Problem 1:

Let's solve each part of the problem step-by-step.

Determine the total number of children
Hannah has invited 14 friends and she herself will be there, making a total of:
14 + 1 = 15

Calculate the number of cakes needed
Each child, including Hannah, needs 3 cakes:
15 x3= 45

Since there are 6 cakes in one packet, the number of packets needed is:
15/6= 7.5

Since they cannot buy half a packet, they need to round up to the nearest whole number:
Packets needed} = 8

Calculate the number of packets of sweets needed
Each child, including Hannah, needs a small bag of sweets:
15x1 = 15 

If there are 5 small packets in one large packet, the number of large packets needed is:
15/5 = 3

Calculate the total cost
- Cost of packets of cakes:
8x £2.99 = £23.92 

- Cost of packets of sweets:
3x £2.89/\text{packet} = £8.67 

- Total cost:
£23.92 + £8.67 = £32.59 

Calculate the change
Hannah's mum pays with two £20 notes:
2x£20 = £40 

The change she gets back:
£40 - £32.59 = £7.41 

 Summary

-a.They need to buy  8 packets of cakes.
- b They need to buy  3 large packets of sweets.
- c. They will spend  £32.59 
- d.Hannah's mum gets  £7.41  in change.

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Problem 2: Let's solve each part of Problem 2 step-by-step.

Determine the total number of pencils needed
John needs to buy 2 pencils for each of the 34 children in his class, including himself:
34x2 = 68 

Calculate the number of packs of pencils needed
If the pencils come in packs of 6, the number of packs needed is:
68/6 ≈11.33 

Since they cannot buy a fraction of a pack, they need to round up to the nearest whole number:
Packs needed= 12

Calculate the number of pencils left over
If John buys 12 packs of pencils, the total number of pencils he will have is:
12x6= 72 

The number of pencils left over is:
72 - 68= 4 

Calculate the total cost
Each pack of pencils costs £1.99:
12 x  £1.99 = £23.88

Calculate the change
John is given £30 to spend. The change he will get back is:
 £30 - £23.88 = £6.12

Summary

- a.John needs to buy 12 packs of pencils. There will be 4 pencils left over.
- b.John will need to spend £23.88. He will get £6.12 in change.