Essential Supplies

Age 14 to 16

ShortChallenge Level Yellow star

Answer:

$170$

Starting with boxes of 12

${2005}\div{12}=167$ remainder

$1$ .
So, if

$167$ of the larger boxes are used

$1$ bar will remain.

If

$166$ larger boxes are used there will be

$13$ bars left (not a mulitple of

$5$ ).
If

$165$ larger boxes are used this will leave

$25$ bars, which can be packed into

$5$ of the smaller boxes.

Total number of boxes is

$165+5=170$ .

Starting with boxes of 5

$2005\div5=401$ so could use

$401$ small boxes

$12$ boxes of

$5$ contain the same number as

$5$ boxes of

$12$

$401\div12 = 33$ remainder

$5$
So

$401$ boxes of

$5 = 33\times(12$ boxes of

$5)$ remainder

$5$ boxes of

$5$

$=33\times(5$ boxes of

$12)$ remainder

$5$ boxes of

$5$

$= 33\times5 + 5 =170$ boxes

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.

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