We were asked to multiply and simplify
if possible.
Here we have a whole number times a
mixed number. We can view 3 times 2 and
five sixths as meaning three copies or
three groups of two and five sixths.
We will first determine the product
using the formal rules outlined below,
but then we'll take a look at a model to
better understand the product. The first
step is to convert the mixed numbers and
whole numbers to improper fractions,
which means we write three as a
fraction with a denominator of 1. So we
have 3 over 1 times, now we convert 2 5/6
to an improper fraction. The
denominator remains 6. To determine the
numerator, we multiply the denominator
and the whole number and then add the
numerator. Six times two is twelve 12
plus 5 equals 17. 2 and 5/6 equals 17/6.
Next we multiply the numerators and
denominators. The numerator is three times 17.
the denominator is 1 times 6. Before we
determine these products though, we will
simplify out any common factors between
the numerator and denominator. Notice 3
and 6 share a common factor of three.
There's 1 3 in three and two threes in
six. The only remaining common factor
between the numerator denominator is 1,
so now we can multiply knowing the
product will be in simplified form.  1 times
17 is 17 and  1 times 2 is equal to 2.
The product as an improper fraction
is 17/2, which is in simplified form.
Let's also express the improper fraction
as a mixed number . A fraction bar means
division and therefore to convert 17/2
to a mixed number, we divide 17 by 2.
There are eight 2's in 17. 8 times 2
equals 16,  subtract the difference is one, which
gives us a quotient of eight and
one-half. To form the fraction it is
always the remainder over the divisor, so
the product
is 17/2 as a improper fraction or
eight and one-half as a mixed number. And
now take a look at a model for this
product to better understand why the
product equals eight and a half. Again
viewing this product as three copies or
three groups of 2 5/6 we would
have the model shown below.
Notice where if the rectangle is one unit,
here we have one copy of two and five
sixths,  two copies and three copies. Now let's
determine how much we have here. We have
1, 2, 3, 4, 5, 6. And here we have 5/6.  One more sixth
would give us 6 sixths or one so five six plus
this 1/6 gives us another unit, which
would be seven.  And here we have 4/6 plus
2/6 here would give us 6/6 or another one,
which gives us eight. And then here we
have 3/6, which we can see is equivalent
to one half. This model shows why the
product equals eight and a half or if we
want 17/2.
I hope you found this helpful.
