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 3 each pair. Large has 7 each pair. White has 5 each pair. Black has 7 each pair.

So

"Small White" =1.5xx2.5=3.75

"Small Black" = 1.5xx3.5=5.25

"Large White" = 3.5xx2.5=8.75

"Large Black" = 3.5xx3.5=12.25

"Total" = 30

Nov 5, 2017

Yes

Explanation:

If the small:large ratio is 3:7
then for some integer p
color(white)("XXX")the number of small balls must be 3p
color(white)("XXX")and
color(white)("XXX")the number of large balls must be 7p
That is, in total, there must be 10p balls (for some integer p.

Similarly:
If the white:black ratio is 5:7
then for some integer q
color(white)("XXX")the number of white balls must be 5q
color(white)("XXX")and
color(white)("XXX")the number of black balls must be 7q
That is, in total, there must be 12q balls (for some integer q).

The total number of balls must be
color(white)("XXX")an integer multiple of 10
color(white)("XXX")and
color(white)("XXX")an integer multiple of 12

The smallest such number is the Least Common Multiple of 10 and 12:
color(white)("XXX")LCM(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) :}