So c’mon … everything that can be said about simplifying fractions has been said … right?
Not quite! Here’s something that might just be original … a hack to smack those fractions down to size.
Suppose you’re staring at an annoying-looking fraction: 96/104, and it’s annoying the heck out of you, particularly because it’s smirking at you!
But it won’t smirk for long. For you open up your bag of hacks (obtained @ mathchat.me) and …
1st) Subtract to get the difference between numerator and denominator. I also like to call this the gap between the numbers. Difference (aka, gap) = 104 – 96 = 8.
NOTE: Turns out that this gap, 8, is the upper limit for any numbers that can possibly go into BOTH 96 and 104. No number larger than 8 can go into both. And this is a … HACK FACT: The gap represents the largest number that could possibly go into BOTH numerator and denominator. In other words, the gap is the largest possible greatest common factor (GCF).
2nd) Try 8. Does 8 go into both 96 and 104? Turns out it does, so smack the numerator and denominator down to size: 96 ÷ 8 = 12, and 104 ÷ 8 = 13.
3rd) State the answer: 96/104 = 12/13.
Is it still smirking? I think … NOT!
Try another. Say you’re now puzzling over: 74/80.
1st) Subtract to get the gap. 80 – 74 = 6. So 6 is the largest number that can possibly go into BOTH 74 and 80.
2nd) So try 6. Does it go into both 74 and 80? No, in fact it goes into neither number.
NOTE: Turns out that even though 6 does NOT go into 74 OR 80, the fact that the gap is 6 still says something. It tells us that the only numbers that can possibly go into both 74 and 80 are the factors of 6: 6, 3 and 2. This, it turns out, is another … HACK FACT: Once you know the gap, the only numbers that can possibly go into the two numbers that make the gap are either the factors of the gap, or the gap number itself.
3rd) So now, try the next largest factor of 6, which just happens to be 3. Does 3 go into both 74 and 80? No. Like 6, 3 goes into neither 74 nor 80. But that’s actually a good thing because now there’s only one last factor to test, 2. Does 2 go into both 74 and 80? Yes! At last you’ve found a number that goes into both numerator and denominator.
4th) Hack the numbers down to size: 74 ÷ 2 = 37, and 80 ÷ 2 = 40.
5th) State the answer. 74/80 gets hacked down to 37/40, and that fraction, my dear friends, is the answer. 37/40 the final, simplified form of 74/80.
O.K., are you ready to smack some of those fractions down to size? I believe you are. So here are some problems that will let you test out your new hack.
As you slash these numbers down, remember this rule. In some of these problems the gap number itself is the number that divides into numerator and denominator. But in other problems, it’s not the gap number itself, but rather a factor of the gap number that slashes both numbers down to size. So if the gap number itself doesn’t work, don’t forget to check out its factors.
Ready then? Here you go … For each problem, state the gap and find the largest number that goes into both numerator and denominator. Then write the simplified version of the fraction.
a) 46/54: gap = 8. Largest common factor (GCF) = 2. Simplified form = 23/27
b) 42/51: gap = 9. Largest common factor (GCF) = 3. Simplified form = 14/17
c) 48/60: gap = 12. Largest common factor (GCF) = 12. Simplified form = 4/5
d) 45/51: gap = 6. Largest common factor (GCF) = 3. Simplified form = 15/17
e) 63/77: gap = 14. Largest common factor (GCF) = 7. Simplified form = 9/11
Josh Rappaport is the author of five math books, including the wildly popular Algebra Survival Guide and its trusty sidekick, the Algebra Survival Workbook. Josh has been tutoring math for more years than he can count — even though he’s pretty good at counting after all that tutoring — and he now tutors students in math, nationwide, by Skype. Josh and his remarkably helpful wife, Kathy, use Skype to tutor students in the U.S. and Canada, preparing them for the “semi-evil” ACT and SAT college entrance tests. If you’d be interested in seeing your ACT or SAT scores rise dramatically, shoot an email to Josh, addressing it to: josh@SingingTurtle.com We’ll keep an eye out for your email, and our tutoring light will always be ON.