Answer:it took so long to solve this so please read my effort completelyStep-by-step explanation:Let T = TNYWR. be the number you will receive. What?Speed = T mph2 hours later the tank is empty. Sounds like my wife (almost empty?) Did she push the car into the gas station? I'd say noFill up 12 gallons 280 minutes at T+10 280/60 = 4.666 hoursarrives with 2 gallons left so she used 10 gal on the last part of her tripgas mileage is constant ? mpg not possible, going faster reduces mpgNow let K = m/n the number of gallons at the start of the trip where m and n are relatively prime positive integers (I don't know what that means) Janet, what are relatively prime positive integers? These are called co-prime numbers, which doesn't help any. Essentially, it is two integers that can have no common factor other than 1. No other integer can divide both numbers evenly.What is the remainder when m+n is divided by 100? ------------------------------------------------------------------------------------T mph times 2 hours = 2T miles fill tank add 12 gallons(T+10) mph times (4.667) hours = 4.667T+ 46.67 miles used 12-2 gallons gasgas mileage mpg = 4.667T+ 46.67 miles / 10 gallons First part of the trip mpg equals second part of the trip mpg 2T miles / unknown gallons = 4.667T+ 46.67 miles / 10 gallonssolve for unknown gallons (2T miles)(10 gallons) / (4.667T + 46.67 miles) = unknown gallons 20T / (4.667T + 46. 67) = K gallons recall T is the speed K = m/n relative prime positive integersNow what? We know K MUST be less than 12 gallonsdrive for 2 hours approx 25 mpg no help m+n might have to be greater than 100, so m+n/100 has a remainder which is wrong ------------------------------------------------------------------------------------ so I use Desmos to graph the function, I also used my old TI-95 Procalc to program the function and I found a rational number I could use for K to find the ratio m/n and be co-primes. I could not figure out an equation to solve for a rational K. A better computer programmer could write a better method and trial and error ------------------------------------------------------------------------------------ 20T / (4.667T + 46. 67) = K gallonsat speed 70 I found this number using logic, normal mph, and trial and error (20)(70) / (4.667(70) + 46. 67) = K gallons 3.75 = K K = 3.75 What co-primes = 3.75 ? the two co-primes I found by using a spreadsheet quickly found K = m/n = 3.75 if m/n = 15/4 m = 15 and n = 4FINALLY ANSWER (m+n)/100 (15+4)/100 = 0 with a remainder of 19check3.75 gallons used to drive 2T (2)(70) miles = 140 / 3.75 = 37.333 mpg10 gallons used to drive (T+10 mph)(4.667 hours) = x miles / 10 gallons (70+10 mph)(4.667 hours) = 373.3333 miles / 10 gallons = 37.333 mpg