A coworker brought in this math problem.
Initially I wanted to take 2/3 of 120. Giving 80 green and 40 red.
That’s wrong because green is 2/3 of the number of red apples. A value we don’t know.
What’s the equation to solve?
A fruit seller got 120 apples. The number of green apples was two thirds of the number of red apples. How many green apples did the fruit seller get?
Answer 48 (according to the test)
It’s a system of two equations:
g = 2/3 * r
r + g = 120
You can solve by plugging in the first equation into the second, solve that, then plugging that result back into the first equation.
Thanks.
Seems simple now. I didn’t think about needing 2 equations.
These internet tests are usually posted as a sixth graders test. But, the students would have seen similar math problems in their textbook and the teacher solved them on the blackboard in class.
They were prepped to take the test.
My brain went to one equation (well, I needed to do a simple calculation at the end to get the number of green apples). x = number of red apples:
x + 2/3x = 120
5/3x = 120
5x = 360
x = 72
120-72 = 48
As a general rule, to solve for such unknowns, you need as many equations as there are unknowns. You then play the substitution game as described above to gradually determine each unknown.
We did a lot of that in physics.
Fundamentally it’s the same thing, you just skipped a step. It’s basically what the two equations would substitute into if you went through the whole process. Same answer regardless.
I appreciate the help.
I’ve been out of school way too long.
Kids in school today would probably struggle even more with it, lol. Our educational situation is, er, not exactly improving…
My way of thinking is we have bins in the ration of 2 to 3. That means 5 equal bins of which 3 hold reds and 2 hold green. Therefore each bin has 120 ÷ 5 or 24 apples. Two bins of greens is thus 48 apples.
ETA: My teachers that required me to show my work hated me.
Whenever I can. Whenever I can…
Word problems almost always have unspoken assumptions, with which you can troll math teachers if that’s your game. I can think of three unspoken assumptions in this problem without breaking a sweat:
(1) the number of each type of apple must be a whole number, so none of the apples are in pieces or a mixture of colors
(2) all of the apples are either red or green, so there are none of any other color, such as yellow
(3) the number of each type of apple is a nonnegative number, so that upon delivery the seller does not incur a debt of, say, 12 red apples
If you accept assumption (1) but dispense with assumptions (2) and (3), you can end up with solutions like 0 red, 0 green, 120 yellow or -90 red, -60 green, 270 yellow (i.e., the seller receives 270 physical yellow apples but incurs a debt of 90 red and 60 green for a net receipt of 120 apples), or any of (by my count) 118 other combinations. If you also dispense with assumption (1), there are an infinitude of solutions, including 0.3 red, 0.2 green, 119.5 yellow.
That’s how I solved it too, but I know the proper method is to write two equations and substitute in, and I’d do that on paper for something more complicated.
My teachers had the same complaint. 
The number of green apples is 2/3 the number of red apples. Green is 2, red is 3, total is 5, actual quantity is 120, a fifth of which is (… opens Excel because I don’t do arithmetic in my head…) 24.
Green apples are twice that or 48. Red apples are 3x that which is 72.
I disagree with the proper method. Any mathematically valid approach is proper.
Yeah, my thought process on it was similar to yours. And speaking as a math teacher, I’d accept any of these answers as correct.
5 is half of 10. So 1/5 of 120 is 1/10 of 120 times 2. And 1/10 is easy, just 12, and 12 times 2 is 24.
That probably made it sound more confusing than it is. To divide by 5, cut off the last digit and double it. 120 => 12 => 24.
True. But the two equations method is more generalisable, and most people aren’t going to remember more than one way to solve a problem, so it makes sense to teach that method.
If a green apple is 2/3 of a red apple, let’s pair then apples into groups of 1 red and 2/3 green. That combines to make 5/3 apples. How many groups do we have? 120 ÷ 5/3 = 360 ÷ 5 = 72 groups. So we have 72 of the 2/3 green apples making 48 green apples.
My days of solving equations are long past but I’m glad to see that I had the correct approach.
The fruit seller should have counted the reds and greens separately. That’s the math lesson to learn here.