Find the value of $\frac{3÷3\times 2-6÷3+7}{5÷5\times 6÷6+3+3}.$

The correct answer is 1.

To find the value of the given expression, we evaluate the numerator and the denominator separately using the order of operations (BODMAS/PEMDAS rule), which dictates that we perform division and multiplication (from left to right) before addition and subtraction (from left to right).

Step 1: Simplify the Numerator

The expression in the numerator is:

$3÷3\times 2-6÷3+7$

First, perform the division operations from left to right:
$3÷3=1$
and
$6÷3=2$

Substitute these values back into the expression:
$1\times 2-2+7$

Next, perform the multiplication:
$1\times 2=2$

Substitute this back into the expression:
$2-2+7$

Now perform the addition and subtraction from left to right:
$2-2=0$
$0+7=7$

So, the numerator evaluates to 7.

Step 2: Simplify the Denominator

The expression in the denominator is:

$5÷5\times 6÷6+3+3$

First, perform the division operations from left to right:
$5÷5=1$

Now update the expression:
$1\times 6÷6+3+3$

Next, perform multiplication and division from left to right:
$1\times 6=6$
$6÷6=1$

Substitute this back into the expression:
$1+3+3$

Now perform the addition:
$1+3+3=7$

So, the denominator evaluates to 7.

Step 3: Calculate the Final Value

Substitute the simplified values of the numerator and the denominator back into the fraction:

$\frac{7}{7}=1$