Fractions in a Box

Age 7 to 11

Challenge Level Yellow star

This was a tricky problem. Well done to those of you who had a go. We had some very clearly explained answers. The key was to work out the size of the booklet first.

Rachel, Ol, Jack and Alex from Moretonhampstead Primary said:

First we worked out how many squares the booklet is. The number has to be a square number and has to be even and

$\frac{1}{4}$ of that number has to be even again. The only number possible for that is

$1$ 6 (four squared).

Then we took

$16$ (that was how big the booklet was) from a

$100$ which is

$84$ (there are

$84$ squares to play the game with). With

$84$ we can answer the first question - how many discs are there altogether? (

$84$ ).

After that we worked out how many discs there would be for the colours. We worked out there would be

$42$ discs (

$\frac{1}{2}$ of

$84$ ),

$21$ black discs (

$\frac{1}{4}$ of

$84$ ) and

$7$ blue (

$\frac{1}{12}$ of

$84$ ).

We put that on the grid as it says on the sheet. Next we used the last full column for blue and green. We know that there are

$7$ blues (because of what we worked out earlier) which means there are

$3$ green discs (

$7 +3 =10$ (which is how many in a column)).

There were five squares left. It says that there is

$1$ white square then the leftovers are yellow so we had

$4$ yellows and

$1$ white disc.

Now we hade completed the grid we could answer question

$2$ and

$3$ . For question

$2$ we counted up all the orange squares (

$6$ ) and the fraction is

$\frac{6}{84}$ but we had to simplify it to

$\frac{1}{14}$ (the answer).

Lastly we did question

$3$ . The fraction of green is

$\frac{3}{84}$ and simplified for the actual answer is

$\frac{1}{28}$ .

The fraction of yellow is

$\frac{4}{84}$ and simplified for the answer is

$\frac{1}{21}$ .

There is only one white square so the answer is

$\frac{1}{84}$ , for white.

Hamish, Rory, Sarah, Jesse and Samuel from Rutherglen Primary also reasoned very clearly and they sent us a picture of the full box which they modelled using cubes:

Sophie and Claire from The Downes School wrote:

$1\times 1$ didn't work because it said that two shortened rows have red discs.
$2\times 2$ didn't work because you need two shortened rows of red and one of orange.
$3\times 3$ didn't work because the total number of discs would be odd and you couldn't halve it. This means all odd numbers didn't work.
$4\times 4$ did work because you had the right amount of shortened rows.
$6 \times 6$ didn't work because you can't divide $64$ by $12$ .
$8 \times 8$ didn't work because you need six whole rows.

Emma, Abi, Matthew B and Yuji from Moorfield Junior School; Keshinie and Sharon at Kilvington GGS Victoria, Australia; Gideon from Newberries Primary School and Hannah, Georgia, Patrick; Hana from Bali International School and Matthew from Brighton College Prep School realised that the number left after taking away the booklet must be a multiple of $12$ . Keshinie and Sharon describe how they continued from there:

So that made it $84$ .
Half of the disks are red so that made the amount of red $42$ .
Then it said that a quarter is black so that made it $21$ .
Then it said that one twelfth is blue so that made it $7$ .
Then it said that one complete row was filled with all of blue and green and the remainder of $10$ if you take away $7$ made it $3$ green.
Then it said that one of the shortened rows is exactly filled with all the orange disks so that makes it $6$ .
Then it said that there was only one white disk.
Then we added all the numbers together making $80$ disks so there was a remainder of $4$ which had to be yellow.
We divided the $84$ disks by the $6$ orange ones that made it $14$ . So the fraction of orange had to be $1$ out of $14$ ( $\frac{1}{14}$ ).
We divided the $84$ disks by the $3$ green disks making the answer $28$ . So the fraction of green had to be $\frac{1}{28}$ .
We already knew that the fraction of white disk was $\frac{1}{84}$ .
We divided the $84$ disks by the $4$ yellow ones making it $21$ so the fraction of yellow had to be $\frac{1}{21}$ .

James from the Charter School explained very well how he went about the problem:

Each of the sides is

$10$ units and I called each of the sides of the booklet

$x$ . this means that the equation for finding the number of discs was

$N=100-x^2$ . (

$N$ being the number of remaining discs).

The amount of Blue discs was

$\frac{N}{12}$ meaning that

$N$ was a multiple of

$12$ . So I then collected all of the multiples of

$12$ :

$12$ ,

$24$ ,

$36$ ,

$48$ ,

$60$ ,

$72$ ,

$84$ ,

$96$ . I then eliminated those that did not fit the earlier equation because there was not a square number that fitted. This left:

$36$ ,

$84$ ,

$96$ .

I eliminated

$96$ because

$x$ had to be more that three for there was one complete row of orange discs and two of red discs.

This left:

$36$ ,

$84$ . From this I deduced that

$x$ had to be

$4$ or

$8$ . This means that the amount of red disks had to end in a

$2$ or a

$4$ , because there are two incomplete rows of red these either have to be a length of

$6$ or

$2$ . Since the amount of red discs is half of

$N$ I halved both my possible

$N$ 's which came up with

$18$ and

$42$ . This means that the amount of reds was

$42$ and

$N$ was

$84$ .

This means that

$x$ is

$4$ and that the amount of orange discs was

$6$ meaning the fraction is

$\frac{1}{14}$ .

The amount of blue was

$\frac{N}{12}$ which was

$7$ .

This means that the amount of green discs was

$3$ because blues + greens =

$10$ . That means the fraction was

$\frac{1}{28}$ .

The amount of white was

$1$ meaning the fraction was

$\frac{1}{84}$ .

Finally the rest were yellow.

Red was

$42$ . Blacks was

$\frac{N}{4}$ which was

$21$ . Blue was

$7$ . Orange was

$6$ . Green was

$3$ . White was

$1$ .

If you take all those away from

$84$ you end up with

$4$ . That is the amount of yellows. This means the fraction is

$\frac{1}{21}$ .

Well done too to Harriet and Harah from Greenacre School for Girls, Ruairidh from St Mary's High School, Anne-Marie, Emma, Katherine, Laura from Gorseland Primary and Eulalie and Holly who go to Lympstone Primary School.

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