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

Answer:

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

Answer:

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) :}#