# 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

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

$\text{Small White} = 1.5 \times 2.5 = 3.75$

$\text{Small Black} = 1.5 \times 3.5 = 5.25$

$\text{Large White} = 3.5 \times 2.5 = 8.75$

$\text{Large Black} = 3.5 \times 3.5 = 12.25$

$\text{Total} = 30$

Nov 5, 2017

Yes

#### Explanation:

If the small:large ratio is $3 : 7$
then for some integer $p$
$\textcolor{w h i t e}{\text{XXX}}$the number of small balls must be $3 p$
$\textcolor{w h i t e}{\text{XXX}}$and
$\textcolor{w h i t e}{\text{XXX}}$the number of large balls must be $7 p$
That is, in total, there must be $10 p$ balls (for some integer $p$.

Similarly:
If the white:black ratio is $5 : 7$
then for some integer $q$
$\textcolor{w h i t e}{\text{XXX}}$the number of white balls must be $5 q$
$\textcolor{w h i t e}{\text{XXX}}$and
$\textcolor{w h i t e}{\text{XXX}}$the number of black balls must be $7 q$
That is, in total, there must be $12 q$ balls (for some integer $q$).

The total number of balls must be
$\textcolor{w h i t e}{\text{XXX}}$an integer multiple of $10$
$\textcolor{w h i t e}{\text{XXX}}$and
$\textcolor{w h i t e}{\text{XXX}}$an integer multiple of $12$

The smallest such number is the Least Common Multiple of $10$ and $12$:
$\textcolor{w h i t e}{\text{XXX}} L C M \left(10 , 12\right) = 60$

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) :}