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DIMENSIONAL ANALYSIS – STEPS TO CONVERT UNITS Dimensional Analysis (Factor Label) Using equivalent relationships to obtain unit conversion factors that help you solve a problem. Unit Conversion Factors – a ratio of two quantities that are equivalent and are used to convert from one unit to another. Two examples of the same conversion factor is shown below: 1 dollar 100 pennies 100 pennies 1 dollar To Solve: 1. Look at what is given in the problem. 2. Look at what is being asked in the problem (answer should be in, what units?). 3. Look for unit conversion factors to get from what is given to what is asked for. 4. For each step, arrange unit conversion factor so that the denominator of your conversion factor is what unit you are starting with, and the numerator is what you are ending with: numerator: what units you want to end with denominator: what units you start with 5. Use a solution map to visualize the route required to solve the problem Starting Units --------------------> Steps ------------> Ending Units Example 1: How many dollars is 5,000 nickels? ANSWER SHOULD BE IN WHAT UNITS? -------------------------------“dollars” HOW MANY NICKELS (what we start with) IS EQUAL TO A DOLLAR (what we end with) 20 nickels = 1 dollar ; We must decide which conversion factor to use below: 20 nickels 1 dollar and 1 dollar 20 nickels Use this one because it has the units our answer should be in “dollars” in the numerator!! numerator: what units you want to end with 5,000 nickels x $1.00 (dollar) 20 nickels = $250!! denominator: what units you start with Example 2: How many nickels is $12,480? Use other conversion factor because it has the units our answer should be in “nickels” in the numerator!! $12,480 x 20 nickels = 249,600 nickels!! numerator: what units you want to end with $1.00(dollar) denominator: what units you start with Dimensional Analysis Worksheet Name___________________________ 1. Convert 42 centimeters to inches. (2.54 centimeters = 1 inch) 2. Convert 50 milliliters to liters. (1000 milliliters = 1 liter) 3. Convert 285 grams to pounds. (454 grams = 1 pound) 4. How many minutes are there in 3 hours? 5. How many minutes are there in 3 days? 6. If you are 2.3 meters tall, what is your height in inches? (39.37 inches = 1 meter) 7. A flea wing weighs 0.00000703 pounds. How many milligrams does it weigh? (1 pound = 454 grams, 1000 milligrams = 1 gram) (Try to answer this one using scientific notation) 2 8. Convert 1.2 x 10 nm to km. 9. If an airline charges 16 cents per mile, how many miles can you travel for $80? Name Date Class Math Skills for Science MATH SKILLS Multiplying and Dividing in Scientific Notation Part 1: Multiplying in Scientific Notation PROCEDURE: To multiply numbers in scientific notation, multiply the decimal numbers. Then add the exponents of the powers of 10. Place the new power of 10 with the decimal in scientific notation form. If your decimal number is greater than 10, count the number of times the decimal moves to the left, and add this number to the exponent. SAMPLE PROBLEM: Multiply (2.6 ⫻ 107) by (6.3 ⫻ 104). 2.6 ⫻ 6.3 ⫽ 16.38 7 ⫹ 4 ⫽ 11 Step 3: Put the new decimal number with the new exponent in scientific notation form. Step 4: Because the new decimal number is greater than 10, count the number of places the decimal moves to put the number between 1 and 10. Add this number to the exponent. In this case, the decimal point moves one place, so add 1 to the exponent. 16.38 ⫻ 1011 1哬 6.38 ⫻ 1011 → 1.638 ⫻ 1012 Try It Yourself! Copyright © by Holt, Rinehart and Winston. All rights reserved. 1. Follow the steps in the Sample Problem carefully to complete the following equations. Multiplying with Scientific Notations Problem Sample problem: (4.4 ⫻ 106) ⫻ (3.9 ⫻ 104) New decimal New exponent Answer 4.4 ⫻ 3.9 ⫽ 17.16 6 ⫹ 4 ⫽ 10 1.716 ⫻ 1011 a. (2.8 ⫻ 108) ⫻ (1.9 ⫻ 104) b. (1.3 ⫻ 109) ⫻ (4.7 ⫻ 10⫺5) c. (3.7 ⫻ 1015) ⫻ (5.2 ⫻ 107) d. (4.9 ⫻ 1024) ⫻ (1.6 ⫻ 105) 2. The mass of one hydrogen atom is 1.67 ⫻ 10 –27 kg. A cylinder contains 3.01 ⫻ 1023 hydrogen atoms. What is the mass of the hydrogen? 1 of 2 MATH SKILLS Step 2: Add the exponents. ▼ ▼ ▼ Step 1: Multiply the decimal numbers. Name Date Class Multiplying and Dividing in Scientific Notation, continued Part 2: Dividing in Scientific Notation PROCEDURE: To divide numbers in scientific notation, first divide the decimal numbers. Then subtract the exponents of your power of 10. Place the new power of 10 with the decimal in scientific notation form. If the resulting decimal number is less than 1, move the decimal point to the right and decrease the exponent by the number of places that the decimal point moved. SAMPLE PROBLEM: Divide (1.23 ⫻ 1011) by (2.4 ⫻ 104). Step 1: Divide the decimal numbers. Step 2: Subtract the exponents of the powers of 10. 1.23 ⫼ 2.4 ⫽ 0.5125 11 ⫺ 4 ⫽ 7 Step 3: Place the new power of 10 with the new decimal in scientific notation form. Step 4: Because the decimal number is not between 1 and 10, move the decimal point one place to the right and decrease the exponent by 1. 0.5125 ⫻ 107 0.5 125 ⫻ 107 → 5.125 ⫻ 106 哬 (1.23 ⫻ 1011) ⫼ (2.4 ⫻ 104) ⫽ 5.125 ⫻ 106 3. Complete the following chart: Problem Sample problem: (5.76 ⫻ 109) ⫼ (3.2 ⫼ 103) New decimal New exponent Answer 5.76 ⫼ 3.2 ⫽ 1.8 9⫺3⫽6 1.8 ⫻ 106 a. (3.72 ⫻ 108) ⫼ (1.2 ⫻ 105) b. (6.4 ⫻ 10⫺4) ⫼ (4 ⫻ 106) c. (3.6 ⫻ 104) ⫼ (6 ⫻ 105) d. (1.44 ⫻ 1024) ⫼ (1.2 ⫻ 1017) 4. The average distance from Earth to the sun is 1.5 ⫻ 1011 m. The speed of light is 3 ⫻ 108 m/s. Approximately how long does it take for light to travel from the sun to Earth? 2 of 2 Copyright © by Holt, Rinehart and Winston. All rights reserved. Dividing with Scientific Notation Intermediate Algebra Skill Multiplying and Dividing Using Scientific Notation Simplify. Write each answer in scientific notation. −6 1) (8.18 × 10 3) (0.8 × 10 4 5) (1.9 × 10 −3 7) 9) 11) 7.8 × 10 )(1.15 × 10 −5 ) )(1.28 × 10 6) )(2 × 10 4 ) −6 )(2 × 10 4 ) 4) (3.8 × 10 −6 )(2.37 × 10 −3 ) 6) (9.2 × 10 5 5.3 × 10 3 4 8) 8 × 10 1 4.6 × 10 2 10) 5.01 × 10 −3 5.5 × 10 −1 5.3 × 10 12) 2 13) (7.87 × 10 15) (4 × 10 2) (5.8 × 10 −1 −6 ) 1 −5 ) 4 19) (6.9 × 10 −6 )(770 × 10 2 ) 1 7.65 × 10 5 7.6 × 10 0 5.4 × 10 −6 2.04 × 10 2 × 10 −1 −2 14) (9.1 × 10 −5 −4 ) 16) (2.19 × 10 17) (9.7 × 10 −3 ) 21) (0.95 × 10 )(4 × 10 −3 ) )(0.4 × 10 −2 ) 23) (5.32 × 10 1 )(2.21 × 10 1 ) 4 −6 18) (5.9 × 10 0 ) ) −2 20) (7.57 × 10 −4 )(8.8 × 10 −1 ) 22) (12 × 10 5 )(2 × 10 2) 24) (8 × 10 −4 ) 6