cilivity 1: rite the following ratios in its simplest form. (a) \( 9: 15 \) (b) \( 21: 35 \) (c) \( 12: 24: 48 \) (d) \( 18: 54: 81 \) silvity 2 Ive the following problems: R360 is shared amongst Paul, Jabu and Pam in order of the ratio \( 4: 3: 2 \). How much money does each one receive.

Alright, I need to simplify the given ratios and then solve the problem about sharing R360 among Paul, Jabu, and Pam in the ratio of 4:3:2. Let's tackle them one by one. **Part 1: Simplifying the Ratios** (a) **9:15** First, I'll look at the ratio 9:15. To simplify this, I need to find the greatest common divisor (GCD) of 9 and 15. The factors of 9 are 1, 3, and 9. The factors of 15 are 1, 3, 5, and 15. The common factors are 1 and 3. The greatest of these is 3. So, I'll divide both numbers by 3. - 9 ÷ 3 = 3 - 15 ÷ 3 = 5 So, the simplified ratio is **3:5**. (b) **21:35** Next, the ratio is 21:35. I'll find the GCD of 21 and 35. The factors of 21 are 1, 3, 7, and 21. The factors of 35 are 1, 5, 7, and 35. The common factors are 1 and 7. The greatest is 7. - 21 ÷ 7 = 3 - 35 ÷ 7 = 5 So, the simplified ratio is **3:5**. (c) **12:24:48** This is a three-part ratio: 12:24:48. To simplify, I'll find the GCD of all three numbers. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The common factors are 1, 2, 3, 4, 6, and 12. The greatest is 12. - 12 ÷ 12 = 1 - 24 ÷ 12 = 2 - 48 ÷ 12 = 4 So, the simplified ratio is **1:2:4**. (d) **18:54:81** This is another three-part ratio: 18:54:81. I'll find the GCD of all three numbers. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54. The factors of 81 are 1, 3, 9, 27, and 81. The common factors are 1, 3, 9, and 27. The greatest is 27. - 18 ÷ 27 = 0.666... (which is not an integer, so I need to check if 9 is the GCD) - 54 ÷ 9 = 6 - 81 ÷ 9 = 9 So, the simplified ratio is **2:6:9**. **Part 2: Sharing R360 in the Ratio 4:3:2** Now, I need to determine how much money each person receives when R360 is shared in the ratio of 4:3:2 among Paul, Jabu, and Pam. First, I'll find the total parts in the ratio: - Paul: 4 parts - Jabu: 3 parts - Pam: 2 parts Total parts = 4 + 3 + 2 = **9 parts** Next, I'll find the value of one part by dividing the total amount by the total number of parts: - R360 ÷ 9 parts = R40 per part Now, I'll calculate each person's share: - **Paul**: 4 parts × R40 = R160 - **Jabu**: 3 parts × R40 = R120 - **Pam**: 2 parts × R40 = R80 **Verification:** R160 (Paul) + R120 (Jabu) + R80 (Pam) = R360, which matches the total amount to be shared. **Conclusion:** - Paul receives R160 - Jabu receives R120 - Pam receives R80