A distributor needs to blend a mix
of Gazebo coffee that normally
sells for $10.20 per pound
with Shade Grown coffee
that normally sells for $11.80 per pound
to create 60 pounds of a coffee
that can sell for $10.33 per pound.
How many pounds of each kind
of coffee should they mix?
Round to the nearest whole pound.
We need to begin by defining a variable
and variable expression for
the amount of Gazebo coffee
and the amount of Shade Grown coffee.
To begin, we know there must
be 60 pounds of the mixture,
so let's record this as 60
equals the pounds of mixed blend
which sells for $10.33 per pound.
Next, let's let the
variable x equal the pounds
of Gazebo coffee which
sells for $10.20 per pound.
And now for the amount
of Shade Grown coffee,
since we know there's a total
of 60 pounds of the blend
and x pounds of Gazebo coffee,
the total of 60 minus x
must give us the pounds
of Shade Grown coffee which
sells for $11.80 per pound.
Again, the total amount
of the mixed blend,
which is 60 pounds, minus x,
the pounds of Gazebo coffee,
must give us the number of
pounds of Shade Grown coffee.
And now from here we
can write an equation.
Our equation will be the cost
of the first coffee times the amount,
plus the cost of the second
coffee times the amount
must equal the mix cost
times the mix amount.
And since we have x
pounds of Gazebo coffee
that sells for $10.20 per pound,
the cost times the
amount is $10.20 times x
or just 10.2 times x.
And then we have plus.
There are 60 minus x pounds
of the Shade Grown coffee
sells for $11.80 per pound,
and therefore the cost times the amount
is $11.80 per pound times
the quantity 60 minus x
or 11.8 times the quantity 60 minus x.
And this must equal the cost
of the mix times the mix amount.
And since we know we have
60 pounds of the mixed blend
that sells for $10.33 per pound,
the cost times the amount
is 10.33 times 60.
We now need to solve this equation for x
and then come back and
determine the number
of pounds of Gazebo coffee
and the number of pounds
of Shade Grown coffee.
To begin solving,
we first need to clear the
parentheses on the left
by distributing 11.8 and then
find the product on the right.
So we have 10.2x,
and then plus 11.8 times 60,
which is equal to 708.
And then we have 11.8 times negative x,
which gives us minus 11.8x equals
on the right 10.33 times 60
is equal to 619.8.
And now we simplify the
left side of the equation
by combining like terms,
10.2x minus 11.8x
is equal to negative 1.6x,
which gives us negative 1.6x
plus 708 is equal to 619.8.
Next step, we isolate the variable term
by subtracting 708 on both sides.
Simplifying, on the left,
we have negative 1.6x is equal
to 619.8 minus 708,
which equals negative 88.2.
The last step to solve for x is
to divide both sides by negative 1.6.
Simplifying on the left,
negative 1.6 divided by
itself simplifies to one.
One times x is x.
We have x equals the
quotient on the right.
Negative 88.2 divided by negative 1.6
is equal to 55.125.
So this is the exact number
of pounds we would need
of the Gazebo coffee
in order to make the desired mix,
but notice how the directions do say round
to the nearest whole number,
which means round to the ones place value.
We have a five in the ones place value,
and to the right we have a
one in the tenths place value,
and therefore we round down to 55.
55.125 is closer to 55 than it is to 56.
So to answer the question,
how pounds of each kind of
coffee should they blend,
we will actually use
x equals 55, or 55
pounds of Gazebo coffee.
Let's answer the question
on the next slide.
So again, we will use x equals
55 to answer the question.
We also need to find 60 minus x,
which will give us the
pounds of Shade Grown coffee.
And 60 minus x is equal to 60 minus 55,
which is equal to five.
So we now know they should
use 55 pounds of Gazebo coffee
and five pounds of Shade Grown coffee
to make 60 pounds of a coffee
that will sell for $10.33 per pound.
Let's write this out
as a complete sentence.
