Confusing maths problem

[Big Pat]

I hesitate to post this here becuase I must be missing something obvious and I'm going to make myself look like an idiot, but here goes. My nine-year-old daughter was stuck on this and asked for help. They've been studying ratio and proportions. It LOOKED easy, and I do know the answer(s) but only by trial and error. How could I have CALCULATED it?

"Chocolate bars cost 26p each and cakes cost 18p each. You spend EXACTLY £5.00. How many of each do you buy?"

My reasoning was that the ratio was 26:18 or simplified 13:9. I added 13 and 9 to make 22 and tried to divide that into 500 but it wont go evenly. That stumped me, but it was probably the wrong move.

Answer A
I know NOW that 4x26 is £1.04, leaving £3.96 which, dividing by 18, gets me 22 cakes. So my answers are 4 and 22. But how could I have worked out that I should start with 4? It was trial and error (mainly error!)

Answer B
I can also spend 13x26, £3.38, leaving £1.62 to divide by 18, to get 9 cakes. This makes sense, because my answers are 13 and 9 (the ratio I identified). But even knowing the ratio in advance, I might have gone with 13x18, £2.34, leaving £2.66 which would not have divided evenly by 26. This would have prompted me to try it the other way round of course, but it still strikes me as trial and error and seems somehow unsatisfactory.

I'm also puzzled by my two sets of answers. 4 & 22 and 13 & 9 don't seem to have the same kind of relationship to each other. And in a ratio problem they should have. I think.

So where have I gone wrong? What utter stupidity have I committed to embarrass myself like this?

Pat

 

[chergh]

Trial and error is pretty much the only way to do this.

 

[boblarson]

Actually, Algebra would do it, but I'm so rusty I am not sure exactly what the starting formula would be.

 

[rainman89]

wouldnt it be

(26p)x+(18p)y=5.00?

then solve for that? cant remember how to solve for mulitple variables, but that seems too complex for a 9 year old

 

[boblarson]

...but that seems too complex for a 9 year old

I don't know - my 10 year old is starting Calculus (just kidding)

 

[rainman89]

whats a p? is it like .1 to 1? like cents to dollars?

 

[chergh]

Yeah algebra does it but its for 9 year olds, unless they start algebra earlier these days.

Using algebra it's easy:

x = choclate, y = cakes

26x + 18y = 500

y = 26/18 * x

y ~ 1.4x

26x + 18/1.4 * x = 500
38.46x = 500
x = 13

26 * 13 + 18y = 500
18y = 162
y = 9

13 choc bars
9 cakes

 

[rainman89]

YAY!! i was right!

 

[chergh]

I already answered the FIRST question a couple of hours ago...
now we have the other two to get through:

So where have I gone wrong?

What utter stupidity have I committed to embarrass myself like this?

No the first question was

How could I have CALCULATED it?

You just gave the answer which she already had.

 

[Big Pat]

Wow. Thanks for all your input folks. Chergh, I followed most of your algebra, but have to admit I stumbled at one point

26x + 18y = 500 FINE
y = 26/18 * x I don't follow. Is this line implied by the first. What happened to the 500?


Secondly: does any of this explain how you could get to the other answer i.e. 4 and 22?

 

[WayneRyan]

Folks,

It is an ACCESS forum after all:

Dim Chocolate As Integer
Dim Cake As Integer

For Chocolate = 0 To 19 ' Max Number of chocolates
   For Cake = 0 To 18   ' Max Number of cakes
      (Chocolate * 26) + (Cake * 18) = 500 Then
         MsgBox ("Chocolate = " & Chocolate & ", Cake = " & Cake)
      End If
      Next Cake
   Next Chocolate

Chocolates = 13
Cakes = 9

Wayne

 

[pbaldy]

I love it, Wayne!

In your copying and pasting, you lost the "If".

 

[boblarson]

Yeah, totally hillarious and yet on topic

 

[chergh]

Wow. Thanks for all your input folks. Chergh, I followed most of your algebra, but have to admit I stumbled at one point

26x + 18y = 500 FINE
y = 26/18 * x I don't follow. Is this line implied by the first. What happened to the 500?


Secondly: does any of this explain how you could get to the other answer i.e. 4 and 22?

Well I have done there requires a slight clarification in defining x and y first.

x and y are the quantities of choclate and cakes.

To solve the equation you need one variable therefore you need to define either x in term of y or y in terms of x, I chose the latter. So essentially what I am doing is calculating the number of cakes that are equal to the price of a choclate bar. To do this I divide the price of a choclate bar by the price of a cake which is 26/18 so for the price of a choclate bar you get ~1.44 cakes which allows me to use substitution so I can solve the equation for the value of x and once I know the value of x i can go back and solve the equation for y also.

 

[Pauldohert]

Yeah algebra does it but its for 9 year olds, unless they start algebra earlier these days.

Using algebra it's easy:

x = choclate, y = cakes

26x + 18y = 500

y = 26/18 * x

y ~ 1.4x

26x + 18/1.4 * x = 500
38.46x = 500
x = 13

26 * 13 + 18y = 500
18y = 162
y = 9

13 choc bars
9 cakes

26x + 18y = 500

from Cherg y = 26/18 * x

So 26x + 18(26/18) * x = 500 (Not 26x + (18/1.4) * x = 500)

26x + 26x = 500

52x = 500

x = 500/52 = 9.6 which is rubbish.

So y = 26(500)/52 * 18 = 13.88888888

-------------------------
Similarly if you switch the original substitution as Cherg would have done

x = 18/26

so 26(18/26)y + 18 = 500

36y = 500
y = 13.888888888 etc

So clearly to me anyhow Chergs method is incorrect, first the calculation in itself is wrong, secondly it doesn't manage to bring out the 2 differing answers. And this is because

y = 26/18 * x

ie 18y = 26x - ie an we know these added together make 500 so - 18y = 26x = 250

so y = 13.888888 and x = 9.61


The quantity of cakes isn't linked to the quantity of chocolates like this at all.

--------------

I am not convinved the answer can be calculated other than by a process of elimination. Cherg wa right to try and find the quantities in terms of each other, but seems to have done it in the wrong way - and then strangely done a calculation which looked like the right answer.

(if we knew something like there are 4 more choclolates than cakes - then
26(Y + 4) + 18y = 500
26y = 18y = 500 - 104
44y = 396
y = 9 so x = 13

or if there are 18 more cakes than chocs

26x + 18(x + 18) = 500
44x = 500 - 324
x = 4 so y = 22

I could be completely wrong on all of this, in which case my apologies in advance.

 

[chergh]

26x + 18y = 500

from Cherg y = 26/18 * x

So 26x + 18(26/18) * x = 500 (Not 26x + (18/1.4) * x = 500)

My bad in the original expression

x = 26/18 * y

Which rearranges to
y = x/ 1.4


So 26x + 18 * x/1.4 = 500

Which is the same as

18/1.4 * x

 

[Big Pat]

(if we knew something like there are 4 more choclolates than cakes - then
26(Y + 4) + 18y = 500
26y = 18y = 500 - 104
44y = 396
y = 9 so x = 13

or if there are 18 more cakes than chocs

26x + 18(x + 18) = 500
44x = 500 - 324
x = 4 so y = 22

I could be completely wrong on all of this, in which case my apologies in advance.

The trouble is that we don't know this. Not in advance. If I had two pieces of information, I could have used simultaneous equations and solved them.

And I **LOVE** the Access solution given above! Brilliant!! But this is trial and error too isn't it?

 

[Pauldohert]

Chergh - You have now corrected your "typo" miscalculation.

Now you need to correct your logic - you have not calculated the answer.

As I demonstrated, you can subsitute in any figure -

ie try 11x = 2y or y = (26/18) x


Why did you choose 18x = 26y - because that brings out the answer you want??


Big Pat - I am trying to explain to chergh - how the method that he used could be applicable - ie simultaneous equations. However - he is either plucking the other equation from thin air, or formulating it knowing the answer.


He is not calculating the solution to the problem.

 

[GaryPanic]

I used to be able to do this stuff in my head, now I just want to eat the chocolate(and the cakes)

g

 

[chergh]

Chergh - You have now corrected your "typo" miscalculation.

Now you need to correct your logic - you have not calculated the answer.

As I demonstrated, you can subsitute in any figure -

ie try 11x = 2y or y = (26/18) x


Why did you choose 18x = 26y - because that brings out the answer you want??

Actually i explained this relly badly.

1 unit of choclate cost 26p

1 unit of cake costs 18p

With these prices you can get 1.44 cakes for 26p

So:

1 * choclate bar = 26p = 1.44 * cakes

1 choclate bar = 1.44 cakes

1 cake = 1 choclate bar / 1.44

x cakes = x choclate bars/1.44

Substitute that into my original equation.