We choose a good number of units to do this question. I am not so sure what primary school students are taught though.

Let June have 6 units of game cards.
Let Kelvin have 1 part of game cards. We use the term "part" rather than "unit" as these represent different numbers, but the usage of the words are very similar.

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June gives away half of her game cards, or 3 units of her game cards, to Kelvin.

At the end,
- June has 3 units of game cards
- Kelvin has (1 part + 3 units) of game cards

This means that Kelvin has 1 part of game cards more than June, which coincidentally is 72 game cards more than June.

So, "1 part" = 72 game cards.

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June gives away 1/6 of her game cards, or 1 units of her game cards, to Kelvin.

At the end,
- June has 5 units of game cards
- Kelvin has (1 part + 1 unit) of game cards

Kelvin still has 16 game cards more than June, so

Kelvin = June + 16
1 part + 1 unit = 5 units + 16

But remember that 1 part = 72 game cards.

72 + 1 unit = 5 units + 16
72 = 5 units + 16 - 1 unit
72 = 16 + 4 units
72 - 16 = 4 units
56 = 4 units
56/4 = 1 unit
14 = 1 unit

So, "1 unit" = 14 game cards.

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Total number of game cards
= 6 units + 1 part (from the start of the story)
= 6 * 14 + 72
= 84 + 72
= 156

(and June must have had 84 game cards while Kelvin must have had 72 game cards)