Math problem

[Les]

My son is a senior in high school. He was chosen along with 8 others in his senior class of approximately 600 to compete in the American Mathematics Competition this year. If he did well enough on the test he will move on up to the next level of competition. He thinks he did pretty fair but there was a question that no one he knows including his pre-calc teacher could figure out. I am going to give you brain whizzes a chance at it. Good luck.

p.s. I guess you could answer anything and I won't know if you are right or not because we don't have the answer yet. If you figure it out, show your work. hehe

Here it is:

A girl has an amount of change with her that consists of pennies, nickels, dimes & quarters & the average equals 20. If she adds one more quarter, the average will be 21. How many dimes does she have?

A. 0
B. 1
C. 2
D. 3
E. 4

Les

 

[giant_robo]

A.

[ February 12, 2004, 02:49 AM: Message edited by: giant_robo ]

 

[Shannow]

Sorry Les, I wouldn't know a nickel from a dime, but think I've an idea what a quarter is.

 

[Gary Allan]

I say "B" ..one.

There may be some smarter way to do it ..or perhaps there's a obvious way that Marilyn Vos Seavette would do it (that means I could be FOS and way off)...but

Since the conditions of the composite of change states that there is "dimes" ..it should be at least "one" ..as well of at least one of each of the diniminations. Since it "averages" 20 we have to conclude that, unless I missed some play on words, that the majority of the coins must be quarters ..and that there are enough of them to have the "average" increase only to 21 with the addition of just one more.

This works ..but I can't believe that this is would employ such a "Neandethralic" techique.

Again ..there must be some "transwarp conduit" to see through this in a far simpler fashion.

10 quarters
1 dime
1 nickel
1 pennie
total 266/13 coins =20.46 AVG=20

11 quarters
1 dime
1 nickel
1 pennie
total 291/14 = 20.785 AVG=21

The answer works ..but there must be some "trick" way of reasoning it without any of the actual long hand calulation. They don't pose questions in that manner for simple (as in simpleton - long and tedious) resolution.

Perhaps I've put too much into this. Perhaps it's as simple as the multiple choice answers. Just because any number less than one negates the conditions of the problem ..and any number greater then 1 renders the "one quarter added" solution null and void.

So ..it's not the question, per se ...but the options of answers.

Please ...make me feel

I can take it

 

[ekrampitzjr]

The answer is A = 0. Here's why. By the way, Shannow, a penny = $0.01 (1 cent), a nickel = $0.05 (5 cents), a dime = $0.10 (10 cents), and a quarter = $0.25 (25 cents).

We don't know how much the total cents amount is. Let that be x. We don't know how many coins are involved. Let that be y. We have two unsolved variables, x and y. So here's what we do know:

x/y = 20, the original average value of the coins.

We also know that if we add one quarter, 25 cents, to x, this will add one more coin to y. So,

(x + 25)/(y + 1) = 21, the new average.

Now we have a system of equations that allows solution for both variables.

[1] x = 20y
[2] x + 25 = 21(y + 1)

Equation 2 reduces to x + 4 = 21y. Now rearrange terms, multiply equation 2 by -1 on both sides, and subtract:

[1] x - 20y = 0
[2] -x + 21y = 4
-------------------------------------
y = 4

And x = 80. Therefore we had 80 cents originally using 4 coins; 80/4 = 20. Adding a quarter made it 105 cents with 5 coins; 105/5 = 21. Simple deduction shows that the original coins were 3 quarters + 1 nickel: 3(25) + 5 = 75 + 5 = 80. For the average to be 20 cents using 4 coins, no dimes can be included in the mix as the average would be incorrect. Therefore, there were no dimes, nor any pennies.

The first set of coins was 3 quarters + 1 nickel. The second set was 4 quarters + 1 nickel.

Thanks for the brain teaser. This was simple algebra and logic. It says a lot about today's teachers that a pre-calculus teacher couldn't figure this out. I've been out of school for 20 years!--Ed

 

[Dan4510]

I did it another way, using a table

p n d q = average

first run gave 80 cents total divded by 4 for average = 20.

What divided by 5 gives 21 as average=105. Add one quarter.

It is pitiful that pre-calculus teacher cant figure this out.

There are no more logic, reason, or critical thinking skills taught anymore in high school. College has a deficit of that also. After teaching for several years, that is what I teach underlying the explicit material of the class. Implicit message is: dont memorize, reason....

Students either do real well im my classes or they fail. Doesnt seem to be much in between.

Dan

[ February 12, 2004, 10:50 AM: Message edited by: Dan4510 ]

 

[Pablo]

a, but what may be stumping some folks is the poor English in the question....if they would have written "mean value" or even just "mean value of the coins Jane has"....so not only are we algebraically challenged we can't even write anymore.....average is a much abused word.

Back to work, where logic never applies.

 

[keith]

What are schools teaching these days?

code:

---

P = penny; N = nickel; D = dime; Q = quarter



w.P + x.N + y.D + z.Q



(w + 5x + 10y + 25z)/(w+x+y+z)=20

(w + 5x + 10y + 25z)=20(w+x+y+z) Eqn. 1



(w + 5x + 10y + 25(z+1))/(w+x+y+z+1)=21

(w + 5x + 10y + 25(z+1))=21(w+x+y+z+1) Eqn. 2



Subtract eqn. 2 - eqn. 1:

25=(w+x+y+z)+21

(w+x+y+z)=4 Eqn. 3



Substitute eqn. 3 into eqn. 1:

(w + 5x + 10y + 25z)=80 Eqn. 4



Conclude:

z80)

z>2 (since (2*25)+(2*10)
z=3





Therefore:

x=1 (only possible solution to eqn. 4)



w=0, x=1, y=0, z=3



Verify: 3*75 + 5 = 80; 80/4 = 20

4*25 + 5 = 105; 105/5 = 21



Answer to original question: A (y=0)

---

I recently took the PPST Math test for fun (Teachers Pre Professional Skills Test). Scored 190 out of a maximum possible 190

Keith.

 

[Brian Barnhart]

As Pablo suggested, the real issue with this problem is the wording. Math majors may understand what the problem is asking, but it's not clear to those with a more balanced education.

It's unclear whether the statement "average equals 20" means the average number of each type of coin is 20, or that average value of each set of coins is 20 cents. Math majors get it right because they are familar with the simple problems they typically devise. However, like the problem, they often forget that the units are frequently just as important as the "number answer".

 

[giant_robo]

quote:

---

Thanks for the brain teaser. This was simple algebra and logic. It says a lot about today's teachers that a pre-calculus teacher couldn't figure this out. I've been out of school for 20 years!--Ed

---

It shocked me too, I hope this teacher was just too busy or preoccupied, he should know how to do this. I remember these questions being much harder in high school, and I don't think this change in perception is only the effect of a continued education. Grade inflation in standardized tests such as SATs and possibly GREs is pretty much a fact. The quality of education and the resources per student are heading downward, plagiarism and cheating are growing problems in schools. Pharma companies are pushing drugs to aid ADD, depression, and obesity on kids that should be out playing, excercising, doing homework instead of watching reality shows on TV, drinking softdrinks, and other behaviour that leads to these conditions (People generally seem to want a quick fix to patch the problem rather than deal with the underlying causes). And kids are learning by example by watching these TV roles how to behave in society (at least I hope they aren't watching our politicians, block cnn your TV set). In fact, I feel like the TV is to our society as the lead aqueducts were to ancient Rome.

[ February 12, 2004, 03:42 PM: Message edited by: giant_robo ]

 

[Les]

Thanks everyone!
Sorry Shannow, the original question was worded explaining that a penny was .01, a nickel .05, etc, I found out later. My son believes if his teacher spent some time with the problem she could have figured it out. She teaches calculus as well as pre-calc and is the school Mathematics Department Chair.

Les

 

[rpn453]

quote:

---

Originally posted by Dan4510:
There are no more logic, reason, or critical thinking skills taught anymore in high school. College has a deficit of that also.

---

Is it any different than it was? The vast majority of those older than me have very little math-related problem solving skills. This is also true of those my age and younger as well though.
For the most part those skills are either there or they aren't. They're not something that can be easily taught, but they should be the focus of a teacher since they can always be improved.

 

[Gary Allan]

quote:

---

As Pablo suggested, the real issue with this problem is the wording. Math majors may understand what the problem is asking, but it's not clear to those with a more balanced education.

---

I agree. Now I never was in the stratosphere of mathmatical realms. Never did well at it ..but enjoyed it. Agl II and trig (only to the limits that it involves electricity/electronics).

Beyond all that ..I would have stated the question this way: A girl has an amount of change with her that may consist of any combination of pennies, nickels, dimes & quarters & the average equals 20. If she adds one more quarter, the average will be 21. How many dimes does she have?

My "looking to the detail of the question for the answer" took for granted that there MUST be at least ONE of each in the answer since it stated so in the question. The answer has no dimes in it ..so the change in her pocket does NOT consist of the described composite.

So ...don't confuse lack of mathmatical ability with lack of "critical thinking".

 

[Dan4510]

quote:

---

So ...don't confuse lack of mathmatical ability with lack of "critical thinking". [/QB]

---

These two may not be the same, but they go hand in hand. If one is missing the other usually is.

The sad part is that most people dont have a clue about critical thinking.

Dan