Solve Nested Division Expression: 3:(4:(5:12)) Using Colon Notation

Question

$3:(4:(5:12))=\text{?}$

00:00 Solve

00:03 Always solve inner parentheses first, even within parentheses the rule applies

00:06 Let's start with the innermost parentheses

00:09 Let's write division since

00:12 Division is also multiplication by the reciprocal

00:20 Let's raise the multiplication to the numerator

00:29 Division is also multiplication by the reciprocal

00:36 Division is also multiplication by the reciprocal

00:43 Let's break down 12 into factors of 4 and 3

00:50 Let's simplify what we can

00:55 And this is the solution to the question

First, we rewrite the multiplication exercise inside parentheses as a fraction:

$3:(4:\frac{5}{12})=$

Now we invert the fraction to create a multiplication exercise:

$3:(4\times\frac{12}{5})=$

We add the 4 to the numerator of the fraction in the multiplication exercise:

$3:\frac{4\times12}{5}=$

Now we invert the fraction to create a multiplication exercise:

$3\times\frac{5}{4\times12}=$

We add the 3 to the numerator of the fraction in the multiplication exercise:

$\frac{3\times5}{4\times12}=$

We break down the 12 into a smaller multiplication exercise:

$\frac{3\times5}{4\times3\times4}=$

We simplify the 3 in the numerator and denominator:

$\frac{5}{4\times4}=\frac{5}{16}$

$\frac{5}{16}$