by Josh Rappaport
Once in a while you notice something very strange, and you just have to tell people about it.
This article describes "one of those things" that I recently noticed.
I was looking at a piece I'd recently written about finding the Least Common Multiple, aka the LCM, and I was staring at the numbers 12 and 16 when the most peculiar thing occurred to me.
First, though, let's back up to recall what the LCM is. The LCM, you remember, is the smallest number that two or more numbers divide into evenly. For example, given the numbers 12 and 16, the LCM has to be 48 since both 12 and 16 divide evenly into 48, and since 48 is the smallest number that they both go into. 12 divides into 48 four times, and 16 divides into 48 three times.
But why should we "give a hoot" about the LCM? Well, as you also may recall, we use the LCM a great deal when adding or subtracting fractions. That's because we can add or subtract fractions only if they have the same denominator. And the LCM is the number that becomes the common denominator for our fractions, if they didn't have one to begin with. So it turns out that knowing how to find the LCM is a good thing, especially when working with fractions.
Anyhow, back to the storyline. So there I was, staring at 12 and 16, when an bizarre fact popped up like a "Jack-in-the-Box." If you take those two numbers and make them into a proper fraction, 12/16, you can simplify that down to get 3/4. Then, if you flip that fraction around, you get 4/3.
But here's the "kicker." I noticed that if you take that flipped fraction, 4/3, and multiply it by your first fraction, 12/16, you get 48/48. You can see this if you look at the problem: 4/3 x 12/16. We just multiply the top numbers together to get 48 and we do the same for the bottom numbers, also getting 48.
So why is this a big deal, you might ask. The big deal is that 48 "just happens" to be the least common multiple (LCM) for 12 and 16. And the even bigger deal is that following these pretty simple steps will — I verified this through a mathematical proof — always lead you to the LCM, like finding the pot of gold at the end of the rainbow.
By now you might be getting tired of all of these words and hopefully ready to see another example. So here goes. Take the numbers, 24 and 30, and imagine that these numbers are the denominators of two fractions we're going to add or subtract, as in the problem: 23/24 – 7/30. Now let's just follow the same steps I laid out above.
Step 1) Take the two denominators, 24 and 30, and put them together to form a proper fraction. That would be the fraction: 24/30.
Step 2) Now simplify this fraction. Divding both 24 and 30 by 6, we see that 24/30 = 4/5.
Step 3) Flipping this simplified fraction, 4/5, we get 5/4.
Step 4) And now, for the "kicker." Multiply the original fraction, 24/30 by this flipped fraction, 5/4. And what do we get?
Well, 5/4 x 24/30 = 120/120, and this tells us that 120 must be the least common multiple (LCM) for 24 and 30.
If you like this approach, there's something you might like even more. Using this approach, it's a cinch to figure out what number you need to multiply the original fractions by to convert them into fractions with denominators of 120. And we need our fractions to have that common denominator so we can add or subtract them, right?
Looking at the multiplication we just did, you'll notice that we multiplied 24 by 5 to get 120; and similarly, we multiplied 30 by 4 to get 120. That means that, to convert the two original fractions, we just multiply 23/24 by (5/5); and similarly, we multiply the fraction 7/30 by (4/4).
Doing this, the original problem gets converted to: 115/120 – 28/120. And the answer to that is 87/120, which simplifies down to 29/40. Voila. All done, and we didn't even break a sweat!
In any case, the main point here is that this approach gives us a handy shortcut for finding the LCM.
If you think back to elementary school, a lot of us get taught that the way to find the LCM is to grind out the multiples for each denominator until you finally find a multiple that matches. That's what we in the math world call a "brute force" approach because, while it does work, it's lacking in sophistication and grace. It turns out that there are several more subtle and powerful ways to find the LCM, and this "flip-the-fraction" approach joins the club of those other approaches. At least I'd like to think so, since I thought it up myself!
But in any case, if you and your children find this "flip-the-fraction" approach intriguing, I suggest you do the following practice problems so you can get even better at it. So get those pencils sharp, and try your hand at these.
FIND THE LCM FOR EACH NUMBER PAIR:
a) 6 and 8
b) 4 and 10
c) 9 and 15
d) 10 and 16
e) 14 and 21
f) 18 and 45
g) 24 and 28
h) 27 and 63
i) 32 and 48
j) 45 and 55
ANSWERS:
a) 6 and 8; LCM = 24
b) 4 and 10; LCM = 20
c) 9 and 15; LCM = 45
d) 10 and 16; LCM = 80
e) 14 and 21; LCM = 42
f) 18 and 45; LCM = 90
g) 24 and 28; LCM = 168
h) 27 and 63; LCM = 189
i) 32 and 48; LCM = 96
j) 45 and 55; LCM = 495
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two teenage children. Josh is the author of the Algebra Survival Guide, and the companion Algebra Survival Guide Workbook, both of which will soon be available for homeschoolers as a computer-based Learning Management System, developed and run by Sleek Corp., of Austin, TX.
Once in a while you notice something very strange, and you just have to tell people about it.
This article describes "one of those things" that I recently noticed.
I was looking at a piece I'd recently written about finding the Least Common Multiple, aka the LCM, and I was staring at the numbers 12 and 16 when the most peculiar thing occurred to me.
First, though, let's back up to recall what the LCM is. The LCM, you remember, is the smallest number that two or more numbers divide into evenly. For example, given the numbers 12 and 16, the LCM has to be 48 since both 12 and 16 divide evenly into 48, and since 48 is the smallest number that they both go into. 12 divides into 48 four times, and 16 divides into 48 three times.
But why should we "give a hoot" about the LCM? Well, as you also may recall, we use the LCM a great deal when adding or subtracting fractions. That's because we can add or subtract fractions only if they have the same denominator. And the LCM is the number that becomes the common denominator for our fractions, if they didn't have one to begin with. So it turns out that knowing how to find the LCM is a good thing, especially when working with fractions.
Anyhow, back to the storyline. So there I was, staring at 12 and 16, when an bizarre fact popped up like a "Jack-in-the-Box." If you take those two numbers and make them into a proper fraction, 12/16, you can simplify that down to get 3/4. Then, if you flip that fraction around, you get 4/3.
But here's the "kicker." I noticed that if you take that flipped fraction, 4/3, and multiply it by your first fraction, 12/16, you get 48/48. You can see this if you look at the problem: 4/3 x 12/16. We just multiply the top numbers together to get 48 and we do the same for the bottom numbers, also getting 48.
So why is this a big deal, you might ask. The big deal is that 48 "just happens" to be the least common multiple (LCM) for 12 and 16. And the even bigger deal is that following these pretty simple steps will — I verified this through a mathematical proof — always lead you to the LCM, like finding the pot of gold at the end of the rainbow.
By now you might be getting tired of all of these words and hopefully ready to see another example. So here goes. Take the numbers, 24 and 30, and imagine that these numbers are the denominators of two fractions we're going to add or subtract, as in the problem: 23/24 – 7/30. Now let's just follow the same steps I laid out above.
Step 1) Take the two denominators, 24 and 30, and put them together to form a proper fraction. That would be the fraction: 24/30.
Step 2) Now simplify this fraction. Divding both 24 and 30 by 6, we see that 24/30 = 4/5.
Step 3) Flipping this simplified fraction, 4/5, we get 5/4.
Step 4) And now, for the "kicker." Multiply the original fraction, 24/30 by this flipped fraction, 5/4. And what do we get?
Well, 5/4 x 24/30 = 120/120, and this tells us that 120 must be the least common multiple (LCM) for 24 and 30.
If you like this approach, there's something you might like even more. Using this approach, it's a cinch to figure out what number you need to multiply the original fractions by to convert them into fractions with denominators of 120. And we need our fractions to have that common denominator so we can add or subtract them, right?
Looking at the multiplication we just did, you'll notice that we multiplied 24 by 5 to get 120; and similarly, we multiplied 30 by 4 to get 120. That means that, to convert the two original fractions, we just multiply 23/24 by (5/5); and similarly, we multiply the fraction 7/30 by (4/4).
Doing this, the original problem gets converted to: 115/120 – 28/120. And the answer to that is 87/120, which simplifies down to 29/40. Voila. All done, and we didn't even break a sweat!
In any case, the main point here is that this approach gives us a handy shortcut for finding the LCM.
If you think back to elementary school, a lot of us get taught that the way to find the LCM is to grind out the multiples for each denominator until you finally find a multiple that matches. That's what we in the math world call a "brute force" approach because, while it does work, it's lacking in sophistication and grace. It turns out that there are several more subtle and powerful ways to find the LCM, and this "flip-the-fraction" approach joins the club of those other approaches. At least I'd like to think so, since I thought it up myself!
But in any case, if you and your children find this "flip-the-fraction" approach intriguing, I suggest you do the following practice problems so you can get even better at it. So get those pencils sharp, and try your hand at these.
FIND THE LCM FOR EACH NUMBER PAIR:
a) 6 and 8
b) 4 and 10
c) 9 and 15
d) 10 and 16
e) 14 and 21
f) 18 and 45
g) 24 and 28
h) 27 and 63
i) 32 and 48
j) 45 and 55
ANSWERS:
a) 6 and 8; LCM = 24
b) 4 and 10; LCM = 20
c) 9 and 15; LCM = 45
d) 10 and 16; LCM = 80
e) 14 and 21; LCM = 42
f) 18 and 45; LCM = 90
g) 24 and 28; LCM = 168
h) 27 and 63; LCM = 189
i) 32 and 48; LCM = 96
j) 45 and 55; LCM = 495
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two teenage children. Josh is the author of the Algebra Survival Guide, and the companion Algebra Survival Guide Workbook, both of which will soon be available for homeschoolers as a computer-based Learning Management System, developed and run by Sleek Corp., of Austin, TX.
At his blog, Josh writes about math education, offering tips and tricks. Josh also authors Turtle Talk, a free monthly newsletter with an engaging "Problem of the Month." You can see a sample issue here or subscribe at this site. Josh also is co-author of the "learn-by-playing" Card Game Roundup books, and author of PreAlgebra Blastoff!, a "Sci-Fi" cartoon math book featuring a playful, hands-on approach to positive and negative numbers.
In the summer Josh leads workshops at homeschooling conferences and tutors homeschoolers nationwide using SKYPE. Contact Josh by email @ josh@SingingTurtle.com or follow him on Facebook, where he poses two fun math Qs/day.
