004-Exact Values: Fractions
Post date: Oct 05, 2011 11:28:34 AM
(For Oct 5/11)
LYN AND SLOPE PRESENT.... FRACTIONS
Lyn: So, as Linear, I've got three number parts. Rise, run and y-intercept.
Slope: Two of those from me!
Lyn: Their values need to be EXACT. No "1/3 is about 0.3" or "isn't five decimals good enough".
Slope: If you were going to get 1/3 of a million dollars, would you be happy with $300 000?
Lyn: So, you're going to need to take your decimal, and make it a fraction.
Slope: In fact, I'm practically a fraction already. Can we assume you know how to reduce them?
Lyn: Most calculators have a key for that if you're stuck. Now, let's look at the decimal cases...
CASE 1: TERMINATING DECIMAL
Lyn: Easy one. Count the number of decimal places, call it d, and your denominator will be 10^d
Slope: Let's say you're charging 0.56 for 5 apples, and you want your unit rate, which is me. You punch in the division, get 0.112
Lyn: That's three decimal places, 10^3 = 1000, so use 112/1000. It reduces to 14/125.
Slope: It's NOT 11 cents each, you got it? Go back to the original five apples with that, and you're being underpaid! And if you think you can't have a fraction of a cent, talk to a gas company. Or Verizon.
CASE 2: CAN'T TELL FROM CALCULATOR
Lyn: If you're looking at a decimal answer on the calculator, odds are you got there by dividing. So use the numbers you were dividing as your fraction instead! Adjust as needed.
Slope: So, you're now charging 0.56 for 13 apples. Because you've got this huge overstock but you also want to scare away this one guy with triskaidekaphobia.
Lyn: Uh, that example's reaching a bit...
Slope: Calculator says 0.0430769231 but you know it's not exact, so your fraction is 0.56/13
Lyn: But since we never mix decimals and fractions, you'll need to multiply both numbers by 10^2=100, to get rid of it. That's 56/1300. It reduces to 14/325.
Slope: Which as a decimal is actually 0.043076923076923076923... etc, etc... you won't get that on the calculator.
Lyn: But that brings us to...
CASE 3: REPEATING DECIMALS
Lyn: Actually, also easy. Count how many numbers are in the repeating pattern, call it d, and your denominator will be (10^d - 1)
Slope: Let's say you've got 0.363636363636... going on. Two decimals repeating, so your denominator is 100-1=99. 36/99 reduces to 4/11. Try dividing them, it works.
Lyn: Where it's a little more interesting is when there's leading numbers. You'll have to split the question up into two parts, as a:
COMBO
Slope: For instance, the value 0.55363636363636... it begins 0.55, not repeating, giving you 55/100. That leaves 0.003636363636....
Lyn: ...which we already know is 36/99 or 4/11, BUT it starts two decimal places late. So multiply it by 1/100. Meaning 4/1100.
Slope: Now add 55/100 to 4/1100. Using the common denominator gives 609/1100. It doesn't reduce, and therefore gives us our 0.55363636363636...
Lyn: So there you have it, a nice exact fraction! Clearly shows rise on the top and run on the bottom.
Slope: Let's do the one from case 2 now!
Lyn: Well, I don't think that...
Slope: 0.043076923076923076923... it starts 0.04, or 4/100 = 1/25, it repeats six digits, so 307692/999999 = 4/13, starting late so use 4/1300. 1/25 + 4/1300 = 56/1300, reducing to 14/325, the same answer as before!
Lyn: ...Yes. I think we're done here.
Slope: Try it at home!