Question

One third of the sum if 3 m and 4 n

Answer 1

Answer:

One-third of some number is 20; what number is 20 one-third of? I have some number of sticks and if I divide those sticks into three equal groups, each such group has 20 sticks; how many sticks do I have total? Hopefully, it’s not too difficult to see that this describes the number 60.

But wait, there’s more: those 60 sticks consist of one group of 12 sticks, and another group of…how many sticks? How many sticks plus 12 sticks totals 60 sticks in all? Well, if I already have 12 sticks, how many more do I need to make 60? The difference between 60 & 12, 60–12 = 48. Thus 48 is the number we seek, because it plus 12 equals 60, and one-third of 60 is 20, as required. (Note: this is probably how the Ancient Greeks would have had to solve this problem.)

Now for the algebraic equivalent: Let y= the number that 20 is one-third of— 13y=20 . Multiply both sides by 3 (to “undo” the multiplication by 1/3) to yield y=3∗20=60 . Now let x= the number whose sum with 12 is 60— x+12=60 ; subtract 12 from both sides (to “undo” the “plus 12”), yielding x=60−12=48 , the answer we seek. (Check: 13(48+12)=13(60)=20 , check! Always do the check step!)

(This alternative algebraic method has a name—simultaneous equations in more than one variable—and is an extremely powerful method, so powerful that it is typically reserved for more complicated—but similar—problems, though I would contend that this problem is kind of “on the line” between problems for which it is vs. isn't warranted.)