Bucket has 4 types of balls: small white small black large white large black. Ratio small:large is 3:7. Ratio white:black is 5:7. Jack says "There must be at least 60 balls in the bucket." Is Jack correct? Show working to justify your answer

2 Answers
Nov 5, 2017

No.

Explanation:

Four Types of balls:

  • Small White
  • Small Black
  • Large White
  • Large Black

Small Has 33 each pair. Large has 77 each pair. White has 55 each pair. Black has 77 each pair.

So

"Small White" =1.5xx2.5=3.75Small White=1.5×2.5=3.75

"Small Black" = 1.5xx3.5=5.25Small Black=1.5×3.5=5.25

"Large White" = 3.5xx2.5=8.75Large White=3.5×2.5=8.75

"Large Black" = 3.5xx3.5=12.25Large Black=3.5×3.5=12.25

"Total" = 30Total=30

Nov 5, 2017

Yes

Explanation:

If the small:large ratio is 3:73:7
then for some integer pp
color(white)("XXX")XXXthe number of small balls must be 3p3p
color(white)("XXX")XXXand
color(white)("XXX")XXXthe number of large balls must be 7p7p
That is, in total, there must be 10p10p balls (for some integer pp.

Similarly:
If the white:black ratio is 5:75:7
then for some integer qq
color(white)("XXX")XXXthe number of white balls must be 5q5q
color(white)("XXX")XXXand
color(white)("XXX")XXXthe number of black balls must be 7q7q
That is, in total, there must be 12q12q balls (for some integer qq).

The total number of balls must be
color(white)("XXX")XXXan integer multiple of 1010
color(white)("XXX")XXXand
color(white)("XXX")XXXan integer multiple of 1212

The smallest such number is the Least Common Multiple of 1010 and 1212:
color(white)("XXX")LCM(10,12)=60XXXLCM(10,12)=60

Since
color(white)("XXX"){: (10,=,2,xx5,,), (ul(12),=,ul(2),ul(color(white)(xx5)),ul(xx6),ul(color(white)(=60))), (LCM,=,2,xx5,xx6,=60) :}