$0.20 type $\to {35}^{\text{lb}}$Explanation: One of the better ways to tackle this problem type is to consider them as a set of fraction that sum to the value of 1. Where 1 represents the whole of the final blend. $\textcolor{b l u e}{\text{Determine the fractional proportion of each type of bean}}$Let the proportion of the$0.20 be $x$

Then the proportion of the $0.68 is $1 - x$Set the target at$0.54

Dropping the unit of measurement ($) for now we have $0.20 x + \left(1 - x\right) 0.68 = 1 \times 0.54$$0.20 x - 0.68 x + 0.68 = 0.54$$- 0.48 x + 0.68 = 0.54$Lets get rid of the decimals for now and multiply everything by 100 $- 48 x + 68 = 54$$48 x = 68 - 54$$x = \frac{14}{48}$$x = \frac{7}{24}$of the blend at$0.20

So the proportion of the $0.68 is $1 - \frac{7}{24} = \frac{17}{24}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $\textcolor{b l u e}{\text{Determine the weight proportion of each bean type}}$$0.68 type $\to \frac{17}{24} \times {120}^{\text{lb") = 85^("lb}}$

$0.20 type $\to \frac{7}{24} \times {120}^{\text{lb")= 35^("lb}}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Check: (35xx0.2)+(85xx0.68)=$64.8

120xx0.54 = $64.8 May 3, 2018 Method 2 of 2 An amazing approach that is not commonly used. Very fast once used to it. 85^("lb")" of$0.68 beans"
35^("lb")" of $0.20 beans" Explanation: This uses the principle of a straight line graph. Let the weight of the$0.20 bean be ${w}_{0.2}$
Let the weight of the $0.68 bean be ${w}_{0.68}$Then by considering only say ${w}_{0.68}$the value of ${w}_{0.2}$is directly inferred by ${w}_{0.2} = 120 - {w}_{0.68}$So it is possible to determine the part of the answer by considering just ${w}_{0.68}$If all ${w}_{0.68}$total value is ................" "120xx$0.68 = $81.60 If all ${w}_{0.2}$total value is ................." "120xx$0.20=$24.00  If all the target blend total value is ." "120xx$0.54 = $64.80  Plotting this on the graph we have: The slope of all is the same as the slope of a part of it color(green)("Slope of all "->("change in y")/("change in x") ->(81.60-24.00)/(120)=57.6/120) $\textcolor{red}{\text{Slope of part: } \frac{64.8 - 24.00}{x} = \frac{40.8}{x} \textcolor{g r e e n}{= \frac{57.6}{120}}}$$x = 40.8 \times \frac{120}{57.6} = {85}^{\text{lb}} = {w}_{0.68}$So we have: 85^("lb")" of$0.68 beans"
120-85=35^("lb")" of \$0.20 beans"