Shaut can do 27 push-ups per minute. Each week he improves his performance by 3 push-ups per minute. Mpho can do 20 push-ups per minute and he increases his performance by 4 push-ups per minute each week. How many push-ups will they each do per minute in the fitness (a) Hownetion which in in 10 weeks' time? (b) After how many weeks will their number of push-ups per minute be (c) the same? (chile training for the competition, Shaun spent 45 minutes doing push-ups per week. How many push-ups does he do altogether in the 10 weeks?

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Let's analyze each part of the problem step by step.

### Given:
- **Shaut:**
  - Initial rate: 27 push-ups per minute.
  - Improvement: +3 push-ups per minute each week.
  
- **Mpho:**
  - Initial rate: 20 push-ups per minute.
  - Improvement: +4 push-ups per minute each week.

---

### **(a) Push-ups per Minute After 10 Weeks**

**Shaut's Push-up Rate After 10 Weeks:**
$$
\text{Initial Rate} + (\text{Improvement per Week} \times \text{Number of Weeks}) = 27 + (3 \times 10) = 27 + 30 = 57 \text{ push-ups per minute}
$$

**Mpho's Push-up Rate After 10 Weeks:**
$$
\text{Initial Rate} + (\text{Improvement per Week} \times \text{Number of Weeks}) = 20 + (4 \times 10) = 20 + 40 = 60 \text{ push-ups per minute}
$$

**Answer:**
- **Shaut:** 57 push-ups per minute
- **Mpho:** 60 push-ups per minute

---

### **(b) Weeks Until Both Have the Same Push-up Rate**

We need to find the number of weeks $$ w $$ where both have the same push-up rate.

$$
\text{Shaut's Rate} = 27 + 3w
$$
$$
\text{Mpho's Rate} = 20 + 4w
$$

Set the two rates equal:
$$
27 + 3w = 20 + 4w
$$
$$
27 - 20 = 4w - 3w
$$
$$
7 = w
$$

**Answer:**
After **7 weeks**, both Shaut and Mpho will have the same number of push-ups per minute.

---

### **(c) Total Push-ups Shaut Does in 10 Weeks**

Shaut trains **45 minutes each week**. To find the total push-ups over 10 weeks, we'll calculate the push-up rate for each week and then sum the total.

1. **Weekly Push-up Rates:**
   - **Week 1:** $$ 27 + 3 \times 1 = 30 $$ push-ups per minute
   - **Week 2:** $$ 27 + 3 \times 2 = 33 $$ push-ups per minute
   - ...
   - **Week 10:** $$ 27 + 3 \times 10 = 57 $$ push-ups per minute

2. **Sum of Push-up Rates Over 10 Weeks:**
   This is an arithmetic series where:
   - First term ($$ a_1 $$) = 30
   - Last term ($$ a_{10} $$) = 57
   - Number of terms ($$ n $$) = 10
   
   $$
   \text{Sum} = \frac{n}{2} \times (a_1 + a_{10}) = \frac{10}{2} \times (30 + 57) = 5 \times 87 = 435 \text{ push-ups per minute}
   $$

3. **Total Push-ups Over 10 Weeks:**
   $$
   \text{Total Push-ups} = \text{Sum of Rates} \times \text{Minutes per Week} = 435 \times 45 = 19,575 \text{ push-ups}
   $$

**Answer:**
Shaut does a total of **19,575 push-ups** over the 10 weeks.

---

### **Summary of Answers:**
- **(a)** After 10 weeks, Shaut does **57** push-ups per minute and Mpho does **60** push-ups per minute.
- **(b)** Their push-up rates become the same after **7 weeks**.
- **(c)** Shaut does a total of **19,575** push-ups in the 10-week training period.
- **After 10 weeks:** - Shaut does 57 push-ups per minute. - Mpho does 60 push-ups per minute. - **Same push-up rate after 7 weeks.** - **Total push-ups in 10 weeks:** 19,575 push-ups.

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