Question posted by YC Mak from Singapore:

Grade/Level: 5th

Question solved by Model Method: Francis, Gif and Hakim shared a sum of money. Hakim decided to give Francis 2/3 of his money. Then Francis gave Gif 3/4 of his money and Gif gave 3/7 of his money to Hakim. In the end, the amount of money they had was in the ratio of 3:6:8. If Hakim had $55 more than Francis at first, how much money did Gif have in the end?

Answer:

Step 1: This question involves the Constant Total Concept as well as the Working Backwards Strategy. It is an internal transfer among 3 parties resulting in no increase in the overall total of the three people. To work backwards, we start by drawing the end ratio among the three people which is 3:6:8.

Step 2: Then, we trace back the last action done which is "Gif gave 3/7 of his money to Hakim". Since Gif gave 3/7 of his money to Hakim, Gif must have 4/7 of his money left. Thus, the 6 parts that Gif had must be equivalent to 4/7. So, we work out the common multiple of 6 and 4 which is equal to 12 and divided the 6 parts into 12 units. And we transfer 9 units (equivalent to 3/7) from Hakim back to Gif.

Step 3: Then, we trace back the second last action done which is "Francis gave Gif 3/4 of his money". Since Francis gave Gif 3/4 of his money, Francis must have 1/4 of his money left. Thus, the 3 parts (or 6 units) that Francis had must be equivalent to 1/4. And so we transfer 18 units (equivalent to 3/4) from Gif back to Francis.

Step 4: Then, we trace back the third last action done which is "Hakim decided to give Francis 2/3 of his money". Since Hakim gave Francis 2/3 of his money, Hakim must have 1/3 of his money left. Thus, the 3 1/2 parts (or 7 units) that Hakim had must be equivalent to 1/3. And so we transfer 14 units (equivalent to 2/3) from Francis back to Hakim. We re-drew Hakim model because it is difficult to overlap the units he gave away and he received.

Thus, at first,

Hakim had 7 units + 14 units = 21 units

Francis had 24 units - 14 units = 10 units

21 units - 10 units = 11 units

11 units ----------> $55

1 unit ----------> $55 / 11 units = $5

12 units ----------> 12 x $5 = $60

Therefore, Gif had $60 in the end.

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