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How many marbles idd each of them have at first?

Juan and Rachel have the same number of marbles. Rachel gives away 10 of her marbles and Juan gives away 22 marbles.  Rachel then has 3 times as many marbles a Juan. How many marbles did each of them have at first?
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2 Answers

Let's start by assigning letters (or variables for each person)
 
j = r
 
Then we have
 
r - 10 (Rachel gave away 10)
j - 22 (Juan gave away 22)
 
What we know about the new status of marbles is that r - 10 = 3(j - 22) which means now Rachel (r-10) has 3 times the amount of Juan (j-22)
 
And since originally r = j, now we can plug in either r or j for the new equation so that we only have one type of variable, and solve for one variable
 
j - 10 = 3(j - 22)
remove parentheses by multiplying
j - 10 = 3j - 66
subtract j from both sides of equation
-10 = 2j - 66
add 66 to both sides of the equation
56 = 2j
divide both sides by 2
28 = j
and since j = r, then we know r = 28
and check the equation to see if 28 is the correct answer for both Rachel and Juan
 
28 - 10 = 3(28 - 22)
28 - 10 = 3(6)
18 = 18
 
let J= # of marbles Juan has
let R=# of marbles Rachel has
J=R to begin with
J-22 and R-10 are the number of marbles each has after giving marbles away
R-10=3(J-22)        (Rachel now has three times as many marbles as Juan)
substitute R for J or J for R because J=R
R-10=3(R-22)
R-10=3R-66
66-10=3R-R
56=2R
56/2=R
28=R
Rachel and Juan each had 28 marbles to begin with
check:28-10=18 and 28-22=6 and 18=3*6