Posted by Rebekah and Esther
The dreaded algebra coin problems have come to Planted by Rivers. Okay, okay, so they really shouldn't be dreaded. The truth of the matter is, if you know 5 simple rules, then solving them should be pretty easy.
Here are the rules of the game.
Penny: 1 ¢
Nickel: 5 ¢
Dime: 10 ¢
Quarter: 25 ¢
Half Dollar: 50 ¢
Important note: Because we are working with cents and not dollars, we want to make sure we convert everything in the problem to cents. This will ensure we get rid of any pesky decimals (you’ll see what I mean).
Rule #2: Choose variables (letters) to represent the number of coins.
In other words, if I am working with dimes and quarters, I could use “d” to represent the number of dimes and “q” to represent the number of quarters. It usually helps to use the first letter in the coin’s name (although you could use any letter).
Important note: Don’t forget that each letter represents the NUMBER of coins!
Rule #3: Learn what phrases like “4 times as many nickels as dimes” means.
Okay, so to some people this may seem pretty straightforward. First choose a letter to represent the amount of nickels (let’s use “n”), and then choose a letter to represent the amount of dimes (let’s use “d”).
So you should end up with an equation that looks like this: n = 4d
But why not d = 4n? If you’re still mystified, keep on reading! (If not, just skip to rule #4)
Now, let’s start simple. Let’s say we have the phrase “Enoch has 4 times as many balls as Daniel.” So then, if “x” represents the amount of balls Daniel has, Enoch must have 4x balls. So, Enoch’s number of balls = 4x
But what if we say, “Enoch has 4 times as many balls as Daniel has cars”? Well, let’s choose the letter “b” for number of balls and “c” for number of cars. Hmmm…the only real difference between this phrase and the above phrase seems to be that now the concentration is between two objects and not two people. Taking this into consideration, one can see that the correct equation for this problem would be: Enoch’s Number of Balls = 4c or b = 4c.
But look! We can also say: “4 times as many balls as cars,” and the equation still remains b = 4c.
So then, phrases like “4 times as many nickels as dimes,” must mean n = 4d.
Rule #4: Know what “more than” and “less than” mean.
|"Coin "Art by Rebekah|
Yes, yes, I know—they’re never phrased so nicely. Usually they’ll go something like this: “there are 7 more nickels than pennies” or “there are 7 fewer nickels than pennies”.
Now, let’s use “n” to represent the number of nickels, and “p” to represent the number of pennies. As a result, “there are 7 more nickels than pennies,” must be
n = p + 7 and “there are 7 fewer nickels than pennies” becomes n = p – 7.
WARNING: Although n = p + 7 is the same as n = 7 + p, n = p - 7 is not the same as n = 7 – p. This is because subtraction is not commutative, and as a result, 7 - p could result in a negative number of nickels!
Rule #5: Do the Algebra!
The best way to explain this is to do actual problems. :) (which I’ll try to post next monday)
To be continued…^_^!