FACTOR LABEL…

1. 10.0 in X X = 0.254 m 2.54 cm 1 in 1m 100 cm FACTOR LABEL…

2. Factor label (also known as dimensional analysis) is a fast, easy way to solve a wide variety of math problems. What exactly is factor label? All you need is an equality…

3. This equality can be made into two fractions: And these two fractions can be used to help us solve specific math problems… For example, you are probably familiar with this equality: 1 ft = 12 in 1 ft 12 in 12 in 1 ft &

4. 54 in 1 If we think of this 54 in as being , then it’s clear that we want to get rid of the unit “in”on top and replace it with a new unit “ft”. Recall that when you multiply fractions, terms that are in the numerator (top) of one fraction will cancel out terms that are in the denominator (bottom) of another. Let’s say you need to convert 54 in into ft. (You probably already know how to do this, but let’s pretend you don’t!) 5 7 7 8 The 7’s will cancel each other out. For example, when you multiply these two fractions… X

5. 54 in 1 54 in 1 (click on your choice) ? X So, if we want to multiply by a fraction that will cancel out the unwanted “in” on top, which of the two fractions below would you pick? 1 ft 12 in 12 in 1 ft

6. 54 in 1 54 in 1 (click on your choice) With “in” on top of both fractions, ? X they will not cancel the way you want them to… So, if we want to multiply by a fraction that will cancel out the unwanted “in” on top, which of the two fractions below would you pick? 1 ft 12 in 12 in 1 ft Click here to try again.

7. 54 in 1 54 in 1 (click on your choice) Good choice! See how the “in”s cancel each other out… ? X So, if we want to multiply by a fraction that will cancel out the unwanted “in” on top, which of the two fractions below would you pick? And see how the units are now “ft” 1 ft 12 in 12 in 1 ft which is just what we wanted.

8. Easy: just take 54 and divide it by 12. 54 in 1 This is the answer = 4.5 ft X We divide by 12 because the 12 is on the bottom of the fraction. So now that we have it all set up, how do we calculate the answer? 1 ft 12 in If the 12 had been on top of the fraction, then you would take the 54 and multiply it by the 12. So remember, anything on top, you multiply by; anything on bottom you divide by.

9. 18.3 mi 1 18.3 mi 1 18.3 mi 1 Again, let’s think of the 18.3 mi as And we’ll make the equality into two fractions: So, which of these do we want to multiply by to get rid of the “mi?” (click on your choice) ? Let’s try another one: Let’s say you had to convert 18.3 mi into km. And you were given the equality that 1 mi = 1.61 km. X 1 mi 1.61 km 1.61 km 1 mi

10. 18.3 mi 1 18.3 mi 1 18.3 mi 1 Again, let’s think of the 18.3 mi as And we’ll make the equality into two fractions: So, which of these do we want to multiply by to get rid of the “mi?” (click on your choice) With “mi” on top of both fractions, ? Let’s try another one: Let’s say you had to convert 18.3 mi into km. And you were given the equality that 1 mi = 1.61 km. X 1 mi 1.61 km 1.61 km 1 mi they will not cancel the way you want … Click here to try again.

11. 18.3 mi 1 18.3 mi 1 18.3 mi 1 Again, let’s think of the 18.3 mi as And we’ll make the equality into two fractions: So, which of these do we want to multiply by to get rid of the “mi?” (click on your choice) Good choice! You made the “mi”s cancel out… ? Let’s try another one: Let’s say you had to convert 18.3 mi into km. And you were given the equality that 1 mi = 1.61 km. X 1 mi 1.61 km 1.61 km 1 mi And now the units are “km” which is what we wanted.

12. We take 18.3 and do what with it? (click on your choice) Divide it by 1.61 Multiply it by 1.61 18.3 mi 1 1.61 km 1 mi And now to calculate the answer… X

13. We take 18.3 and do what with it? (click on your choice) Divide it by 1.61 Multiply it by 1.61 18.3 mi 1 Not quite... Notice that the 1.61 is on top of the fraction. Click here to try again 1.61 km 1 mi And now to calculate the answer… X

14. We take 18.3 and do what with it? (click on your choice) Divide it by 1.61 Multiply it by 1.61 18.3 mi 1 That’s right, we multiply by 1.61 because the 1.61 is on top… That gives us an answer of… 1.61 km 1 mi And now to calculate the answer… = 29.5 km X

15. Sometimes, we do not have enough information to go directly from one unit to another. When this happens, it may be necessary to use factor label as a series of steps – strung together like cars in a train. When this is done, the entire “train” can be set up and then calculations done all together at the end. This can be a big time saver.

16. 1 qt = 1.057 L 1 L = 1000 mL 1 qt = 16 oz Listed below are the six possible fractions that can be made from these equalities: Let’s say you wanted to convert 83.4 mL into oz, and you were given the three equalities at right 1 qt 1.057 L 1 L 1000 mL 1 qt 16 oz 1.057 L 1 qt 1000 mL 1 L 16 oz 1 qt

17. 1 qt = 1.057 L 1 L = 1000 mL 1 qt = 16 oz 84.3 mL 1 83.4 mL 1 Listed below are the six possible fractions that can be made from these equalities: So, which of these do we want to multiply by to get rid of the “mL?” (click on your choice) ? X Let’s say you wanted to convert 83.4 mL into oz, and you were given the three equalities at right 1 qt 1.057 L 1 L 1000 mL 1 qt 16 oz 1.057 L 1 qt 1000 mL 1 L 16 oz 1 qt

18. Sorry, that wouldn’t work. Click here to try again.

19. 1 qt = 1.057 L 1 L = 1000 mL 1 qt = 16 oz 84.3 mL 1 83.4 mL 1 Listed below are the six possible fractions that can be made from these equalities: So, which of these do we want to multiply by to get rid of the “mL?” (click on your choice) ? X Let’s say you wanted to convert 83.4 mL into oz, and you were given the three equalities at right Good; now the “mL”s cancels out. what’s next? (click on your choice) 1 qt 1.057 L 1 L 1000 mL 1 qt 16 oz 1.057 L 1 qt 1000 mL 1 L 16 oz 1 qt

20. 1 qt = 1.057 L 1 L = 1000 mL 1 qt = 16 oz 84.3 mL 1 83.4 mL 1 Listed below are the six possible fractions that can be made from these equalities: So, which of these do we want to multiply by to get rid of the “mL?” (click on your choice) ? X X Very good; that cancels out “L.” What’s next? (click on your choice) Let’s say you wanted to convert 83.4 mL into oz, and you were given the three equalities at right 1 qt 1.057 L 1 L 1000 mL 1 qt 16 oz 1.057 L 1 qt 1000 mL 1 L 16 oz 1 qt

21. 1 qt = 1.057 L 1 L = 1000 mL 1 qt = 16 oz 84.3 mL 1 83.4 mL 1 Listed below are the six possible fractions that can be made from these equalities: So, which of these do we want to multiply by to get rid of the “mL?” (click on your choice) 1 qt 1.057 L ? X X X Let’s say you wanted to convert 83.4 mL into oz, and you were given the three equalities at right 1 L 1000 mL 1 qt 16 oz 1.057 L 1 qt 1000 mL 1 L 16 oz 1 qt Excellent! That cancels out the “qt”s and gets us into “oz”s

22. 1 qt = 1.057 L 1 L = 1000 mL 1 qt = 16 oz 84.3 mL 1 83.4 mL 1 Listed below are the six possible fractions that can be made from these equalities: So, which of these do we want to multiply by to get rid of the “mL?” (click on your choice) 1 qt 1.057 L ? X X X = 1.26 oz Let’s say you wanted to convert 83.4 mL into oz, and you were given the three equalities at right 1 L 1000 mL 16 oz 1 qt Now that the factor label fractions are all lined up, we’re ready to calculate the answer: First we enter “83.4” into the calculator, then we divide by “1000,” and finally, we multiply by “16.” This gives us an answer of… then we divide by “1.057,”

23. This concludes the tutorial. Try the problems on the factor label worksheet, checking your answers with the ones listed.

24. 1 qt = 1.057 L 1 L = 1000 mL 1 qt = 16 oz Convert 4.53 qt into mL. Get it all set up on scrap paper before continuing. 4.53 qt 1 Is this what your set-up looks like: = 4790 mL X X Now try one on your own using the same equalities listed at right: 1000 mL 1 L 1.057 L 1 qt If so, good. Now calculate the answer. Is this what you came up with?