Ratios and Proportions PDF

6.1 Ratios Name ________________________________________________ _ What is a ratio? A ratio, often expressed as a fraction, compares two numbers. Examples: Eight people want equal shares of one pie. You c...

6.1 Ratios Name ________________________________________________ _ What is a ratio? A ratio, often expressed as a fraction, compares two numbers. Examples: Eight people want equal shares of one pie. You could set up a ratio. When you remove the words, you can see that each person should get of the pie. You can compare any two numbers with a ratio. For example: Shaundra read 5 books last month, and Carmen read 4. So the ratio of Shaundra’s reading to Carmen’s reading was 5:4 (Say “five to four”). You can also express this as a mixed number. Shaundra read times as many books as Carmen. Exercises SOLVE Indicate (True or False) whether the ratios are equal. State the ratio as a fraction. Paul is making a plaster mixture for his sculpture class. If he mixes 5 ounces of plaster with 4 ounces of water, what is the ratio of plaster to water? ______________________________________ Erika’s mom separates the laundry into sets. If she puts two sheets and three pillow cases into each set, what is the ratio of pillowcases to sheets? ______________________________________ Floyd is setting tables for a sports banquet. For each place setting, he puts two forks to the left of the plate and a knife and  a spoon to the right of the plate. What is the ratio of forks to knives? ______________________________________ Jean is making pizza. She adds 4 slices of pepperoni and 3 olives to each slice of pizza. What is the ratio of pepperoni to olives? ______________________________________ What is the ratio of olives to pepperoni? ______________________________________ 6.2 Proportions and Cross- Multiplying Name ________________________________________________ _ Do you ever have to use more than one ratio to solve a problem? Yes, a proportion is a problem that contains two ratios that are equal. Example: Suppose you want to give a party for 20 people. You know that two quarts of potato salad will feed 10 people, but how much potato salad will you need if you are feeding 20 people? When you have two ratios, but don’t know the value of one of the numerators or one of the denominators, set up a proportion problem. This kind of problem is called an equation, a mathematical statement that two things are equal. You can use q to stand for the unknown number of quarts. Cross-multiplying is the way to find the missing number. Multiply the numerator of the first fraction by the denominator of the second fraction and write that on one side of the equation. Then multiply the denominator of the first fraction by the numerator of the second fraction and write that on the other side of the equation. To get the answer, look at the side of the  equation that has both a known number and the unknown number. Divide both sides of the equation by that known number. Step 1: Set up your equation. Step 2: Cross-multiply. 2 × 20 = 10 × q Step 3: Find the side of the equation with the unknown number. Then look at the known number on that side. (In this equation, it’s 10.) Divide both sides of the equation by that known number. 40 ÷ 10 = 4 q = 4 You will need four quarts of potato salad for 20 people! Exercises SOLVE Indicate (True or False) whether the ratios are equal. Solve for the unknown variable. 6.3 Rates Name ________________________________________________ _ A rate is a fixed ratio between two things. It is solved exactly like a proportion problem.  Example: Maria drives at a rate of 65 miles per hour. How many hours does it take her to drive 195 miles? Step 1: Express the proportion problem using two ratios. In this problem, let’s use h for the unknown number of hours. Step 2: Cross-multiply. 195 = 65 × h Step 3: Divide each side by 65 to solve for h: So it will take Maria 3 hours to drive 195 miles! Exercises SOLVE George likes to sweeten his ice tea. When he is drinking a 20ounce ice tea, he adds two teaspoons of sugar. If he makes a gallon of ice tea, how many teaspoons of sugar should he add (there are 128 ounces in a gallon)? ______________________________________ Peter wants to make scrambled eggs for the customers at the diner. His recipe calls for 12 eggs and is enough for 5 people. How many eggs will he need if he has to feed 75 customers? ______________________________________  A person needs to drink 3 quarts of water for every hour of running time. If a runner plans to complete a marathon in 5 hours, how many quarts of water should she drink during the race? ______________________________________ Diane’s car uses 5 gallons of gasoline to travel 125 miles. How far will Diane be able to travel on 20 gallons of gasoline? ______________________________________ When Frank goes on a 4-day vacation with his family, he packs 4 t-shirts and 3 pairs of shorts. If he is going on an extended vacation for 12 days, how many t-shirts and shorts will he need to pack? ______________________________________ There were 90 customers at the restaurant on Friday, 60 of whom ordered the vegetarian meal. If there are 195 customers on Sunday, how many vegetarian meals would you expect to sell? ______________________________________ 6.4 Proportions and Percent Name ________________________________________________ _ What if you wanted to answer the question, “What percent of 80 is 48?” Remember that the word percent means “per 100”. Think of percent as a proportion where the denominator is always 100. So if you wanted to know what percent 48 is of 80, then you would set the problem up as a proportion.  Examples: What percent of 80 is 48? Step 1: Express the problem using two ratios. Use x for the unknown. “48 is to 80 as x is to 100” or Step 2: Cross multiply: 4800 = 80x Step 3: Divide each side by 80 to solve for x: 4800 ÷ 80 = 60 x = 60 So 48 is 60% of 80. What if you wanted to know what number is 50% of 90? Just like the first example, you would solve the problem using two ratios. Step 1: Set up the problem using two ratios. Remember that percent is just a ratio with 100 as the denominator: Step 2: Cross-multiply: 100x = 4500 Step 3: Divide each side by 100: 4500 ÷ 100 = 45 x = 45 So 45 is 50% of 90. Exercises CALCULATE USING PROPORTIONS 40% of 50 is _____________ 72 is 18% of _____________ 14 is ______________ % of 35 12% of 85 is _____________  50 is 40% of _____________ 18 is _____________% of 270 70% of 70 is _____________ 66 is 30% of _____________ 1.5 is _____________% of 75 140% of 95 is _____________ 13 is 65% of _____________ 12 is _____________% of 32 6.5 Percent Change, Mark-up, and Discount Name ________________________________________________ _ When calculating percent change, you are determining the percentage change from a starting point. If you want to determine the percent your height has changed, say from 65 inches to 70 inches, you want to know how much it has changed relative to your starting height of 65 inches. Simple subtraction tells us your height has changed 5 inches, but to calculate the percent change, you must set up an equation:  Example: Starting height = 65 inches Change in height = (70 in. − 65 in.) = 5 inches Step 1: Set up your equation: Step 2: Calculate: 500 = 65x 7.69 = x Your height has increased approximately 7.69% Mark-up is an amount that you want to add to something you sell. Say your store’s headquarters has determined that you should sell all your products at cost plus 25%; the 25% is called the mark-up. One way to calculate mark-up is to take the cost, calculate 25% of the cost, and add it to the cost. This is your selling price. If you have a product that costs $200, what is the mark-up and selling price? To calculate the mark-up you must set up a proportion. Example: Step 1: Set up your equation: Step 2: Calculate: 100x = 5000 x = 50 Your mark-up is $50, so you would sell the product for $200 + $50 = $250 Discount is much like a mark-up, but instead of increasing the price, you are reducing the price of an item. You are at a sale and the sign states to take 30% off all items. You see an MP3 player that has a price of $150. How much is it after the discount?  Example: Step 1: Set up your equation: Step 2: Calculate: 100x = 4500 x = 45 The discount is $45, so the sale price would be $150 − $45 = $105 Exercises SOLVE What is the selling price for an item that costs $50 and has a mark-up of 40%? ______________________________________ What is the mark-up amount for an item that costs $125 and has a mark-up of 35%? ______________________________________ What is the selling price for an item that costs $70 and has a discount of 40%? ______________________________________ What is the discount amount for an item that costs $160 and has a discount of 45%? ______________________________________ 6.6 Percents and Fractions Name ________________________________________________ _ Can you express percents in ways other than as a decimal? Yes, you can change percents into fractions. The denominator of a percent is always 100, and the numerator will be the number of the percent. Some fractions can be turned into simple percents. If the denominator of the fraction can divide evenly into 100, find the quotient. Then multiply the numerator by the quotient, and add the percent sign. If the denominator of the fraction cannot divide evenly into 100, the fraction cannot be converted into a simple percent. Example: 30% of 50 = ? Step 1: Convert the percent to a fraction: 30% = Step 2: Simplify the fraction if you can: Step 3: Multiply: Step 4: Simplify the product: so 30% of 50 = 15 Example: = ?% Step 1: Divide 100 by the denominator: 100 ÷ 5 = 20  Step 2 Multiply the numerator by the product: 2 × 20 = 40 Step 3: Add the percent sign. 40% So = 40% Exercises CALCULATE USING MULTIPLICATION 18 is _____________% of 60 20% of 85 is _____________ 30% of 90 is _____________ 15% of 60 is _____________ 35% of 220 is _____________ 40% of 65 is _____________ 65% of 40 is _____________ 6.7 Multiplying Percents and Fractions Name ________________________________________________ _ Can you multiply percents and fractions? Of course! You’ve learned that percents can be expressed as fractions. So just convert the percent to a fraction and multiply the two fractions. But remember these two important points: you find a percent of a fraction, the product will be a fraction. you find a fraction of a percent, the product will be a percent. Example: What is 50% of Step 1: Convert the percent to a fraction: Step 2: Multiply the fractions: 50% of Example: What is of 96%? Step 1: Convert the percent to a fraction: Step 2: Multiply the fractions: Step 3: Convert back to a percent: Another way to find a fraction of a percent is to multiply the percent, as if it were a whole number, by the fraction. The product will almost always be an improper fraction. Change that fraction into a mixed number and add the percent sign. Exercises CALCULATE 6.8 Percents and Decimals Name ________________________________________________ _ How can you convert decimals with thousandths, ten thousandths, and even smaller places into percents? Just move the decimal point two places to the right and then add the percent sign. Example: Rename.46072 as a percent. Exercises CONVERT Convert to a percent..7612 ________________.01543 ________________ 1.59 ________________.5721 ________________.0012 ________________.000134 ________________ 10.45 ________________ 1.89 ________________.569 ________________.9999 ________________ 0.0011 ________________ 3.1345 ________________ 99.99 ________________.175555578 ________________.187 ________________.87 ________________ 6.9 Simple and Compound Interest Name ________________________________________________ _ SIMPLE INTEREST What does simple interest mean when you’re talking about a loan or a bank account? How do you calculate it? The amount you borrow or put into the bank is called the principal. Simple interest is a percent of the principal that has to be paid, by you, if you borrow money, or by the bank, to you, if you deposited money. The interest is money that is added to the principal. Example: Principal = $500 Rate of Interest for one year = 4% Step 1: Interest (i) = Principal (p) × Rate of Interest (r): $500 × 4% = i Step 2: Convert the Rate of Interest to a fraction: Step 3: Calculate: If you wanted to pay the loan back at the end of the year, you would have to pay both the principal and the interest. p + i = $500 (the principal) + $20 (the interest) = $520 If you deposited this money into a savings account, the bank would have to add $20.00 interest to your deposit at the end of a year. Exercises CALCULATE How much interest would you earn if you put $500 in a bank for 15 years and received simple interest of 8%? ______________________________________ Calculate the simple interest on a bank account where you deposit $500 and earn 12% a year for 5 years. ______________________________________ Calculate the ending balance of your savings account if you deposit $400 and earn simple interest of 7% for 5 years. ______________________________________ Calculate the ending balance of your savings account if you deposited $1,000 and earned simple interest of 6% for 6 years. ______________________________________ COMPOUND INTEREST What is the difference between compound and simple interest? Compound interest pays interest on the principal and the interest, while simple interest pays interest only on the principal. Examples: Let’s look at an example where you put $100 in Bank A that pays compound interest of 10% each year, and $100 in Bank B that pays simple interest of 10% each year. We will examine what happens over 3 years.  The difference in balances is due to compound interest. If you want to calculate the balance you will have after n years, the formula is: [ to the n th power] Starting Balance × (1 + interest rate as a decimal) Calculate the balance at Bank A after 3 years at a compounded interest rate of 10%. Step 1: Convert the interest rate to a decimal: 10% =.1 3 Step 2: Set up an equation: $100 × (1 +.1) = x 3 Step 3: Calculate: $100 × (1 +.1) = $133.10 The balance at Bank A after 3 years is $133.10 Exercises CALCULATE Calculate the interest earned over a 5-year period when you deposit $2,000 and earn compound interest of 8% per year. ______________________________________ How much interest would you earn if you put $500 in a bank for 20 years and received a compound interest rate of 4%? ______________________________________ How much money would you owe if you borrowed $2,000 for 5 years, with a compound interest rate of 28%, and did not make any payments during that period? ______________________________________ Is it better to receive compounded interest for 7 years at 12% on your balance of $500, or to receive the same rate of simple interest for 9 years on that same balance? ______________________________________ Unit Test Name ________________________________________________ _ Lessons 1-6 Solve. Edie’s local newspaper has 4,476 pages of advertising each year. If the magazine is published once a week, about how many pages of advertising are in each issue? (Calculate using 52 weeks in a year.) ______________________________________________ _________________________ How many pages exactly? ______________________________________________ _________________________ Add or subtract. Change each to a mixed number. Determine if the following proportions are equal. (Write Yes or No.) Solve for x. Solve. Walter looked at the list of nutrients in the fruit juice he bought. He noticed that there were a total of 4 grams of carbohydrates and 3 grams of sugar in every bottle of juice. Compare the amount of sugar to carbohydrates in the fruit juice. ______________________________________________ ______________________ Will rides his unicycle at an average speed of 8 miles per hour. How far will he travel in hours? ___________________________ Jack makes 22 muffins for every 3 batches he bakes. How many batches of muffins will he need to bake in order to sell 242 muffins? ______________________________________________ ______________________ Priscilla drinks an average of quart of water for each mile she walks. How many quarts of water will she drink if  she walks miles? ______________________________________________ ______________________ Jenny changes the oil in her car every 2,250 miles. How many times will she need to change the oil in her car if she takes a trip that is 9,000 miles in length? ______________________________________________ ______________________ 40% of 440 43% of.705 84% of 1.906 Pete’s Pet Emporium is having a sale on birdcages. Pete is selling his $50 cages at a 20% discount, his $75 cages at off, and his $100 cages at 60% off the original price. What are the new sale prices for the 3 cages? $50 cage  __________ $75 cage ___________ $100 cage ___________ Tom is selling wristbands for $4.50. He has to charge sales tax of 6% on each wristband. What is the cost to the customer, including sales tax, for one wristband? ______________________________________________ ______________________ Ursula put $200 into a money market account that pays 3% simple interest. How much will she have in her account at the end of 1 year if she does not deposit any more money in the account? ______________________________________________ ______________________ How much will she have at the end of 2 years? ______________________________________________ ______________________ Pam put $400 into a savings account that pays 2.5% in compound interest. How much will she have in the account at the end of 2 years, if she does not deposit any more money in the account? ______________________________________________ ______________________ How much will she have at the end of 5 years? ______________________________________________ ______________________ 7.1 Place Value and Rounding  Name ________________________________________________ _ Understanding place value can help you work with decimals. Look at the chart. Suppose you are asked to round a decimal to its highest whole number. You can do that by looking at the digit in the tenths place. If that digit is less than 5, round down and keep the whole number that is already there. If the digit in the tenths place is 5 or greater, add 1 to the whole number. You can also round a decimal to its nearest tenth, its nearest hundredth, its nearest thousandth, and so on. Just look at the digit to the right of the rounding place. If that digit is less than 5, keep the digit you see in the rounding place. If that digit is 5 or greater, add 1 to the digit in the rounding place. Exercises ROUND Round to the nearest whole number. 48.6 98.3 156.67 3026.92 189.41233 2244.66795  279.99556 428.5 Round to the nearest tenth. 124.5755 175.5133 349.49888 313.35664 375.77454 44.00913 566.9943 61.15 Round to the nearest hundredth. 1536.3357 32.4589 118.9977 523.75776 1099.989877 1.11881 33.43718 555.555 Round to the nearest thousandth. 729.239788 409.13391 8056.708035 549.594959 99.80007 177.555901 2012.20507 901.901445 7.2 Changing Fractions to Decimals Name ________________________________________________ _ What is the difference between a decimal and a fraction? Actually, a decimal is a fraction expressed in a different way. However, decimals are expressed only in tenths, hundredths, thousandths, and so on. So some fractions cannot be converted to simple decimals.  Example: Every fraction represents its numerator divided byits denominator. So Set up a simple division problem. Add a decimal point and as many placeholder zeros as you need in your dividend. As you can see, or six hundred twenty-five thousandths Exercises CONVERT TO DECIMAL Round to the nearest thousandth. 7.3 Changing Decimals to Fractions Name ________________________________________________ _ Can you also change decimals to fractions? Yes, and it is much easier to do than changing fractions into decimals. Begin by looking at the place value farthest to the right. Use that as your denominator. The decimal number becomes the numerator. After you have changed the decimal into a fraction, you might even be able to reduce it. Example: Exercises CONVERT TO FRACTION 1.3 _____________.60 _____________.588 _____________ 3.875 _____________ 6.75 _____________ 1.125 _____________ 3.26 _____________.625 _____________ 4.2 _____________ 12.101 _____________ 2.009 _____________.3125 _____________ 1.046875 _____________ 2.64 _____________ 5.55 _____________ 22.222 _____________ 5.8 _____________ 33.99 _____________ 3.5 _____________ 8.18 _____________ Chad had.625 gallons of gas left in his lawnmower at the end of summer. Restate the amount as a fraction. _________________________________________ Wanda toured the milk processing plant with her class. The guide said there were over.75 miles of conveyor belts in the plant. Restate that number as a fraction. _________________________________________ 7.4 Comparing and Ordering Decimals Name ________________________________________________ _ If you look at place values, you can compare and order decimals just as you can compare and order whole numbers. Remember to line up your decimals so that the decimal points are all in the same column. As with whole numbers, each digit is one place value higher than the digit to its immediate right. When comparing whole numbers with decimals, always look at the whole numbers first. If two whole numbers are the same, then compare by moving right from the decimal point. To compare decimals like.07 and.072, you can imagine a placeholder zero to make them both fill the same number of places. So.07 =.070. That is less than.072! Example: Order these decimals:5.62 6.186 0.2.07 5.65.071.009 The order, reading from greatest to least is: 6.186, 5.65, 5.62, 0.2,.071,.07,.009  When comparing numbers with decimals, always look at the whole numbers first. If two whole numbers are the same, then compare the numbers to the right of the decimal point. Exercises COMPARE Order from least to greatest. 1.3, 1.031, 1.322, 13.1,.1332, 1.5, 1.55, 1.505 __________________________________________.751,.75, 7.51,.705,.075,.34, 1.675, 1.68 __________________________________________.17, 1.7,.017,.00175,.01695,.107, 1.07 __________________________________________.45,.625,.405,.420,.415,.451, 1.4 __________________________________________ Order from greatest to least..25,.333,.15,.155,.125,.33 __________________________________________.332,.3334,.334,.3,.3033,.0335, 1.0001 __________________________________________.6667,.7501,.6,.75,.751,.707,.667 __________________________________________.68,.55,.6,.63,.6665,.06665,.59996 __________________________________________ 8.1 Adding Decimals Name ________________________________________________ _ Is there anything special you need to know in order to add decimals? Yes, you need to make sure to line up your addends by place value. Once you do that, adding decimals is exactly the same as adding whole numbers. Example: You also need to put a decimal point in its proper column in the total! Exercises ADD  Troy plays for the football team as a punter. Last week Troy made three punts of 11.324 meters, 12.6742 meters, and 10.227 meters. What was the total length of these three punts? _________________________________________ Kate measured the amount of rain that fell during the last three rainstorms. She measured.903 inches of rainfall for the first storm, 1.6778 inches for the second storm, and 1.2655 inches for the third storm. What was the total rainfall for the three storms? _________________________________________ 8.2 Subtracting Decimals Name ________________________________________________ _ Is subtracting decimals similar to subtracting whole numbers? Yes, but you have to remember to line up the decimals. Insert placeholder zeros if necessary. Example: The value of a number does not change if you add a decimal point at the end and insert placeholder zeros. You can use as many placeholder zeros as you need to complete your calculations. Exercises SUBTRACT  The winner of the pole vault recorded a best vault of 5.8833 meters. The second place winner recorded a best vault of 5.4993 meters. What was the difference between the winning vault and the second-place vault? _________________________________________ Annie measured the depth of the water in the school’s fountain and found there were 9.774 inches of water. Annie measured the depth of the water again the next day and noted that the water level was 1.7456 inches lower. What was the new depth of the water? _________________________________________ 9.1 Multiplying with Decimals Name ________________________________________________  _ When multiplying decimals, you do not line up the decimal points. You just multiply as if there were no decimal points at all. When you are finished multiplying, count the total number of decimal places in the factors, the numbers you have multiplied. Then, starting from the right of your product, count that number of places, and put your decimal point to the left of the last place you counted. Example: Exercises MULTIPLY  Pauline’s mom drives the soccer team to and from each of their away games. If the team has 12 away games, and on average Pauline’s mom uses 2.775 gallons of gas to make the round trip, how much gas will she use for the whole season? _________________________________________ The delivery truck driver has 64 packages to deliver to the school. The average weight of a package is 13.7552 pounds. If the delivery truck has a load capacity of 900 pounds, can the delivery driver deliver all 64 packages in one load? _________________________________________ 9.2 Dividing with Decimals Name ________________________________________________ _ When dividing a decimal by a whole number, only the dividend has a decimal point. To calculate correctly, you must line up a decimal point in the quotient with the decimal point in the dividend. When dividing either a decimal or a whole number by a decimal, you must move the decimal point of the divisor all the way to the right. So you must multiply the divisor by whatever power of 10 will do that. Then, you have to also multiply the dividend by that same power of 10. Examples: 92.4 ÷ 7 Step 1: Complete the division problem, and make sure the decimal in the quotient aligns with the decimal in the dividend:  27 ÷.08 Step 1: Multiply both numbers by 100:.08 × 100 = 8 27 × 100 = 2700 Step 2: Complete the division problem: Exercises DIVIDE Unit Test Name ________________________________________________ _ Lessons 7-9 Round to the nearest tenth. 3406.997 ______________________ 334,782.099 ______________________ 65,529.0887 ______________________ Round to the nearest hundredth. 2,467,891.3554 ______________________ 97.009 ______________________ Round to the nearest ten thousandth. 17.99986  ______________________ 99.11115 ______________________ Identify the place value of the underlined number. 1,683,679.57344 _______________________ 1,499,667.5773 _________________________ Convert the decimal to a fraction..8 = __________________.875 = __________________.08 = __________________ Convert the fraction to a decimal. Put the decimals in order from least to greatest..122,.1145,.616,.6165,.513,.3132,.2126,.819 ______________________________________________ ________________________________.217,.0217,.0133,.0487,.1243,.20413,.5257,.05257,.05205  ______________________________________________ ________________________________ Add or Subtract. Multiply or Divide. Chad and his service club collected a total of 954.75 pounds of canned food for the local animal shelter. There are 15 people in the club. If each member collected the same amount of canned food, how many pounds did each member collect? ______________________________________________ ______ pounds Jane went to the store to buy food for a party of 6 neighbors. For each guest, she spent $2.55 for salad, $1.75 for a cold beverage, and $1.25 for a fruit cup. How much did she spend in total to buy the food? ______________________________________________ ______ Jane has $35. Does she have enough money to buy everything she needs? ______________________________________________ ______ If so, then how much change will Jane get back? ______________________________________________ ______ Rodney runs on a cross-country course that is 2.35 miles in length. During the week he ran the course times. How far did he run that week?  ______________________________________________ ______ Stan is preparing pots for planting flowers. He has 23.5 pounds of potting soil. If Stan fills each pot with 1.35 pounds of potting soil, how many pots can he fill? ______________________________________________ ______ Billy charges 45¢ per square foot to varnish patio decks. If the patio deck he is varnishing is 220.25 square feet in area, how much should Billy charge for the job? ______________________________________________ ______ Tracy mixed cold beverages for all of the teams participating in a local baseball tournament. For each batch, she mixed 3.25 gallons of lemonade with 1.3 gallons of iced tea. If Tracy made batches, how many gallons of cold beverage did she make? ______________________________________________ ______ gallons 10.1 Multiplying and Dividing Exponents Name ________________________________________________ _ What if you want to multiply 3 × 3 × 3? Or 5 × 5 × 5 × 5? Is there a simple way you can write that? Yes, you can use an exponent. The number you keep multiplying by itself is called the base. The exponent (written as a small number next to and slightly above the base) tells how many times you multiply the base by itself. Exponents are also called powers. So 103 is 10 to the third power. Any number can be a base. For example, 64 is 4 to the third power. Often, when a base is raised to the second power, we use the word “squared.” 252 can be expressed as “25 to the second power,” or “25 squared.” When a base is raised to the third power, we often use the word “cubed.” So 123 can be expressed as “12 cubed.” Is there a simple way to multiply and divide bases that have exponents? Yes. Can you figure out how to do that by looking at the following information? Can you see how to do it? To multiply a base raised to a power by the same base raised to a power, simply add the exponents. To divide a base raised to a power by the same base raised to a power, simply subtract the exponents. Exercises CALCULATE Express your answer using a base and an exponent. 45 × 4 5 75 ÷ 73 316 ÷ 34 1222 × 125 117 × 115 1232 ÷ 1210 105 × 104 237 ÷ 236 1616 ÷ 162 155 ÷ 153 1111 ÷ 112 44 × 4 7 10.2 Powers Name ________________________________________________ _ What if you have a problem with 2 exponents separated by parentheses? Is there a rule for how to calculate that expression? Yes, let’s take a look at the example. 33 Example: (5 ) This expression means 53 multiplied by itself 3 times. (5 × 5 × 5) × (5 × 5 × 5) × (5 × 5 × 5) = 59 The rule for this type of exponential expression is: (Am) n = Am × n What about an expression that looks like (Am)n, except the parentheses are left out: Amn? In this case, the order of operations says that you calculate exponents first, so it would be A raised to the mn power. So 5333 = 527 Exercises CALCULATE Express your answer using a base and an exponent. (54)3  (87)5 (1410)7 (320)8 764 824 1983 1593 (83)8 (25)6 (74)15 (135)3 1828 27764 (333)15 (34)5 10.3 More about Exponents Name ________________________________________________  _ All the work that you have done with exponents has been with a positive number as the exponent, but there are also negative exponents. What number does 5−3 represent? If we multiply 53 × 5−3 and use the properties of exponents that you already learned, then 53 × 5−3 = 53 + (−3) = 50 = 1 5−3 is the multiplicative inverse of 53, so When you encounter a negative exponent, you simply apply the same rules of exponents you already know: Am × A−n = Am−n Exercises CALCULATE Convert to a fraction. 4−3 3−3 6−4 5−5 7−2 4−1 9−5  2−8 Convert to exponential form. Multiply. 44 × 4−2 57 × 5−4 712 × 7−6 1424 × 14−20 10.4 Squares and Square Roots Name ________________________________________________ _ To square a number means that you take the number and multiply it by itself. If you square 6, the result is 36. This would be written as 62 = 36. Numbers that result from squaring an integer are called perfect squares. The numbers that are perfect squares and less than 200 are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196. The square root of a number is the number that, when multiplied by itself, is equal to that number. The square root of 196, written as. The is called the radical sign. Exercises CALCULATE Identify the square root. Square these numbers. 12 22 32 42 52 62 72 82 92 102 112 122 10.5 Rational Numbers Name ________________________________________________ _ A rational number is any real number that can be made by dividing two integers. In other words, any number that can be expressed as a fraction. Rational numbers include the set of whole numbers, the set of integers, plus fractions. Examples: The whole number 1 is rational because you can write it as. The negative integer −6 is rational because you can write it as. The mixed number is rational because you can write it as. The decimal 0.8 is rational because you can write it as. The repeating decimal is rational because you can write it as. Any rational number can be written as a decimal that is finite (ends) or eventually repeats. To convert a repeating decimal to a fraction, set the repeating decimal equal to x. Find a power of 10 that you can multiply the number by to capture the repeating section, then subtract the original number and solve for x.  Example: Convert 3.151515… to a fraction. First, set the whole number 3 aside to isolate the repeating decimal:.151515… Let x =.151515… The repeating part is 15. You will need to go out 2 decimal places to capture the repeating part, so multiply by 102 to capture the repeating section. 100x = 15.1515… Subtract x and the repeating decimal disappears. Now add back the whole integer 3 that we set aside in the beginning: Exercises CALCULATE Circle each group the number belongs to (there can be more than one). Change each number to a fraction. −23 __________________ 0.156 __________________ 19 __________________ −8 __________________ 1.945 __________________ 78 __________________ 76.38 __________________ −302 __________________ 10.6 Irrational Numbers Name ________________________________________________ _ An irrational number is any number that belongs to the set of real numbers, but cannot be written as a fraction made by dividing two integers. Unlike rational numbers that can be written as decimals that end or eventually repeat, irrational numbers are decimals that go on and on forever without ever repeating. Example: π is irrational because it equals a decimal that does not end or repeat: 3.14159265359… What about the integers that are between the perfect squares? We know that their square roots are not integers. The number 45 is not a perfect square, but you know that its square root must be greater than 6 and less than 7 but closer to 7 as 62 = 36 and 72 = 49. The square root of a number that is not a perfect square is an irrational number. is irrational because it equals a decimal that does not end or repeat: 1.41421356237… Exercises CALCULATE Estimate the square root of a number.  Noah’s garden is a square with an area of 73 square feet. The length of each side is between which two whole numbers? ________________ Isabella wants to add a wallpaper border along one wall of her room. If her room is a square with an area of 154 square feet, and the wallpaper border is sold only in whole numbers of feet, how many feet of border should she purchase? ________________ 10.7 Scientific Notation Name ________________________________________________ _ Is it difficult to write and read long numbers like 4,500,000,000 or 61,020,000? Is there a simpler way to express long numbers? Yes, you could use scientific notation, a way to express any number as a product of 10 and a decimal greater than 1. Examples: 4,500,000,000 = 4.5 × 109  61,020,000 = 6.102 × 107 When you use scientific notation, notice that the decimal is always greater than 1, but less than 10. You might think the difficult part is figuring out which power of 10 to use. However, that is not so hard. Look at the number in standard, or regular, notation. If the number does not have a decimal point, put one at the far right of the number. You want to move the decimal point left or right until you create a number that is greater than 1 but less than 10. Count the number of places you had to move the decimal point to do that. If you moved the decimal point to the left, the power will be positive. If you moved the decimal point to the right, the power will be negative. Exercises CONVERT Write each number using scientific notation..0013 ______________ 810.114 ______________ 4.0095 ______________.00005 ______________.5851 ______________  220.467 ______________ 426.7 ______________ 11901.55 ______________.0606544 ______________.8852 ______________ 1488.951 ______________ 200001.990 ______________.0006660 ______________.002679 ______________ 1.1110 ______________ 3007.5 ______________ Write each number in standard form. 2.6699 × 105 ______________ 1.4455 × 103 ______________ 9.6603171 × 106 ______________ 3.0302 × 104 ______________ 2.77 × 10−3 ______________ 3.919181 × 105 ______________ 1.588 × 103 ______________ 1.0801 × 10−2 ______________ 10.8 Estimation and Comparison Name ________________________________________________ _ Scientific notation can be very useful to estimate the size of very large or very small things. Example: A grain of sand measures 0.00212 inches in diameter. We can convert this to scientific notation by writing 2.12 × 10 −3. We can estimate the value to be 2 × 10−3. You can also use scientific notation to compare the size of things. You can perform operations on numbers in scientific notation just as you would with numbers in standard notation. To find out how many times larger one number is than another, we divide. We can say that 8 is 2 times larger than 4 because 8 ÷ 4 = 2. The same can be done with numbers in scientific notation. Example: 6 × 105 is how many times larger than 2 × 103? Set it up as a division problem.  The first part is easy: To divide exponents, you can just subtract the exponents if they have the same base. Expanded out, we have: = 100. That’s the same as subtracting the exponents: 105−3 = 102,. The full answer then is 3 × 102 or 300. 6 × 105 is 300 times larger than 2 × 103. Exercises Use this chart to answer the questions. Write each distance in the chart in scientific notation. How many times larger is Uranus than Saturn? How many times larger is Jupiter than Earth? Estimate the distance from the Earth to the Sun in scientific notation. 11.1 Order of Operations Name ________________________________________________ _ What happens if you see a long string of mathematical calculations to perform? Is there some way to know where to begin? Yes, you can use a rule called the order of operations. It tells you in what order you should do calculations for equations in a long string. There is even a simple word that can help you remember the order of operations: PEMDAS. This stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. Example: Solve: 20 − 5 × 2 + 36 ÷ 32 − (9 − 2) = ? We can solve this problem using the order of operations. Remember PEMDAS. 1. Parentheses (9 − 2) = 7 20 − 5 × 2 + 36 ÷ 32− 7 = ? 2. Exponents 32 = 9 20 − 5 × 2 + 36 ÷ 9 − 7 = ? 3. Multiplication and Division 5 × 2 = 10 36 ÷ 9 = 420 − 10 + 4 −7=? 4. Addition and Subtraction 20 − 10 = 10 10 + 4 = 14 14 − 7 = 7 Exercises CALCULATE (5 + 2) × (5 − 3) − (3 × 3) + 2(5 − 2) (6 − 5) × (6 − 4) − 23 +6 (7 − 4)3 + (7 − 2)2 + 5 − 2 + 3(5 − 3) (8 + 2) × (8 − 5) + 42 − (7 − 4)3 (8 − 6)3 − (7 − 5)3 + 9 − (5 − 2) (4)2 − (23 − 5) + (22 + 2) − 23 (7 − 2) + (8 − 5) − (4 − 1) − 2 (4 + 3) × (5 − 2) × (2 − 1)2 11.2 Commutative and Associative Properties Name ________________________________________________ _ Numbers behave in specific ways. Each kind of number behavior is called a property. Commutative Property of Addition: You can add addends in any order without changing the sum. 7 + 3 + 6 = 6 + 3 + 7 Commutative Property of Multiplication: You can multiply factors in any order without changing the product. 5 × 2 × 9 = 9 × 5×2 Associative Property of Addition: You can group addends any way you like without changing the sum. (7 + 8) + 3 = 7 + (8 + 3) Associative Property of Multiplication: You can group factors any way you like without changing the product. (3 × 12) × 4 = 3 × (12 × 4) Exercises IDENTIFY Identify which property is represented in the example (Commutative or Associative). 8×4×3=3×4×8 ___________________________________________  (2 + 9) + 22 = 2 + (9 + 22) ___________________________________________ 3×4+4×2=4×3+2×4 ___________________________________________ 4×2+3×4=3×4+4×2 ___________________________________________ 3+2+4=4+2+3 ___________________________________________ 12 + (7 + 1) = (12 + 7) + 1 ___________________________________________ (2 + (8 + 6) + 4) = (2 + 8) + (6 + 4) ___________________________________________ 6×4×2=2×4×6 ___________________________________________ (8 + 2) + 9 + 14 = 8 + (2 + 9) + 14 ___________________________________________ 9 × 4 + 27 + 4 × 9 = 4 × 9 + 27 + 9 × 4 ___________________________________________ 7+9+6+4=4+6+9+7 ___________________________________________ (9 + 9) + 6 + 9 = 9 + (9 + 6) + 9 ___________________________________________  29 + 2 + 1 + 29 = 2 + 1 + 29 + 29 ___________________________________________ 28 + 20 + 28 + 20 = 20 + 20 + 28 + 28 ___________________________________________ 11.3 Distributive and Identity Properties Name ________________________________________________ _ Distributive Property of Multiplication: To multiply a sum of two or more numbers, you can multiply by each number separately, and then add the products. To multiply the difference of two numbers, multiply separately and subtract the products. Some numbers in a problem do not affect the answer. These numbers are called Identity Elements. With adding, the identity element is 0, because any addend or addends + 0 will not change the total. In multiplication, the identity element is 1, because any factor or factors × 1 will not change the product. However, subtraction and division do not have identity elements. Examples: 9 × (2 + 5) = (9 × 2) + (9 × 5) 5 × (12 − 10) = (5 × 12) − (5 × 10) You can also use the Distributive Property for dividing, but only if the numbers you are adding or subtracting are in the dividend. (28 + 8) ÷ 4 = (28 ÷ 4) + (8 ÷ 4) However, you cannot use the Distributive Property when the numbers you are adding or subtracting are in the divisor. 28 ÷ (4 + 2 ) does not = (28 ÷ 4) + (28 ÷ 2) Exercises IDENTIFY THE PROPERTY 0+6=6 ___________________________________________ 4 (4 + 1) = 4 × 4 + 4 × 1 ___________________________________________ 7+0=7 ___________________________________________ 9 × 12 + 9 × 9 = 9 (12 + 9) ___________________________________________ 4(1 + 1) = 4(1) + 4(1) ___________________________________________ 15 + (4 + 0) = 15 + 4 ___________________________________________ 4 (11 + 9) = 4 × 11 + 9 × 4 ___________________________________________ 6×5+0=6×5 ___________________________________________ (34 × 0) + (7 × 0) = 0(34 + 7) ___________________________________________  (35 − 1) + (3 − 3 + 0) = (35 − 1) + (3 − 3) ___________________________________________ Rewrite the equation using the Distributive Property. 4 (5 + 7) ___________________________________________ 6×2+8×2 ___________________________________________ 11.4 Properties of Equality and Zero Name ________________________________________________ _ Zero Property of Multiplication: Any number multiplied by zero will be zero. 16 × 0 = 0 Equality Property of Addition: You can keep an equation equal if you add the same number to both sides. (6 + 4) + 3 = (9 + 1) + 3 Equality Property of Subtraction: You can keep an equation equal if you subtract the same number from both sides. (6 + 4) − 5 = (9 + 1) − 5 Equality Property of Multiplication: You can keep an equation equal if you multiply both sides by the same number. (6 + 4) × 10 = (9 + 1) × 10 Equality Property of Division: You can keep an equation equal if you divide both sides by the same number. (6 + 4) ÷ 2 = (9 + 1) ÷2 You may never divide a number by zero. Exercises SOLVE 5×0  (2 + 4) 0 0 × 3.56 2 (5 − 5) Identify which Equality Property is being displayed. If 8 + 1 = 6 + 3, then does 4 (8 + 1) = 4 (6 + 3)? _____________________ If 3 × 15 = 9 × 5, then does 6 + 3 × 15 = 6 + 9 × 5? ___________________ If 20 − 3 = 12 + 5, then does 20 − 3 + 11.5 = 12 + 5 + 11.5? _____________________ If 3 + 4 + 1 = 11 − 3, then does 3 + 4 + 1 + 8 = 11 − 3 + 8? _____________________ 12.1 Negative Numbers Name ________________________________________________ _ Negative numbers are numbers that are less than zero. You identify them by adding a minus sign to the front of a number. So − 1 is 1 less than 0. − 53.5 is 53.5 less than 0. There are special symbols used in comparing the number value parts. < means “less than”  > means “greater than” ≤ means “less than or equal to” ≥ means “greater than or equal to” = means “equal to” Zero (0) is neither positive nor negative. Examples: Look at the number line. Notice that −3 is three spaces to the left of 0 on the negative side. Also note that 3 is three spaces to the right of 0, on the positive side. The Property of Additive Inverse: When you add a negative number to its inverse (its exact opposite on the other side of the number line), the total is 0. For example, −7 + 7 = 0. Exercises CALCULATE  On the number line, place and label the number values as points on the number line. A (−1), B (1), C (−2.5), D (1.5), E (−8), F (−8.5), G (−4.5)Then order values from least to greatest. __________________________________________________ ____ Order the number values from greatest to least.4.6, −6.6, −6, −6.7, −4.3, 3, 3.3, −3.3, 8, −8.1 __________________________________________________ Use > (is greater than), < (is less than), or = (is equal to) in comparing the number value pairs. −4.5 ____________ 4.5 −3.3 _______________ −3.0 7.5 _____________ −7.5 −7.5 _____________ 1.5 −6.1 ____________ −6.25 −5 _____________ −4.95 13.9 _______________ 13.9 −9 _____________ 9 −8 _____________ −.888 12.2 Adding and Subtracting Negative Numbers Name ________________________________________________ _ When you add a positive number and a negative number, compare the numbers as if they do not have positive or negative signs. If the positive number is larger, just subtract. Example: 6 + (−4) = ? Step 1: Remove the + sign and the parentheses. Step 2: Subtract: 6 − 4 = 2 If the negative number is larger, subtract the smaller number from the larger one. Then put a minus sign in front of the difference. Example: 1 + (−3) = ? Step 1: Subtract the smaller from the greater number: 3 − 1 =2 Step 2: Write a minus sign in front of the difference: 1 + (−3) = −2 When adding two negative numbers, ignore the minus sign and add. Then write a minus sign in front of the total.  Example: (−1) + (−6) = ? Step 1: Ignore the minus signs and add: 1 + 6 = 7 Step 2: Write a minus sign in front of the total: (−1) + (−6) = −7 Exercises ADD OR SUBTRACT 171 + (−32) = _________________ (−145) + 61 = __________________ 111 + (−112) = __________________ 715 + (−316) = __________________ 1101 + (−561) − 114 = ______________ 295 + (−365) + (−111) = ____________ (−210) + (−210) − 427 = ________ 42 + 22 + (−67) − (−31) = _______ 12 − (−29) − 29 = _____________ 12.3 Multiplying and Dividing Negative Numbers Name ________________________________________________ _ Negative numbers can also be multiplied and divided. You just need to remember these two rules: When two numbers have the same sign, either negative or positive, their product, or quotient, is positive. When two numbers have different signs, one negative and one positive, their product, or quotient, is negative. Exercises MULTIPLY OR DIVIDE −5 × (−3) = _____________ 15 × (−10) = _____________ −100 ÷ 12 = _____________ −25 × −3 = _____________