Eighth Grade Algebra

I like algebra, and I usually like word problems, but this problem from my 13yo's pre-algebra class had me stumped for over 30 minutes. No wonder she gets frustrated! So how many resident CHM geniuses can solve it?

Jim runs a lawn care business for which he charges $15 to mow and an additional $5 to trim.  One week he earned $415, trimming 5 more yards than he mowed. Write an equation to determine how many yards Jim trimmed, then solve it.

I am sure SOMEone will get the right answer, but if no-one does I'll post the solution in a few days.

BTW, my other daughter, a junior in college, got it wrong. So, are you smarter than an 8th grader and a college student!?

I came up with 2 ways:

(415 - (5*5)) / 15 = x

(5*5) + 15x = 415

I think it'd be this:

415 = 15(x-5) + 5x
415 = 15x - 75 +5x
415 = 20x - 75
340 = 20x
x = 17 yards trimmed

I solved for lawns mowed, not trimmed.  Here's a rewrite:

{(415 - (5*5)) / 15} + 5 = x

The question is, do we assume that all lawns were mowed and trimmed, and that five additional lawns were trimmed?

Driven wrote:

The question is, do we assume that all lawns were mowed and trimmed, and that five additional lawns were trimmed?

You think like my oldest daughter.  She thought the problem indicated that x lawns were cut while y lawns were only trimmed.  However, the problem states "he charges $15 to mow and an additional $5 to trim".  This implies that he cuts all of them, but only trims the ones that pay $5 more.

kerrick wrote:

I think it'd be this:

415 = 15(x-5) + 5x
415 = 15x - 75 +5x
415 = 20x - 75
340 = 20x
x = 17 yards trimmed

You're heading the right direction, except for two mistakes.  #1, like Driven and my college-aged daughter you seem to be assuming that some yards are cut while others are only trimmed. (see post just above).  And #2, you subtracted 75 (lines 3 to 4) from both sides rather than adding 75; if you follow your equation correctly and add 75 line 4 then becomes 490 = 20x, making x 24.5 (the same answer my oldest daughter came up with), and incorrect as we can probably assume he didn't trim half a yard (well, if he was a kid he probably did. LOL!)

Xid wrote:

I solved for lawns mowed, not trimmed.  Here's a rewrite:

{(415 - (5*5)) / 15} + 5 = x

Computer programmer?    That's not an algebra equation, at least not one a 13 year old would know how to write, and the answer is incorrect anyway. Sorry.

Oh whoops!  I did subtract from both sides...    Send me back to 8th grade...  In my defense, I hadn't had my coffee yet haha! 

So Jim is trimming the same yards he mowed, plus five more yards?  I see, that makes sense.  That changes it...  I've gotta get back to work, but this will take a little more time...

Ok what about this?

   mowed            5
|--------------|------|
|<-----trimmed---->|

trimmed (total) = x
mowed = x - 5

415 = (15+5)*5 + 15(x-5)
415 = 100 + 15x - 75
390 = 15x
x = 26 yards?

BearDad wrote:

Xid wrote:

I solved for lawns mowed, not trimmed.  Here's a rewrite:

{(415 - (5*5)) / 15} + 5 = x

Computer programmer?    That's not an algebra equation, at least not one a 13 year old would know how to write, and the answer is incorrect anyway. Sorry.

I took a little programming back in the day and used to be pretty good at Algebra. 

I think I see the error of my ways:  It's a trick question so he mowed all of them and only trimmed 5.

kerrick wrote:

Oh whoops!  I did subtract from both sides...    Send me back to 8th grade...  In my defense, I hadn't had my coffee yet haha! 

So Jim is trimming the same yards he mowed, plus five more yards?  I see, that makes sense.  That changes it...  I've gotta get back to work, but this will take a little more time...

No, he's cutting x number of yards, and trimming all but five of them (x - 5).  And yes, it took me some time as well.

Ok… so lemme get this straight… original post said "trimming 5 more yards than he mowed", but here, you're saying "trimming all but five of them". It's the latter, right? Just trying to make things clear here. I'm a bit confused.

Xid wrote:

BearDad wrote:

Xid wrote:

I solved for lawns mowed, not trimmed.  Here's a rewrite:

{(415 - (5*5)) / 15} + 5 = x

Computer programmer?    That's not an algebra equation, at least not one a 13 year old would know how to write, and the answer is incorrect anyway. Sorry.

I took a little programming back in the day and used to be pretty good at Algebra. 

I think I see the error of my ways:  It's a trick question so he mowed all of them and only trimmed 5.

No, it's not a trick question. I probably screwed it up, as I didn't word it exactly the way it was in the problem (I don't have it with me; my daughter has it at school). Like I told Kerrick, he mowed all the yards, and he trimmed all but five of them.  ("he charges $15 to mow and an additional $5 to trim")

{(415 - (5*15)) / 20} = x
(415 - 75) / 20 = x
340 / 20 = x
x = 17

Driven wrote:

you're saying "trimming all but five of them". 

Yeah, I think I screwed up there. Let's break this down:

"Jim runs a lawn care business for which he charges $15 to mow and an additional $5 to trim."

This indicates that every yard service includes a mow, but only includes a trim for an additional $5.

"One week he earned $415, trimming 5 more yards than he mowed."

I'll admit, this makes it sound like some yards he only trimmed while others he mowed, which is probably what screwed me up so bad last night. However, I believe it indicates of all the yards he services, he mowed and trimmed five more than he simply mowed. Don't thing of "trimmed" meaning only a trim, but being a mow and trim. In other words:


yards serviced = yards mowed + yards mowed and trimmed
yards serviced = yards mowed + (yards mowed + 5)   <-- solve for number of yards mowed

or

yards serviced = (yards mowed and trimmed - 5) + yards mowed and trimmed  <-- solve for number of yards mowed and trimmed


and finally, 

money earned = yards serviced
money earned = (yards mowed and trimmed - 5) + yards mowed and trimmed   <-- plug in from above

So, in algebraic format, where t represents yards trimmed and m represents yards mowed

415 = t + m
415 = t + (t - 5)  <-- big hint there.

BearDad wrote:

Driven wrote:

you're saying "trimming all but five of them". 

Yeah, I think I screwed up there. Let's break this down:

"Jim runs a lawn care business for which he charges $15 to mow and an additional $5 to trim."

This indicates that every yard service includes a mow, but only includes a trim for an additional $5.

"One week he earned $415, trimming 5 more yards than he mowed."

I'll admit, this makes it sound like some yards he only trimmed while others he mowed, which is probably what screwed me up so bad last night. However, I believe it indicates of all the yards he services, he mowed and trimmed five more than he simply mowed. Don't thing of "trimmed" meaning only a trim, but being a mow and trim. In other words:


yards serviced = yards mowed + yards mowed and trimmed
yards serviced = yards mowed + (yards mowed + 5)   <-- solve for number of yards mowed

or

yards serviced = (yards mowed and trimmed - 5) + yards mowed and trimmed  <-- solve for number of yards mowed and trimmed


and finally, 

money earned = yards serviced
money earned = (yards mowed and trimmed - 5) + yards mowed and trimmed   <-- plug in from above

So, in algebraic format, where t represents yards trimmed and m represents yards mowed

415 = t + m
415 = t + (t - 5)  <-- big hint there.

Wait so am I right?

kerrick wrote:

Ok what about this?

   mowed            5
|--------------|------|
|<-----trimmed---->|

trimmed (total) = x
mowed = x - 5

415 = (15+5)*5 + 15(x-5)
415 = 100 + 15x - 75
390 = 15x
x = 26 yards?

t=trim & mow - m=mow only - Therefore m=t-5

15m + 20t = 415
15(t-5)+20t = 415
15t - 75 + 20t = 415
35t - 75 = 415
35t - 75 + 75 = 415 + 75
35t = 490
35t / 35 = 490 /35
t=14

Xid wrote:

 t=trim & mow - m=mow only - Therefore m=t-5

15m + 20t = 415
15(t-5)+20t = 415
15t - 75 + 20t = 415
35t - 75 = 415
35t - 75 + 75 = 415 + 75
35t = 490
35t / 35 = 490 /35
t=14

Check: 

(14 * 20) + ((14 - 5) * 15)
280 + (9 * 15)
280 + 135
415

kerrick wrote:

BearDad wrote:

Driven wrote:

you're saying "trimming all but five of them". 

Yeah, I think I screwed up there. Let's break this down:

"Jim runs a lawn care business for which he charges $15 to mow and an additional $5 to trim."

This indicates that every yard service includes a mow, but only includes a trim for an additional $5.

"One week he earned $415, trimming 5 more yards than he mowed."

I'll admit, this makes it sound like some yards he only trimmed while others he mowed, which is probably what screwed me up so bad last night. However, I believe it indicates of all the yards he services, he mowed and trimmed five more than he simply mowed. Don't thing of "trimmed" meaning only a trim, but being a mow and trim. In other words:


yards serviced = yards mowed + yards mowed and trimmed
yards serviced = yards mowed + (yards mowed + 5)   <-- solve for number of yards mowed

or

yards serviced = (yards mowed and trimmed - 5) + yards mowed and trimmed  <-- solve for number of yards mowed and trimmed


and finally, 

money earned = yards serviced
money earned = (yards mowed and trimmed - 5) + yards mowed and trimmed   <-- plug in from above

So, in algebraic format, where t represents yards trimmed and m represents yards mowed

415 = t + m
415 = t + (t - 5)  <-- big hint there.

Wait so am I right?

kerrick wrote:

Ok what about this?

   mowed            5
|--------------|------|
|<-----trimmed---->|

trimmed (total) = x
mowed = x - 5

415 = (15+5)*5 + 15(x-5)
415 = 100 + 15x - 75
390 = 15x
x = 26 yards?

Not quite

Incredibly poorly worded problem.  This is the reason that many kids can't get word problems because the teachers/text book writers can't seem to communicate effectively with words.  I have had this same problem with my kid's homework from public school.  On one occasion I gave my daughter three different solutions to the same word problem (in college) and told her to give them all to the teacher and tell him one of them had to be correct or else the textbook writer is a complete moron and is wasting everybody's time. 

In the real world if a client gave me a problem like this I would likely spend an hour and fifty minutes getting them to word the problem correctly and an additional ten minutes solving the problem....then they would wonder why I was charging them for 2 hours.

I find this amusing...

Staybrite wrote:

In the real world if a client gave me a problem like this I would likely spend an hour and fifty minutes getting them to word the problem correctly and an additional ten minutes solving the problem....then they would wonder why I was charging them for 2 hours.

+1

Staybrite wrote:

Incredibly poorly worded problem.  This is the reason that many kids can't get word problems because the teachers/text book writers can't seem to communicate effectively with words.  I have had this same problem with my kid's homework from public school.  On one occasion I gave my daughter three different solutions to the same word problem (in college) and told her to give them all to the teacher and tell him one of them had to be correct or else the textbook writer is a complete moron and is wasting everybody's time. 

In the real world if a client gave me a problem like this I would likely spend an hour and fifty minutes getting them to word the problem correctly and an additional ten minutes solving the problem....then they would wonder why I was charging them for 2 hours.

Yep. That's what more-or-less started this whole thing. My wife was so frustrated she gave up and said, "I'll bet Courtney couldn't get that" ... and she was right! So I decided to see if anyone here could. The difference is you guys had the benefit of having someone that had already worked it out to explain it and break it down, and while I didn't have that I at least have experience with poorly worded requirements (computer programmer, remember.  ).  A 13 year old kid that already hates math is never going to be able to find the solution! it makes me want to just do her math homework for her, because she will never pass on her own.

This little problem of yours was bugging me all day.  When it comes to puzzles I get a little OCD leaving them unsolved.