3. Talia and Mutinta are given an order to make 5 dresses for a matric dance. It takes thours for Talia to make the dresses working on her own, and Mutinta would take 8 hours longer if she did the job on her own. Working together, they could complete the dresses in 28 hours. Calculate how long it would take each of them

Let t be the number of hours Talia takes by herself to make the 5 dresses. Then Mutinta, working alone, would take (t + 8) hours. When someone works alone, their work‐rate is (total work)/(time). Here the work is “5 dresses.” • Talia’s rate: 5/t dresses per hour. • Mutinta’s rate: 5/(t + 8) dresses per hour. Working together, they complete the 5 dresses in 28 hours, so their combined work‐rate is 5/28 dresses per hour. Thus, we can set up the equation for their combined rates:   (5/t) + (5/(t + 8)) = 5/28 Step 1. Divide both sides by 5:   (1/t) + (1/(t + 8)) = 1/28 Step 2. Combine the fractions on the left:   (t + 8 + t) / [t(t + 8)] = 1/28   (2t + 8) / [t(t + 8)] = 1/28 Step 3. Cross‐multiply:   28(2t + 8) = t(t + 8) Step 4. Expand both sides:   56t + 224 = t² + 8t Step 5. Rearrange the equation:   t² + 8t − 56t − 224 = 0   t² − 48t − 224 = 0 Step 6. Solve this quadratic equation by using the quadratic formula:   t = [48 ± √(48² − 4(1)(−224))]/2 Calculate the discriminant:   48² = 2304 and 4×224 = 896, so   √(2304 + 896) = √3200 Notice that 3200 = 16×200, so:   √3200 = 4√200 = 4×10√2 = 40√2 Thus,   t = (48 ± 40√2)/2 = 24 ± 20√2 Since time cannot be negative, we take the positive solution:   t = 24 + 20√2 hours Step 7. Now, Mutinta’s time is:   t + 8 = (24 + 20√2) + 8 = 32 + 20√2 hours Thus, the answers are:  • Talia takes exactly 24 + 20√2 hours (approximately 52.3 hours),  • Mutinta takes exactly 32 + 20√2 hours (approximately 60.3 hours). You can check the work rates:  Talia’s rate = 5/(24 + 20√2),  Mutinta’s rate = 5/(32 + 20√2), and together these sum to 5/28 dresses per hour, as required. So, the final answer is:  Talia: 24 + 20√2 hours,  Mutinta: 32 + 20√2 hours.