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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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

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