Answer:
it took so long to solve this so please read my effort <em>completely</em>
Step-by-step explanation:
Let T = TNYWR. be the number you will receive. What?
Speed = T mph
2 hours later the tank is empty. Sounds like my wife (almost empty?)
Did she push the car into the gas station? I'd say no
Fill up 12 gallons
280 minutes at T+10 280/60 = 4.666 hours
arrives with 2 gallons left so she used 10 gal on the last part of her trip
gas mileage is constant ? mpg not possible, going faster reduces mpg
Now let K = m/n the <em>number of gallons</em> 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 gas
gas 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 gallons
solve 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 integers
Now what?
We know K MUST be less than 12 gallons
drive for 2 hours approx 25 mpg no help
m+n <em>might</em> 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 gallons
at 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 = 4
FINALLY ANSWER
(m+n)/100 (15+4)/100 = 0 with a remainder of 19
check
3.75 gallons used to drive 2T (2)(70) miles = 140 / 3.75 = 37.333 mpg
10 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
