How many quarts of olive oil did they use? (see below) Thanks!

#color(blue)("Building the model")#

To maintain a high level of precision I will stick with fractions.
Rounding decimals can lead to accumulated errors. So if you wish to use decimals convert the fractions at the very end.

Let the volume of 30% oil be #v_30#
Let the volume of 79% oil be #v_79#
Let the volume of the blended be #v_b = 19/20" "#quarts

For now ignore the units of quarts.

When blended for total volume we have:

#v_30+v_79=19/20 " "......................Equation(1)#

When blended for % of oil volume content we have (in quarts):

#[30/100xxv_30]+[79/100xxv_79] =[(40 6/19)/100 xx19/20]....Equation(2) #
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#color(blue)("Answering the question")#

In #Eqn(2)# we have 2 unknowns. If we can express one of these in terms of the other we end up with just 1 unknown. Thus solvable.

I choose to substitute for #v_79#

From #Equation(1)#

#v_79=19/20-v_30" "....................Equation(3)#

Using #Equation(3)# substitute for #v_79# in #Equation(2)#

#[30/100xxv_30]+[79/100xx(19/20-v_30)] =[(40 6/19)/100 xx19/20]#

#color(white)(d"d")(30v_30)/100color(white)("dd.d")+color(white)("ddd")1501/2000-(79v_30)/100 color(white)("ddd")= color(white)("ddd")(766)/2000#

Having a common denominator of 2000 gives:

#(600v_30)/2000 +1501/2000-(1580v_30)/2000=766/2000#

Multiply both sides by 2000

#600v_30+1501-1580v_30=766#

#-980v_30=-2267 #

#color(green)(v_30=(-2267)/(-980) = 2 307/980 "quarts "...................Equation(3))#
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Using #Equation(3)# substitute for #v_30" in "Equation(1)#

I will let you finish that off.