Cake word problem: Michael is baking 240 cakes for the school fair

[patelsm]

Hi all,

Once again I am stuck on a question which I am not able to work it out for my child.

"Michael is baking 240 cakes for the school fair and Zara is decorating them. She ices every 4th cake in pink icing.
She puts silver sprinkles on every 5th cake and every 3rd cake she adds white chocolate star.
How many cakes in total will have pink icing AND silver sprinkles AND white chocolate star?"

Below is how I was aiming to trying to make my child visualize the pattern, and assumed that to get the answer I need to find the line where it has all the three (i.e star, pink and silver sprinkles).
I began with 12 cakes, hoping it will be smaller number and proportionally equal to 240. By looking at this we can work out individual type of cakes for example 4 cakes with star (12/3), 4 cakes with pink (12/3) and 2.5 ? cakes with sprinkles (12/5)....but I just cannot figure out how to get number of cakes with all of them.

cake 1 - first>>>>>>>>first>>>>>>>>>>first
cake 2 - second>>>>>>second>>>>>>>>second
cake 3 - third (star)>>>>third>>>>>>>>>third
cake 4 - first>>>>>>>>fourth(pink)>>>>>fourth
cake 5 - second>>>>>>first>>>>>>>>>>fifth (silver sprinkles)
cake 6 - third (star)>>>>second>>>>>>>>first
cake 7 - first >>>>>>>third>>>>>>>>>>second
cake 8 - second >>>>>fourth (pink)>>>>>third
cake 9 - third (star)>>>first>>>>>>>>>>fourth
cake 10 - first >>>>>>second>>>>>>>>fifth (silver sprinkles)
cake 11 - second>>>>>third>>>>>>>>>first
cake 12 - third (star)>>fourth(pink)>>>>>second

any help could do please,

saf

 

[Deleted member 4993]

Hi all,

Once again I am stuck on a question which I am not able to work it out for my child.

"Michael is baking 240 cakes for the school fair and Zara is decorating them. She ices every 4th cake in pink icing.
She puts silver sprinkles on every 5th cake and every 3rd cake she adds white chocolate star.
How many cakes in total will have pink icing AND silver sprinkles AND white chocolate star?"

Below is how I was aiming to trying to make my child visualize the pattern, and assumed that to get the answer I need to find the line where it has all the three (i.e star, pink and silver sprinkles).
I began with 12 cakes, hoping it will be smaller number and proportionally equal to 240. By looking at this we can work out individual type of cakes for example 4 cakes with star (12/3), 4 cakes with pink (12/3) and 2.5 ? cakes with sprinkles (12/5)....but I just cannot figure out how to get number of cakes with all of them.

cake 1 - first>>>>>>>>first>>>>>>>>>>first
cake 2 - second>>>>>>second>>>>>>>>second
cake 3 - third (star)>>>>third>>>>>>>>>third
cake 4 - first>>>>>>>>fourth(pink)>>>>>fourth
cake 5 - second>>>>>>first>>>>>>>>>>fifth (silver sprinkles)
cake 6 - third (star)>>>>second>>>>>>>>first
cake 7 - first >>>>>>>third>>>>>>>>>>second
cake 8 - second >>>>>fourth (pink)>>>>>third
cake 9 - third (star)>>>first>>>>>>>>>>fourth
cake 10 - first >>>>>>second>>>>>>>>fifth (silver sprinkles)
cake 11 - second>>>>>third>>>>>>>>>first
cake 12 - third (star)>>fourth(pink)>>>>>second

any help could do please,

saf

Hint:

3, 4 & 5 are relatively prime numbers (no common factor/s).

Lowest common multiple of (3,4,5) is 60.

Continue....

 

[patelsm]

Really sorry...I don't get it...how does the LCM of the indexes 3,45,5 help us out? I'm sure it does but I'd need to explain it in such a way.... apologies

 

[Dr.Peterson]

Once again I am stuck on a question which I am not able to work it out for my child.

"Michael is baking 240 cakes for the school fair and Zara is decorating them. She ices every 4th cake in pink icing.
She puts silver sprinkles on every 5th cake and every 3rd cake she adds white chocolate star.
How many cakes in total will have pink icing AND silver sprinkles AND white chocolate star?"

Below is how I was aiming to trying to make my child visualize the pattern, and assumed that to get the answer I need to find the line where it has all the three (i.e star, pink and silver sprinkles).
I began with 12 cakes, hoping it will be smaller number and proportionally equal to 240. By looking at this we can work out individual type of cakes for example 4 cakes with star (12/3), 4 cakes with pink (12/3) and 2.5 ? cakes with sprinkles (12/5)....but I just cannot figure out how to get number of cakes with all of them.

Here's the idea.

We have a pattern like this (I've turned your table on its side, and simplified it):

[FONT=courier new]
cake:       1  2  3  4  5  6  7  8  9 10 11 12
[/FONT][FONT=courier new]star:             *        *        *        * ...
[/FONT][FONT=courier new]icing:               *[/FONT][FONT=courier new][FONT=courier new]           *           * ...
[/FONT]sprinkles:              *              *       ...
[/FONT]

A cake gets a star if its index is a multiple of 3; icing if it is a multiple of 4; and sprinkles if it is a multiple of 5.

Any cake that gets all three will be a multiple of all three numbers, 3, 4, and 5 -- in particular, the first will be the Least Common Multiple. What is that? (You've already been told the answer.) In my table, the first one that got both star and icing was #12, the LCM of 3 and 4.

Then the pattern will repeat. So the number of cakes with all three will be the number of times this pattern repeats -- namely, 240 divided by the LCM (assuming, as it does, that this divides evenly).

 

[patelsm]

4 cakes with all the three toppings!
Thank you ever so much. I now totally understand this concept and will enjoy explaining to my child who will be chuffed.
Working out LCM is one thing, but to actually understand on how it is used to work out such problems is the core of it - this will help in further building my child's confidence.
Thanks again and am extremely grateful.

Here's the idea.

We have a pattern like this (I've turned your table on its side, and simplified it):

[FONT=courier new]
cake:       1  2  3  4  5  6  7  8  9 10 11 12
[/FONT][FONT=courier new]star:             *        *        *        * ...
[/FONT][FONT=courier new]icing:               *[/FONT][FONT=courier new][FONT=courier new]           *           * ...
[/FONT]sprinkles:              *              *       ...
[/FONT]

A cake gets a star if its index is a multiple of 3; icing if it is a multiple of 4; and sprinkles if it is a multiple of 5.

Any cake that gets all three will be a multiple of all three numbers, 3, 4, and 5 -- in particular, the first will be the Least Common Multiple. What is that? (You've already been told the answer.) In my table, the first one that got both star and icing was #12, the LCM of 3 and 4.

Then the pattern will repeat. So the number of cakes with all three will be the number of times this pattern repeats -- namely, 240 divided by the LCM (assuming, as it does, that this divides evenly).

 

[Dr.Peterson]

4 cakes with all the three toppings!
Thank you ever so much. I now totally understand this concept and will enjoy explaining to my child who will be chuffed.
Working out LCM is one thing, but to actually understand on how it is used to work out such problems is the core of it - this will help in further building my child's confidence.
Thanks again and am extremely grateful.

Yes, you got it. And your point is exactly right: knowing how to recognize when a particular concept is relevant is far more important than the procedure.

Note that I used a visual model, and a smaller problem (3 and 4 only) to help make it easier to see. These are both valuable things to do on one's own to help understand a problem.

 

[patelsm]

Thank you so much

Yes, you got it. And your point is exactly right: knowing how to recognize when a particular concept is relevant is far more important than the procedure.

Note that I used a visual model, and a smaller problem (3 and 4 only) to help make it easier to see. These are both valuable things to do on one's own to help understand a problem.