Question

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

Answer 1

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`

Answer 2

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) :}#