I received an interesting question from alert reader Ivasallay a couple of days ago … about fractions.
Responding to my post about the fraction “hack” of using the gap between fraction numbers, Ivasallay wrote: “What if the numerator is smaller than the gap?”
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Good question, and thanks for sharing it. My answer: Yes, the numerator could be smaller than the gap, and if it is, that can help us simplify fractions, too.
Now we could have a fraction like 15/6, in which the lesser of the two numbers is the denominator, so to keep our discussion general I’m going to talk, not about the numerator, but rather about the “smaller fraction number,” whether numerator or denominator.
The way this matters is as follows: like the gap, the smaller fraction number provides an upper limit, a greatest possible value, for the GCF of the fraction’s two numbers. So if the fraction is 12/90 (smaller number being 12), that means that the GCF can be no larger than 12. If the fraction is 3/1011, with lesser number 3, the GCF can be no larger than 3.
The reason should be obvious, and when I say this I really mean it. Take the fraction 6/792, for example. Could a number larger than 6 go into both 6 and 792? Well there may be a number larger than 6 that goes into 792 evenly, but nothing larger than 6 can go into 6 itself, right? A large peg can’t go through a tiny hole, right? So there you go. Nothing larger than 6 can go into both 6 AND 792. QED.
So what does this mean for you, the math student, or parent of a math student, or the teacher of math students? … I means you want to keep in mind that in actuality two different numbers will help you nail down the size of the GCF. One is the gap between the fraction numbers, and the other is our “new friend,” the smaller of the two fraction numbers.
And here’s another … hack fact. (Whenever I say that, you know we’re heading into ‘nerd-land,’ right?) For both limiting numbers, the gap and the smaller fraction number, the only numbers that can possibly go into both fraction numbers are the FACTORS of those limiting numbers. So for example, if your fraction is 6/50, with the smaller number of 6, the only numbers that can possibly go into 6 and 50 are the factors of 6: i.e., 6, 3, or 2.
A nice rule of thumb: see which is smaller, the gap or the smaller fraction number. Then use that smaller number as your largest possible GCF. To nail this down, let’s do two example problems.
Example 1: 8/44. What’s smaller? 8 or the gap, 36. Obviously 8! So use 8. Test the factors of 8, which are 8, 4, 2. Notice that 8 doesn’t go into both 8 and 44. But 4 does, so 4 is the GCF, and using 4, the fraction simplifies down to 2/11.
Example 2: 22/36. What’s smaller? 22 or the gap, 14. Here the gap is smaller. So test the gap’s factors: 14, 7, 2. 14 doesn’t go into 22 and 36; nor does 7. But 2 does. So 2 is the GCF, and using 2, the fraction simplifies to 11/18.
Time for you all to try your hands at this fun practice, which catapults your “number sense” to new heights.
For each problem, 1) identify the fraction’s smaller number and the gap. 2) Say which of those two numbers is smaller. 3) Using that number’s factors, find the GCF. 4) Finally, using the GCF, simplify the fraction. Answers follow.
SIMPLIFY THE FRACTIONS:
a) 8/42: 1) smaller # = 8; gap = 34. 2) 8 < 34. 3) GCF = 2. 4) 4/21
b) 12/20: 1) smaller # = 12; gap = 8. 2) 8 < 12. 3) GCF = 4. 4) 3/5
c) 36/60: 1) smaller # = 36; gap = 24. 2) 24 < 36. 3) GCF = 12. 4) 3/5
d) 18/96: 1) smaller # = 18; gap = 78. 2) 18 < 78. 3) GCF = 6. 4) 3/16
e) 21/91: 1) smaller # = 21; gap = 70. 2) 21 < 70. 3) GCF = 7. 4) 3/13
Josh Rappaport is the author of five math books, including the wildly popular Algebra Survival Guide and its trusty sidekick, the Algebra Survival Workbook. And FYI: the 2nd Edition of the Survival Guide was just released in March, so get it while it’s hot off the press! If you’d like to get tutored by Josh, you can. Josh and his remarkably helpful wife, Kathy, use Skype to tutor students in the U.S. and Canada in a wide range of subjects. They also prep students for the “semi-evil” ACT and SAT college entrance tests. If you’d be interested in seeing your ACT or SAT scores soar, shoot an email to Josh, sending it to: josh@SingingTurtle.com We’ll keep an eye out for your email, and in our office, our tutoring is always ON … except on Saturdays.