Word Translations 4.2 Ed. - Chapter 8 - In Action #2, pg 139

[j.scott.sandifer]

*Note: This question makes references to the previous question on the same page (#1), but still contains all the information required to solve it. (At least I think so!)

2. Shaggy has to learn the same 71 hiragana characters, and also has one week to do so; unlike Velma, he can learn as many per day as he wants. However, Shaggy has decided to obey the advice of a study-skills professional, who has advised him that the number of characters he learns on any one day should be within 4 of the number he learns on any other day.

a. What is the least number of hiragana that Shaggy could have to learn on Saturday?

b. What is the greatest number of hiragana that Shaggy could have to learn on Saturday?

The answer key suggests "...trial and error, considering different minimum values and noting the consequences."

Where I'm stuck, is figuring out where to begin picking values.
I have difficulties picking "smart numbers" elsewhere in the guides, but here in this problem the possibilities are absolutely bewildering.
I'm having trouble seeing how trial and error would be the most effective tactic on a problem such as this - especially in a timed setting.

[j.scott.sandifer]

Ok.
I'll try asking in a different way.

Under timed circumstances, what is the best method to approach this problem?
If the best method is to pick numbers. What numbers do you pick first and why?

If the best method is setting up an inequality, could you explain how?

[StaceyKoprince]

Good questions. (And sorry it has taken so long to get a response. We've been swamped.)

Let's try both ways.

Shaggy has 7 days. He needs to learn 71 things. And he wants to follow this "within 4" rule.

Then the ask us to find bothe the maximum and the minimum that he could learn on any one day. So how would we do that?

Well, to maximize or minimize something, I have to try to find extreme values. Often, thinking about the "average" helps me figure out what I can do for extremes.

So if he has to learn 71 things and has 7 days, if he spreads them all pretty evenly, he could learn 10 on 6 days and 11 on one day - that's 71.

So he'd probably need to be within 4 of one of those... either 10 or 11. Let's try 6 and 7 for the minimum (which are within 4 of either 10 or 11). If he does 6 on one day, then the most he can do on another day is 10. 6 days * 10 per day = 60, plus 1 day * 6... doesn't get us to 71, so that can't be it. Try 7... and hey, that one works.

We could do the same thing for the maximize one. For that one, we might try 14 first (10+4) but then when we do the math we realize that's too many! So we go down one and try 13, and bingo, that one works.

Now, what about the inequality method?

If I'm trying to minimize the amount I can do on one particular day, that means I also want to maximize the number that I do on the other 6 days, right? So, on all 6 of those days, I want to do +4, the maximum difference I'm allowed.

Let's call my minimized number x. That's one day, so on the other 6 days, I want to do x+4. Add those all up and that has to get me to at least 71. That gives us x + 6(x+4) >= 71

That last part is tricky. You can just remember that when you're *minimizing* you have to get to *at least* whatever number you want - the reverse is true for maximizing. Alternatively, just set it equal to 71. You may get a decimal as an answer. Round up.

x + 6x + 24 = 71
7x = 47
x = 47/7 = between 6 and 7. Round up to 7.

For the maximized number, follow the same process. Max number = y for one day, so I have to do y - 4 for the other six days. I need to get AT MOST 71 (because I'm maximizing one day), or I set equal to zero and round DOWN this time.

y + 6y - 24 = 71
7y = 95
y = 95/7 = between 13 and 14. Round down to 13.

Which way makes more sense to you? Use that way.

Also, this is REALLY hard and you probably won't have to do something like this on the real test - though you may see a maximize or minimize problem that is not quite this complicated.

[raffi.soulonfire]

It is stated 'the number of characters he learns on any one day should be within 4 of the number he learns on any other day.'

The 'within' should literally translate to 'less than'. That means if he reads 7 characters on one day the maximum he can read on any other day is 10 as 10 is less than 11(7/4).

Am I misinterpreting tihs?

Thanks.

[tim]

you are misinterpreting this..