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

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

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

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!) 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” in 1 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. For example, when you multiply these two fractions… X The 7’s will cancel each other out.

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? 54 in 1 1 ft 12 in 1 ft 54 in 1 X ? (click on your choice)

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? 54 in 1 1 ft 12 in 1 ft 54 in 1 X ? (click on your choice) With “in” on top of both fractions, they will not cancel the way you want them to… Click here to try again.

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? 54 in 1 1 ft 12 in 1 ft 54 in 1 X ? (click on your choice) Good choice! See how the “in”s cancel each other out… And see how the units are now “ft” which is just what we wanted.

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

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. Again, let’s think of the 18.3 mi as 18.3 mi 1 And we’ll make the equality into two fractions: 1 mi 1.61 km 1 mi 18.3 mi 1 X ? (click on your choice) So, which of these do we want to multiply by to get rid of the “mi?” 18.3 mi 1

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

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

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

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

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

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.

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

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

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

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

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

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

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