12 inches 4 feet = 48 inches

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1 Free Pre-Algebra Lesson 9! page Lesson 9 Converting Between Units How many minutes in a year? How many feet in sixteen and one-quarter miles? The need to convert between units arises so frequently that you can type your current and desired units into the Google search box and get an instant answer: Google 4 feet 9 inches -> inches Search About 4,50,000 results (0.44 seconds) Advanced search Everything More 4 feet 9 inches = 57 inches More about calculator. In this lesson, we take over from Google to do these conversions ourselves. The idea is simple and powerful, and brings together a lot of topics we ve already covered. Measurement Systems There are two main systems of measurement in use today. People living in the United States use the American system of measurement in everyday life, and the metric system for science. The American system measures length in inches, feet, yards, and miles, temperature in degrees Fahrenheit, weight in ounces and pounds, and liquid volume in cups, quarts, and gallons. The metric system measures length in meters, temperature in degrees Celsius, and weight in grams. The metric system is used in everyday life everywhere else besides the U.S. It s helpful to be familiar with both systems. When converting measurements, we may need to convert within a system or between systems. Conversions within a system are designed to be simple. For example, there are twelve inches in a foot, so converting between feet and inches is a matter of multiplying or dividing by twelve. But since meters and inches come from different systems, there is no simple whole number conversion factor there are approximately inches in a meter. Here we concentrate on conversions within the American system, since it is more familiar to most U.S. students. But no matter which system or what the conversion factor is, so long as one type of unit is some multiple of another we can use the method in this section. This method, called dimensional analysis, or the unit-factor method, is frequently used in science classes. The method uses an equation relating two units to create a fraction equal to, called a unit fraction. The units in the numerator and denominator of the fraction are set up to cancel with units in the problem to find the answer in the new more desirable units. For example, to change 4 feet to inches, we would do the multiplication below. inches 4 feet = 48 inches foot Here s the step-by-step explanation. Step : Making Unit Fractions If a unit of measurement is a multiple of another unit, we can write an equation relating the units. If we want to convert feet to inches, we write an equation relating feet and inches. There are one inches in one foot, I write foot = inches. From this equation, I can make a fraction, by dividing both sides of the equation by foot: foot foot = inches foot. inches The numerator and denominator on the left hand side of this equation cancel completely, so are equal to =. foot 00 Cheryl Wilcox

2 Free Pre-Algebra Lesson 9! page If I had divided both sides by inches, foot foot foot one), but it would still be equal to : inches = = inches foot, I would have a different fraction (the reciprocal of the first The great thing about the number is that it has the easiest times table: when you multiply anything by, it doesn t change. So if I multiply anything by either of the fractions above (which are equal to ), the form may change, but the amount stays the same. These unit fractions are the key to converting between units. Example: Write two unit fractions from the equation yard = feet. yard feet = = feet yard Example: Write two unit fractions from the equation day = 4 hours. day 4 hours = = 4 hours day Step : Setting Up the Cancellation There are two unit fractions from each equation relating units. The next step is to decide which to use. The goal is to cancel the original units of the problem, so the original units must go in the denominator of the fraction. The new units, the ones we are trying to find, go in the numerator. To Change Feet to Inches: Cancel feet in the original by putting feet in the denominator: 4 feet inches foot To Change Inches to Feet: Cancel inches in the original by putting inches in the denominator: 48 inches foot inches Example: Set up the unit fraction to change 8 feet to yards. Example: Set up the unit fraction to change days to hours. 8 feet yard feet days 4 hours day 00 Cheryl Wilcox

3 Free Pre-Algebra Lesson 9! page Step : Multiply Once you ve chosen the unit fraction you can just go ahead and multiply. The answer is there waiting for you. To Change Feet to Inches: Cancel feet in the original by putting feet in the denominator: 4 feet inches = 48 inches foot To Change Inches to Feet: Cancel inches in the original by putting inches in the denominator: 4 48 inches foot inches = 4 feet Complete Walkthrough Example: How many yards in 8 feet? ( feet = yard) Use the equation to make unit fractions: yard feet = = feet yard Set up the unit fraction to cancel the original units: 8 feet yard feet Multiply to find the new units: 6 8 feet yard feet = 6 yards Example: How many hours in days? (4 hours = day) Use the equation to make unit fractions: day 4 hours = = 4 hours day Set up the unit fraction to cancel the original units: days 4 hours day Multiply to find the new units: days 4 hours = 7 hours day Example: How many ounces in /4 of a pound? (6 ounces = pound) Use the equation to make unit fractions: pound 6 ounces = = 6 ounces pound Set up the unit fraction to cancel the original units: 4 pounds 6 ounces pound Multiply to find the new units: ounces = ounces It was convenient to just get rid of the cancelled units here because of the fraction. 00 Cheryl Wilcox

4 Free Pre-Algebra Lesson 9! page 4 More than One Step Unit fractions can be chained so that we convert several levels of units at once. Example: How many minutes in days? (60 minutes = hour; 4 hours = day) We re trying to get from days to minutes, but there is an intermediate unit, hours. We have two conversion equations, so there will be two unit fractions. We go in order of size of unit. days! hours! minutes We could do this in two separate one-step equations like the ones we just did: Days to Hours days 4 hours = 7 hours day 7 Hours to Minutes 7 hours 60 minutes = 40 minutes hour Or, the two unit fractions can be chained, one after the other: days 4 hours day 60 minutes = 40 minutes hour Example: How many miles is 0,000 inches? ( inches = foot; 580 feet = mile). My plan is to change inches to feet, then feet to miles. Begin by cancelling inches: 0,000 inches foot inches I can see that I ll need to multiply again to cancel feet and end up with miles. So I just continue the chain: Now multiply and cancel: 0,000 inches foot inches mile 580 feet 0,000 inches foot inches mile 580 feet = 0,000 6,60 miles 0,000 6,60 = = 5 79 miles While this answer is very proper, we d probably write it as a decimal (coming soon) in real life. At any rate, we ve found out that 0,000 inches is less than one mile. If you round the numerator and denominator to the nearest hundred, you get 00/800 or about /8 of a mile. 00 Cheryl Wilcox

5 Free Pre-Algebra Lesson 9! page 5 Using Conversion Factors From a Table In some situations you have a table of conversion information and you need to find what s relevant to your problem. Example: Find the number of fluid ounces in and / gallons. My plan is to go down in size, gallons! quarts! cups! ounces I don t need the information about pints, since I can go directly from quarts to cups. gallons 4 quarts gallons 4 cups quarts 8 ounces cup U.S. Units of Liquid Volume gallon = 4 quarts quart = pints quart = 4 cups pint = cups cup = 8 fluid ounces gallons 4 quarts gallons 4 cups quarts 8 ounces cup =! $ % & 4 = ( )( 4) ( 8) ounces ounces = 448 ounces! 00 Cheryl Wilcox

6 Free Pre-Algebra Lesson 9: Converting Between Units Worksheet Lesson 9! page 6a Name. Write two unit fractions from the equation 60 seconds = minute a. How many minutes is 00 seconds? b. How many seconds is 8 minutes?. Write two unit fractions from the equation 00 centimeters = meter a. How many meters is 00 cm? b. How many cm is 5 / meters?. Write two unit fractions from the equation gallon = 4 quarts a. How many quarts are in 9 / 8 gallons? b. How many gallons is 8 quarts? 4. A square foot has length inches and width inches, so the area is: a. Write two unit fractions from the equation sq foot = 44 sq in b. How many square feet is 080 square inches? 00 Cheryl Wilcox

7 Free Pre-Algebra Lesson 9! page 7a 5. Use the equations 60 seconds = minute 60 minutes = hour to find the number of seconds in 4 hours. 6. How many hours is one million seconds? 7. How many days is one million seconds? 8. The amazing vehicle traveled 500 feet per second for minutes. How far did it travel? a. Change minutes to seconds. b. Calculate the distance traveled, in feet. c. There are 580 feet in mile. How many miles did the vehicle travel? Estimate the number of seconds in one month, to the nearest million. Estimate the number of seconds in one year, to the nearest million. 00 Cheryl Wilcox

8 Free Pre-Algebra Lesson 9! page 8 Lesson 9: Converting Between Units Homework 9A Name. Measure the line. Add another line / 4 inch long to the end of the line. Find the total length. and 9/6 inches. Find the blue shaded area. Measurements are in inches. and /6 inches Write and solve the associated addition problem.. Solve with a multiplication. a. What is / of 66 days? 4. Solve with a division. a. How many /4 oz. pieces can be cut from a bar weighing 7 oz? b. What is / of $,000,000? b. How many / 8 inch lengths can be cut from a ribbon 8 inches long? 5. Divide the rectangles so they have the same number and size parts. Write equivalent fractions for the shaded areas with a common denominator. xxxxxxxxxxxx 4 xxxxx 6. After the potluck, Mahmood had /4 tray of brownies left and Darshani had / tray of brownies left. Darshani gave hers to Mahmood, and he ate them all. How much did he eat? xxxxx Did he eat more or less than / a tray? 00 Cheryl Wilcox

9 Free Pre-Algebra Lesson 9! page 9 7. Find equivalent fractions with a common denominator Find equivalent fractions with a common denominator Simplify. 4a 5 + 7a 0 7a. Add the fractions a. Subtract the fractions. 5! 0. Simplify.! 40 7b 0 + $ 8% & 7b. Add the mixed numbers b. Subtract the mixed numbers. 5 5!. Evaluate!x + x when x = 4.. How many inches is 00 feet? ( foot = inches). How many days is 00 hours? ( day = 4 hours) 4. How many hours is 00,000 seconds? ( hour = 60 minutes, minute = 60 seconds) 00 Cheryl Wilcox

10 Free Pre-Algebra Lesson 9! page 0 Lesson 9: Converting Between Units Homework 9A Answers. Measure the line. Add another line / 4 inch long to the end of the line. Find the total length. and 9/6 inches and /6 inches /4 inches. Find the blue shaded area. Measurements are in inches. Write and solve the associated addition problem.. Solve with a multiplication. a. What is / of 66 days? = = 5 6 = = days = days 4. Solve with a division. Square Rectangle =! $! $ 4% & 4% & '! $! $ % & 8% & = 9 6 ' 6 = 8 6 = Area is / square inch. a. How many /4 oz. pieces can be cut from a bar weighing 7 oz? 7 4 = = 6 pieces b. What is / of $,000,000? 500,000 $,000,000 = $500,000 b. How many / 8 inch lengths can be cut from a ribbon 8 inches long? 8 8 = = = 6 lengths 5. Divide the rectangles so they have the same number and size parts. Write equivalent fractions for the shaded areas with a common denominator. 4 = = 4 6. After the potluck, Mahmood had /4 tray of brownies left and Darshani had / tray of brownies left. Darshani gave hers to Mahmood, and he ate them all. How much did he eat? 4 + = + 4 = 7 He ate 7/ of a tray. Did he eat more or less than / a tray? More, 7/ > 6/ = / 00 Cheryl Wilcox

11 Free Pre-Algebra Lesson 9! page 7. Find equivalent fractions with a common denominator. 45 = 5 = = = a. Add the fractions = = = 45 = 5 7b. Add the mixed numbers = 5 + ( ) + = = 8 5! $ 45% & 8. Find equivalent fractions with a common denominator. 5 = 5 = 5 = 5 5 = 0 5 8a. Subtract the fractions. 5! = 5! 0 5 =! 7 5 8b. Subtract the mixed numbers. 5 5! = 5 5! 0 5 = 4 8 5! 0 5 = Simplify. 4a 5 + 7a 0 = 8a 0 + 7a 0 = 5 a = a 0 0. Simplify. 40 7b 0 + 5! $ 8% & = 40! 7b $ & 0 % + 40! $ & = 4b %. Evaluate!x + x when x = 4. =! % $ 4& ' + % $ 4& ' =! % $ 6& ' + 4 =! = 6. How many days is 00 hours? ( day = 4 hours) 5 00 hours day 4 hours = 5 days = days. How many inches is 00 feet? ( foot = inches) 00 feet inches =,600 inches foot 4. How many hours is 00,000 seconds? ( hour = 60 minutes, minute = 60 seconds) ,000 sec min 60 sec hour 60 min = hours = hours 00 Cheryl Wilcox

12 Free Pre-Algebra Lesson 9! page Lesson 9: Converting Between Units Homework 9B Name. Measure the line. Add another line / inch long to the end of the line. Find the total length. and 9/6 inches. Find the blue shaded area. Measurements are in inches. and /6 inches Write and solve the associated addition problem.. Solve with a multiplication. a. What is /4 of $,000,000? 4. Solve with a division. a. How many 5/8 oz. pieces can be cut from a bar weighing 5 oz? b. What is / of 48 months? b. How many 6 / 4 inch lengths can be cut from a string 500 inches long? 5. Divide the rectangles so they have the same number and size parts. Write equivalent fractions for the shaded areas with a common denominator. 6. Mercedes had /4 of a cup full of coffee, and Romelia poured in another / cup. How much coffee was there then? 4 xxxxx How much more coffee would be needed to make cup? xxxxxxxxxxx 00 Cheryl Wilcox

13 Free Pre-Algebra Lesson 9! page 7. Find equivalent fractions with a common denominator Find equivalent fractions with a common denominator Simplify. 7y 5 + 5y 7 7a. Add the fractions a. Subtract the fractions. 4! 0. Simplify.! 0 7b 0 + $ 4% & 7b. Add the mixed numbers b. Subtract the mixed numbers. 5 4!. Evaluate x! x when x =.. How many miles is,000,000 yards? ( mile = 760 yards). How many cups is 0 quarts? ( quart = 4 cups) 4. How many days is,000 minutes? ( hour = 60 minutes, day = 4 hours) 00 Cheryl Wilcox