Aaron and Henry had 52 grapes together. If Aaron ate 11 grapes and gave 5 to Henry, he would have 19 fewer grapes than Henry. How many grapes did Aaron have in the beginning?

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a=# of grapes Aaron has

h=# of grapes Henry has

a+h=52

Aaron ate 11 and gave away 5, so Aaron has a-11-5 grapes or a-16

Henry got 5 so Henry has h+5

after all this Aaron has 19 fewer grapes than Henry

(a-16) is 19 less than (h+5)

(a-16)+19=(h+5)

a-16+19=h+5

a+3=h+5, and a+h=52

a+h=52, so a=52-h

replace a in a+3=h+5 with 52-h

52-h+3=h+5

55-h=h+5

55=h+h+5

55-5=h+h

50=2h

50/2=h

25=h

Henry has 25 grapes and Aaron has 52-25=27 grapes

check:27-11-5=11, 25+5=30 and 11 is 19 fewer than 30

you could also use a+3=h+5 and a+h=52 this way:

a-h=5-3

a-h=2

a+h=52

now add the two equations

2a=54 and a=54/2=27 grapes for Aaron and 25 grapes for Henry

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