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?

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.