the trees must be planted, only seven people show up to work. How many trees must each person plant? When a hole is dug for one tree, 3 kg of mulch and $$ 4,5 \mathrm{~kg} $$ of fertiliser must be placed in the hole before the tree is planted. Another $$ 0,5 \mathrm{~kg} $$ of mulch must be spread around the tree. a. How much fertiliser and how much mulch must be bought for 120 trees? b. If another 15 trees are added to the number of trees to be planted, how much extra fertiliser and mulch must be bought? c. Is this an example of direct or inverse proportion?

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Let's break down the problem step by step.

### Known Conditions:
1. **Number of people**: 7
2. **Number of trees**: Initially 120 trees, then 135 trees (120 + 15).
3. **Mulch and Fertiliser per tree**:
   - For each tree:
     - Mulch: 3 kg (for the hole) + 0.5 kg (around the tree) = 3.5 kg
     - Fertiliser: 4.5 kg

### Part a: Calculate the total amount of fertiliser and mulch for 120 trees.

1. **Total mulch for 120 trees**:
   $$
   \text{Total mulch} = 120 \times 3.5 \text{ kg}
   $$

2. **Total fertiliser for 120 trees**:
   $$
   \text{Total fertiliser} = 120 \times 4.5 \text{ kg}
   $$

Now, let's calculate these values.

### Part b: Calculate the extra amount of fertiliser and mulch for 15 additional trees.

1. **Total mulch for 15 trees**:
   $$
   \text{Extra mulch} = 15 \times 3.5 \text{ kg}
   $$

2. **Total fertiliser for 15 trees**:
   $$
   \text{Extra fertiliser} = 15 \times 4.5 \text{ kg}
   $$

Now, let's perform the calculations for parts a and b.
Calculate the value by following steps:
- step0: Calculate:
  $$120\times 4.5$$
- step1: Multiply the numbers:
  $$540$$
Calculate or simplify the expression $$ 15 * 4.5 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$15\times 4.5$$
- step1: Multiply the numbers:
  $$67.5$$
Calculate or simplify the expression $$ 120 * 3.5 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$120\times 3.5$$
- step1: Multiply the numbers:
  $$420$$
Calculate or simplify the expression $$ 15 * 3.5 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$15\times 3.5$$
- step1: Multiply the numbers:
  $$52.5$$
### Part a: Total Amount of Fertiliser and Mulch for 120 Trees

1. **Total mulch for 120 trees**:
   $$
   \text{Total mulch} = 120 \times 3.5 = 420 \text{ kg}
   $$

2. **Total fertiliser for 120 trees**:
   $$
   \text{Total fertiliser} = 120 \times 4.5 = 540 \text{ kg}
   $$

### Part b: Extra Amount of Fertiliser and Mulch for 15 Additional Trees

1. **Extra mulch for 15 trees**:
   $$
   \text{Extra mulch} = 15 \times 3.5 = 52.5 \text{ kg}
   $$

2. **Extra fertiliser for 15 trees**:
   $$
   \text{Extra fertiliser} = 15 \times 4.5 = 67.5 \text{ kg}
   $$

### Summary of Results:
- For **120 trees**:
  - Total mulch needed: **420 kg**
  - Total fertiliser needed: **540 kg**
  
- For the **additional 15 trees**:
  - Extra mulch needed: **52.5 kg**
  - Extra fertiliser needed: **67.5 kg**

### Part c: Proportion Type
This situation is an example of **direct proportion** because the amount of mulch and fertiliser needed increases directly with the number of trees planted. As the number of trees increases, the total amount of mulch and fertiliser required also increases in a consistent ratio.
For 120 trees, you need 420 kg of mulch and 540 kg of fertiliser. For the additional 15 trees, you need 52.5 kg more mulch and 67.5 kg more fertiliser. This is a direct proportion because the amount of mulch and fertiliser increases with the number of trees.

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