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# fractions

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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?

Jul 9, 2021

#1
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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.

Jul 9, 2021
#2
+26
+3

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.

Jul 9, 2021