid are larger than those in a solution but smaller than those in a suspension. Colloids are like suspensions because they are unevenly mixed. Many common substances you deal with everyday are colloids. The table below describes the different types of colloids and gives an example of each. Types of Colloids Name What is it? Example(s) emulsion   a colloid composed of two immiscible liquids   mayonnaise, milk foam   a colloid composed of a gas trapped in a liquid or solid   whipped cream, marshmallows fog   a colloid composed of droplets of water vapor suspended in air   clouds, mist gel   a colloid composed of liquid particles trapped in a solid   butter, gelatin, cheese smoke   a colloid composed of solid and/or liquid particles trapped in air   smoke sol   a colloid composed of a solid suspended in a liquid   paint, blood           The light is scattered by the colloid on the right and passes through the solution on the left. This is called the Tyndall effect. One way to distinguish between a colloid and a suspension is to let it sit. A suspension will always settle. The solute particles of the different substances only temporarily mix with each other. Eventually, they separate into layers. Colloids can also be distinguished from solutions by sight. Remember that solutions are transparent, so they do not scatter light. The particles that make up the colloid are larger than those in a solution; they are large enough to reflect light. If you shine a light on a solution, the beam cannot be seen because the particles are too small to reflect the light. If you shine a light on a colloid, you will be able to see the light as it passes through. The scattering of light as it passes through colloids is called the Tyndall effect. The Tyndall effect gets its name from an Irish scientist named John Tyndall. He was the first person to describe the scattering of light due to colloids in the 19th century.
Making Connections
Salt accumulates on the shores of the Dead Sea. The Dead Sea is a salt lake located in the Middle East between Israel and Jordan. The Dead Sea is a gigantic salt water solution that is very concentrated. The water is about nine times as salty as the ocean. There is so much salt present that the water is unable to dissolve it all. The mineral salts precipitate and accumulate on the bottom of the sea as well as along the beaches. What is so special about the Dead Sea? First, while there is plenty of wildlife that lives in the area around the sea, the Dead Sea cannot support life. Its salt content is too high. Second, mud from the Dead Sea is thought to have healing properties. Scientists and doctors study its composition and effects as well as the climate in the area to find out why this seems to be the case. Third, people are able to actually float in the water with almost no effort. (No, the Dead Sea is not deadly to people!) A tourist demonstrates buoyancy in the Dead Sea. How is this possible? The extremely salty water has a greater density than regular water in pools, rivers, and oceans. It is so much greater that our bodies are able to float without any effort. The water pushes the body upwards, and instead of sinking, a person just bobs up and down. This is due to buoyancy, the upward force on an object produced by the surrounding fluid. This lesson introduced you to solutions. You learned about solubility, dissolving, suspensions, colloids, and different types of solutions. In the next lesson, you will learn about solutions composed of acids and bases.

Lesson 2: Math Concepts for Physical Science

Introduction
The recipe for this cake required knowledge of measurement and the use of fractions. You are planning a party, and a friend asks you how how far away your house is from hers. How would you explain the distance to your home? You could describe the number of miles it is, tell her how many minutes it takes to get there, or even use the number of blocks. No matter how you explain it, you have to use numbers. Scientists rely on numbers and math to help them describe their ideas and observations. Math is critical to science, from the study of gravity to chemical equations to climate change. In this lesson, you will review basic math concepts that will be helpful to you in this course. You will also learn how to express very large and very small numbers using scientific notation. Use the Practice Problems after each section to test your understanding of each concept.
What Math Concepts Are Used in Physical Science?
In science, numbers are used for comparisons and measurements. Below, you can review some of the different ways to compare numbers, such as mean and decimals. You will also have a chance to review how to do some basic calculations involving numbers with decimals, fractions, and/or percentages as well as review basic algebra skills.
Mean Mean, sometimes called average, gives the "middle of the road" for a set of data. To find the mean, take the sum of all the data in a set and divide it by the total number of items of data. Steps for Finding the Mean       1. Find the sum of all the numbers in the set.      2. Count how many numbers are in the set. 3. Divide the sum of all the numbers by how many numbers there are in the set. For example: Find the mean of the set. 2, 2, 3, 0, 4, 0, 6, 5, 17, 10, 9, 2, 1, 9 Find the sum of all the numbers in the set.   2 + 2 + 3 + 0 + 4 + 0 + 6 + 5 + 17 + 10 + 9 + 2 + 1 + 9 = 70                               Count how many numbers are in the set.   2, 2, 3, 0, 4, 0, 6, 5, 17, 10, 9, 2, 1, 9 →14 numbers          Divide the sum of all the numbers by how many numbers there are in the set to find the mean, which is 5.   70 ÷ 14 = 5
Decimals Greater Than and Less Than The smaller side of the sign points to the smaller number 4 < 9 3 > 2 4 is less than 9 3 is greater than 2 Numbers with decimals are used often in science. They allow for very precise values when collecting or comparing data and taking measurements. It is important to understand the value of decimal places and to be able to compare decimal numbers. For example, which of the following numbers is greater? 1.45         1.513 Compare decimal numbers by comparing one digit at a time, from left to right. Make sure to compare digits in the same place values. Using a place value chart is helpful when doing this. Look at the place value charts below. In the ones place, the numbers are equal.   hundreds tens ones . tenths hundredths thousandths                      100 10 1 .1 .01 .001 1.45 → 1 . 4 5 1.513 → 1 . 5 1 3     In the tenths place, 1.513 has the greater digit.   hundreds tens ones . tenths hundredths thousandths                      100 10 1 .1 .01 .001 1.45 → 1 . 4 5 1.513 → 1 . 5 1 3 You do not have to compare any more digits. You now know that 1.513 is the greater number. 1.45 < 1.513
Reviewing Decimal Computations In this course it is important to be able to add, subtract, multiply, and divide numbers with decimals. Use the table to review the steps and solve the problems below. Adding and Subtracting Decimals 1. Line up the decimal points in the numbers. 2. If the numbers end in different place values, add zeros to make them end in the same place value. 3. Add or subtract the numbers. Place the decimal point in the answer directly beneath where it is in the problem. For example: Solve. 4.75 + 0.38 Write the numbers with the decimal points lined up. Use a place value chart if it helps. 10 1 .1 .01 .001 4 . 7 5   0 . 3 8               Complete the addition. Carry numbers as when adding whole numbers. Pull the decimal point straight down into the answer. 1 1 4 . 7 5   + 0 . 3 8 5 . 1 3 Multiplying decimals is a lot like multiplying whole numbers. There are just a few extra steps at the end. Multiplying Decimals 1. Line up the numbers to the right. Do not line up the decimal points. 2. Multiply as with whole numbers. 3. Count the total number of digits to the right of the decimal point in both numbers you multiplied. In your answer, the product, count that number of places from the right and place the decimal. Notice that the first step of multiplying is not the same as for adding and subtracting. With multiplying, only think about the decimal point after completing the multiplication. For example: Solve. 3.14 × 1.8 Write the numbers with their digits lined up on the right. Pay no attention to the decimal points yet.     3.14 ×   1.8                                        Multiply the numbers. Remember the problem is not done until the decimal point is placed.      1  3    3.14 ×   1.8   2512   3140    5.652       Count the number of digits to the right of the decimal point in both the numbers you multiplied together. Then count the same number of digits from the right of the answer. This is where the decimal goes to complete the problem.    3.14 ×   1.8   5.652
Fractions Numbers may also be expressed as fractions. A fraction is a portion of a whole number. Recall that the numerator is the number on top and the denominator is the number on the bottom. Use fractions to show rates and proportions, which are different ways to compare numbers. A rate is a comparison of two values using units. For example, a rate is used when comparing the number of miles a car has driven to the number of gallons of gas it has used. A unit rate is a rate with a denominator of 1, such as a mile per gallon. Fractions are most often expressed in simplest form. When a fraction is in simplest form, the numerator and denominator have no common factor except 1. Simplest form is obtained when all of the factors shared by the numerator and denominator have been cancelled out by factoring, reducing, or simplifying. For example: Reduce the fraction below to its simplest form. 15 25 Find a factor that is common to both the numerator and the denominator. Write the numerator and denominator as products of their factors. 15 = 3 × 5 25 5 × 5                                   Cancel out the factors that are in both the numerator and the denominator. 3 ×  5  5 ×  5        3  is the simplest form of   15 . 5 25 3 5 Reviewing Fraction Computations It is important to know how to multiply and divide fractions because they are used in this course to convert between different units of measurement. Follow these steps to multiply fractions. Steps for Multiplying Fractions      1. Multiply one numerator by the other. This is the numerator of the product.        2. Multiply one denominator by the other. This is the denominator of the product.   a  ×   c  =  a × c  =  ac b d b × d bd Always multiply straight across. Never multiply the top by the bottom or the bottom by the top.  For example: Solve. 4  ×  2 5 9 Multiply the first numerator by the second numerator. 4  ×   2  =   4 × 2  =   8 5 9 5 × 9                                                   Multiply the first denominator by the second denominator. 4  ×   2  =   4 × 2  =   8 5 9 5 × 9 45       Make sure the answer is in simplest form. 8 45 Dividing fractions is almost the same as multiplying them. To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is the inverse. To find the reciprocal, flip over the fraction. The denominator becomes the numerator and the numerator becomes the denominator. For example, the reciprocal of 2/3 is 3/2. Once you find the reciprocal, rewrite the problem as multiplication. Steps for Dividing Fractions       1. Rewrite the problem as a multiplication problem by taking the reciprocal of the second fraction.         2. Multiply one numerator by the other and one denominator by the other.   a  ÷   c  =   a  ×   d  =   ad b d b c bc For example: Solve. 7 ÷ 1 15 4 Write the first fraction. 7 15             Add a multiplication sign. 7  ×   15       Write the reciprocal of the second fraction. 7  ×   4 15 1                                                Multiply. Make sure the answer is in simplest form. 7  ×   4  =  7 × 4  =   28 15 1 15 × 1 15
Practice Problems Solve each problem on your paper. Then click on the "Check Answers" button to check your answers. 1. Polly has taken 5 clarinet playing tests in band class this semester. Her scores are shown below. What is Polly's mean test score? Test # Score 1 114 2 115 3 98 4 115 5 107 2. Solve. 5 - 1.53 3. Solve. 6.41 × 1.2 4. Solve. 7  ×   3 9 5. Solve. 2  ÷  5 9 8
Percentages In physical sciences like chemistry and physics, percentages are used to describe and compare many different types of data. A percentage is a way of expressing a number as a fraction of 100. Percentages can also be expressed as small numbers less than 1, as shown below. 23%  =    23  =  0.23  100 The different ways of expressing percentages can be used to solve problems. Calculate the percentage     1. Convert the percentage into a decimal.       2. Multiply the decimal by the whole number.   How do you calculate percentage? For example: A teacher buys a 6-foot sub sandwich for a classroom celebration. He cuts it into 24 equal pieces. By the end of class, 75% of the sandwich has been eaten. How many pieces did the students eat? Convert the percentage to a decimal. 75% = 0.75                                                         Multiply the decimal by the total number of slices that make up the whole sandwich to find the number of pieces eaten.   0.75 × 24 = 18 The students ate 18 pieces. To convert a fraction to a percentage, divide the numerator by the denominator. The result is a decimal number. Convert it to a percentage by multiplying by 100. For example: A teacher buys a 6-foot sub sandwich for a classroom celebration. He cuts it into 24 equal pieces. By the end of class, 18 pieces of the sandwich have been eaten. What percentage of the sandwich did the students eat? Write the numbers out as a fraction. The students ate 18 out of the total 24 pieces.   18 24           Simplify.   18 = 3 24 4           Divide the numerator by the denominator.   3  =  0.75  4           Convert the fraction to a percent by multiplying by 100.   0.75 × 100 = 75%   The students ate 75% of the sandwich.
Algebra Many concepts in physical science are explained and solved using basic algebra. For this, an understanding of variables is necessary. This section is a review of those basic algebra skills. Tips for Solving Algebraic Equations      1. Isolate the variable by using the opposite operation.     2. Always perform the same operation on both sides of the equation.   3. Remember the order of operations. 4. When the variable is by itself, the equation is solved. For example: Solve for m. m + 5 = 2 Isolate m by subtracting 5. Remember to do the same thing to both sides.   m + 5 - 5 = 2 - 5                                                 Complete the operations. Notice that the 5 is cancelled out by subtracting 5.   m = -3
Scientific Notation 100  =   1  = 1  101  =   10  =   10  102  =   10 × 10  =   100  103  =   10 × 10 × 10  =   1,000  104  =   10 × 10 × 10 × 10  =   10,000           10-1  =   0.1  =  0.10 10-2  =   0.1 × 0.1  =  0.01 10-3  =   0.1 × 0.1 × 0.1  =  0.001 10-4  =  0.1 × 0.1 × 0.1 × 0.1  =  0.0001 Scientific notation is a shorthand method used to express very large or very small numbers. In scientific notation, the numbers are written as values between 1 and 10, and multiplied by a power of 10. Look at the table. Each power of 10 adds another zero, and moves the decimal point one place to the right. If the exponent is negative, the decimal point moves that number of times to the left. Each power of 10 adds another zero between the number and the decimal. Remember that any number to the first power is equal to itself. Any non-zero number to the zero power is equal to 1. When reading numbers in scientific notation, look at the exponent on the 10. Moving the decimal to the right or left that number of times produces the original number. 2.345 × 102     ®     234.5 4.16 × 107     ®     41,600,000 2.7 × 1011     ®     270,000,000,000 9.996 × 10-3     ® 0.00996 6.75 × 10-9     ®  0.00000000675 To convert a number into scientific notation follow the steps in the table below. Convert Very Small or Very Large Numbers to Scientific Notation     1. Move the decimal point to the right or left until there is only one non-zero digit to the left of the decimal point. If there is no decimal point, add a decimal point at the end of a number, and move it in the correct direction.       2. Count the number of times the decimal was moved from its original position. The number of places the decimal has to be moved is the value of the exponent on the 10.   How do you write 810,000,000 in scientific notation? 1. Rewrite the number by adding a decimal point on the end: 810,000,000.0 2. Move the decimal as shown to find the first number. 3. Count the number of decimal places and then write the 10 with that value as the exponent. ®  8.1 × 108
Think About It Why would it be especially useful for astronomers to use scientific notation?
For example: The planet Mercury is about 57 million km from the Sun. What is this distance in scientific notation? Write the number out. Be sure to include the decimal point.   57,000,000.0                                                   Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point.   57,000,000.0 ® 5.7       Count the number of times the decimal was moved from its original position to find the exponent on the 10.     5. 7 0 0 0 0 0 0 .0   7 6 5 4 3 2 1         Write the number in scientific notation.   5.7 × 10^7