+103 Darius, Ellen and Farman each had some marbles. They traded marbles with one another. In the 1st round, Darius gave 1/5 of his marbles to Ellen. In the 2nd round, Ellen gave 1/4 of his marbles to Farhan. In the 3rd round, Farhan gave 1/2 of his marbles to Darius. Darius had 45 marbles, Ellen had 30 marbles and Farhan had 25 marbles left in the end. How many marbles had each of them at first?
+313 Information given:
At the end,
Number of marbles Darius has = 45 marbles
Number of marbles Ellen has = 30 marbles
Number of marbles Farhan has = 25 marbles
Calculations: Let’s say Darius has x number of marbles, Ellen has y number of marbles, and Farhan has z number of marbles at the beginning.
1st round
Darius gave 1/5 marbles to Ellen.
=> Number of marbles Darius has = x – x/5 = 4x/5
Number of marbles Ellen has = y + x/5
2nd round
Ellen gave 1/4 marbles to Farhan
=> Number of marbles Ellen has = (y + x/5) – 1/4 (y + x/5) = 3/4 (y + x/5)
Number of marbles Farhan has = z + 1/4 (y + x/5)
3rd round
Farhan gave 1/2 of his marbles to Darius.
=> Number of marbles Farhan has = 1/2 [z + 1/4 (y + x/5)]
Number of marbles Darius has = 4x/5 + 1/2 [z + 1/4 (y + x/5)]
So, it can be written as,
1/2 [z + 1/4 (y + x/5)] = 25
4x/5 + 1/2 [z + 1/4 (y + x/5)] = 45
3/4 (y + x/5) = 30
Simplifications:
4x/5 + 25 = 45
=> x = 5(45-25)/4 = 25
3/4 (y + x/5) = 30
=> (y + 25/5) = (30)4/3 = 40
=> y + 5 = 40
=> y = 40 - 5 = 35
1/2 [z + 1/4 (y + x/5)] = 25
=> [z + 1/4 (y + x/5)] = 2 (25) = 50
=> z + 1/4 (35 + 25/5) = 50
=> z + 1/4 (35 + 5) = 50
=> z + 1/4 (40) = 50
=> z + 10 = 50
=> z = 50 - 10 = 40
Conclusions:
Darius has 25 number of marbles, Ellen has 35 number of marbles, and Farhan has 40 number of marbles at the beginning.
+26 Let's solve this problem by working backwards.
Third Round:
Since Farhan gave half of his marbles last round and he now has 25 marbles, he had 25 x 2 = 50 marbles in the third round.
Darius has 45 - 25 = 20 marbles in the third round as well.
Second Round:
Ellen gave a fourth of her marbles during the second round and now has 30 marbles left. This means that 3/4 of her previous marbles = 30, so she has 30 x 4/3 = 40 marbles in the second round.
Ellen also gave 10 marbles to Farhan during this round, so Farhan had 50 - 10 = 40 before then.
First Round:
Darius gave 1/5 of his marbles and ended up with 20 marbles. This means that Darius originally had 20 x 5/4 = 25 marbles.
Ellen received 5 marbles and ended up with 40 marbles, so she originally had 40 - 5 = 35 marbles.
Let's now write down the number of marbles each person has after working backwards:
Ellen has 35 marbles.
Darius has 25 marbles.
Farhan has 40 marbles.
