Mary's farm stand sold a total of 165 pounds of apples and peaches. She sold apples for $1.75 per pound and peaches for $2.50 per pound. If she made $337.50 how many apples and how many peaches did she sell?

The weight of apples is: 100 The weight of pairs is: 65

Let the mass in pounds of apples be##" " a## Let the mass in pounds of pairs be ##" "p##

Let total income from apples be ##" "T_a##

Let the total income from pairs be ##" "T_p##

##color(blue)("Breaking the question into parts to build the equations:")##

Sold a total of 165 pound of apples##" "->a+p=165## ....(1)

Sold apples for $1.75 per pound ##" "-> T_a=1.75xxa##....(2) Sold pairs for $2.50 per pound##" "->T_p=pxx2.50##....(3)

She made $337.50 ##" "->T_a+T_p=337.50##.....(4) '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

##color(blue)("Determine the value of p")##

From equation (1) ##color(brown)(a=165-p" "..........(1_a))##

Substituting equations (2) and (3) into (4)

##T_a+T_p=337.50" "->" "1.75color(green)(a)+2.5p=337.50##

But from ##(1_a) " "color(green)(a)=165-p## giving

##T_a+T_p=337.50" "->" "1.75(color(green)(165-p))+2.5p=337.50##

##288.75-1.75p+2.5p=337.5##

##288.75 +0.75p=337.5##

##p=(337.5-288.75)/(0.75)##

##color(red)(p= 65)##.......................................(5) '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ##color(blue)("Determine the value of a")##

Substitute equation (5) into equation (1)

##a+color(green)(p)=165" "->" "a+color(green)(65)=165##

##a=165-65 = 100##

##color(red)(a=100)## .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Check: ##$1.75(100) + $2.50(65)" "=" "$175+$162.5" "=" "$337.5 " "-> color(red)("True")##