## Thursday, June 30, 2011

### am I right?

Remember the Math Beast and his weekly challenge over at the Manhattan GRE website? Here's the beast's latest poser:

This Week's Problem: "Spinach Rows"

Q: Pea, tomato, and spinach plants are planted in a field. For every 2 pea plants, there are 3 tomato plants, and for every 5 pea plants, there are 6 spinach plants. Spinach is sown 18 plants to a row and tomatoes 12 plants to a row, with no partial or mixed-crop rows. What is the minimum number of spinach plants in the field?

A:

(A) 36
(B) 72
(C) 120
(D) 144
(E) 180

I took a stab at this one, but am still unsure of my answer. Work the problem out for yourself, then look at my answer in the comments to this post to see whether we reasoned it out the same way. I think I may have gotten this wrong, but I'm not sure.

_

Kevin Kim said...

I say the answer is A: 36. Here's why I think so.

First, to set the problem up, I named the variables by the first letters of the names of the vegetables in question. So:

pea plants = p
tomato plants = t
spinach plants = s

I then established the following equations:

2p = 3t

and

5p = 6s

In solving for p, I got this:

p = (3/2)t

and

p = (6/5)s

Transitively, then,

(3/2)t = (6/5)s

So far, so good? We solve for t, and get:

t = (4/5)s

We know that all three types of plants are planted in this field. At first, I was freaked out because it seemed to me that we would need to know the size of the field in order to solve this problem. But we don't: what we need is already provided, namely, (1) the fact that all three types of plants are in this field, and (2) the fact that there's at least one row of spinach and one row of tomatoes.

If we have at least one row of tomatoes (and we do), then we have at least [(3/2)*12] pea plants, i.e., at least 18 pea plants.

Also, if we have at least one row of spinach plants, then we have [(6/5)*18] pea plants, i.e., at least another 21.6 pea plants.

Add 18 and 21.6 together, and you see we have at least 39.6 pea plants in this field. So returning to our pea-to-spinach formula, we get the following:

39.6 = (6/5)s, or

s = (39.6*5)/6, or

s = 33

Since spinach is sown 18 plants to a row, with no incomplete rows, then 33 can't be the actual answer: the closest answer that corresponds to there being at least 33 spinach plants is A, or 36, i.e, two rows of spinach.

QED...?

Kevin Kim said...

I guess the Math Beast was too PC to do a "corn rows" problem.

Charles said...

Wait... t=(4/5)s? Isn't that backward? It's been a really long time since I've done anything like this (solving for whatever), but: if there are 10 pea plants, that would mean there are 15 tomato plants and 12 spinach plants, so s=(4/5)t, or t=(5/4)s. Right? Or has it been so long since I've done math that I can't even read mathematical notation any more?

Here's my messy "solution" (I use quotes because it's not really a valid proof).

The only use for the peas is to give us the ratio of spinach to tomato, so after I multiplied 2 and 5 to get 10 and then used that as my base get the ratio of spinach (2*6=12) to tomato (3*5=15), or 4:5, I moved on.

So, spinach is planted in rows of 18 plants and tomato is planted in rows of 12 plants. We need a figure for spinach that will also produce a valid figure for tomato. This is where things get a little fuzzy. I know I'm not doing this the right way because it's been a long time since I've done this, but basically I just started experimenting with the numbers. The lowest valid number of spinach alone (keeping the ratio in mind and using 4 as my starting point) that would produce rows of 18 plants is 36. However, this would give us 45 tomato plants, which does not leave us with a whole number when divided by 12; instead we get 3.75 rows. So I started going by multiples of 36 to see what would happen. 72 spinach would give us 90 tomato: 7.5 rows. 108 spinach would give us 135 tomato: 11.25 rows. 144 spinach would give us 180 tomato: 15 rows. (Actually, I saw the pattern after 72 and just jumped straight to 144.)

So the answer is 144. I have no idea why, though--that is, I wouldn't know how to do the mathematical proof. Which is kind of embarrassing, but again, it's been a while.

daeguowl said...

I think the answer is 144. If there are 144 spinach plants (8 rows of 18), then there will be 120 tomato plants. (six spinach plants for 5 pea plants) If there are 120 pea plants then there must be 180 tomato plants,or 15 rows at 12 to a row (3 tomatoes for 2 peas). 144 is the first answer that allows us to have full rows of tomatoes and spinach in accordance with the planting formula.

daeguowl said...

You went wrong right from the beginning...there are 3 tomato plants for every 2 pea plants therefore P = 2/3*T not 3/2*T. Your first row of tomato plants gives you 8 pea plants which means you have at least 9.6 spinach plants. The forumla to spinach to peas is similarly wrong.

I also think it was an error to add the number of peas for one row of spinach and the number of peas for one row of tomatoes. I think the same peas can be counted for each ratio.

Kevin Kim said...

Daeguowl,

Yes, I see the problem. Bad set-up on my part. Egads. But I have to say, it's hard to see how I went wrong. After all, if there are 3 tomato plants for every 2 pea plants, then there are 1.5 tomato plants for every one pea plant, which makes it tempting to write this as "p = 1.5t" (or p = [3/2]t, as I originally put it).

But as you say, this is wrong (and Charles noted that something was backward, too) because my formulation makes it sound as if the number of pea plants is greater than the number of tomato plants, which is exactly backwards. In reality, the number of pea plants is indeed two-thirds the number of tomato plants.

Rookie mistake.

Kevin Kim said...

Someone put this problem on a math thread, and a commenter also arrived at 144-- very rapidly, I might add. See here. The setup used by this commenter is absurdly simple:

p : t : s
2 : 3
5 : x : 6

10 15 12

120 180 144 (multiply with 12, it is common to both as spinach ratio and tomatoes plants in a row)

ans. D


It took a moment for me to see how this worked, but I got it: the row after the "2:3" row is merely the first row multiplied by 2.5. So x would be 7.5, which is how the commenter got the ratio of 10:15:12. His explanation in the parenthetical takes us the rest of the way. Wow.

Unknown said...

I think you're definitely rusty. To figure out how many tomato plants per spinach plant you have, you multiply 3/2 by 5/6. That gives you 5/4. Then you just quickly multiply each choice by 5/4 and pick the smallest one that's divisible by 12. That's 144 which gives you 180 tomato plants or 15 rows.

I believe the GMAT has the same math as the GRE. IIRC, the trick is to not overthink it. If you spend more than sixty seconds thinking about the problem without calculating, you should start over. They like to throw out problems that look like you need to remember "real" math, but that's a mirage.

Kevin Kim said...

Mellow Yellow,

Yes, the trick is usually in the rapid conception and rapid set-up of the problem. The algebra itself isn't hard; it's the fact that we have to wrap our minds around what's being asked for.

Not hopeful about tomorrow. But this has been instructive, because it's highlighted a flaw in the way I visualize ratios. I interpreted "3 tomato plants for every 2 pea plants" to mean "3t = 2p," but what I should have done is to set this up as "t/p = 3/2," which gives us "t = (3/2)p," or "p = (2/3)t."

And this is why, at my current job, I'm rarely called to tutor SAT Math. Anonymous said...

Now I'm feeling insecure, because my answer is different from all the above, and embarrassed about that, because I'm supposed to be all mathy and all. But here goes.

I didn't see this as an algebra problem at all--it's essentially a ratio and loosest common multiple problem. No partial rows allowed, and no partial plants allowed (you either have a tomato plant or you don't, you can't plant .6 of a tomato plant.) And it clearly states that the field has all three kinds of plants in it. If each tomato row has to have 15 plants and each spinach row has to have 18 plants in it, then you have to have something that divides evenly into both 15 and 18==the LCM of 15 and 18 is 90. But when you check that with the peas, you'd have 75 pea plants, which isn't allowed--there has to be an even # of peas, so double 90 to get 180.

When I started to write this up, got all insecure again, so figured it more carefully:

You have two sets of ratios to adhere to, ratios among #s of plants and one ration given for # of plants in a row. Plant ratios work out to 10 peas, 15 tomatoes, and 12 spinach, as a previous commenter showed. The row ratio is 12 T to 18 S.

The LCM for 10, 15, and 12 is 60 (you need 2 factors of 2, two factors of 3, and one factor of 5--multiply those out and you get 60). The LCM of 12 and 18 is 36. Now we need the LCM of 60 and 36, because both # of rows and # of plants have to come out to whole #s. 60 is missing a factor of 3, and 36 is missing a factor of 5. 60 X 3 = 180, and 36 X 5 = 180, so 180 satisfies all the requirements.

At this point I'm satisfied that 180 is the answer, though I may not have explained it well. Work it through for yourself, with that constraint that we have to have whole #s of both rows and plants; label everything carefully as you go. Mixing up numbers that apply to rows and #s that apply to plants is yielding a lot of those distractors, I suspect. I'd eschew the algebra--doing it with equations seems harder to me, but use them if they work for you. Also work it backwards, with the whole # requirements in mind--which answer works for all three plants?

John said...

I got lost on the three plant species. Why the FUCK would the farmer plant such diverse species in one field? Harvest time must be a real nightmare. So... another problem. pea plants would outgrow the others and smother them...The tomatoes would need to be pratected from birds by netting (which peas hate) How can that STUPID FUCKING FARMER stay ahead of his mortgage payments!!!!

Charles said...

@Addofio: The flaw in your first calculation is a misread of the question: tomato rows have to have 12 to a row, not 15. I have to admit that I could't really follow your second explanation, but 144 satisfies all the requirements as well: 144 spinach plants in 8 rows, 180 spinach plants in 12 rows, and 120 pea plants (in however many rows the farmer wants). Since we're looking for the minimum number, 144 wins out over 180.

@Kevin: that solution you posted from the math thread is pretty much how I worked it out, I just didn't know how to express it. It actually took me far longer to write out my comment than it did to figure out the problem.

After reviewing all the comments, though, I think we're all wrong--John obviously has the right answer.