Ashwin and Donald decided to set out from two towns on their bikes, which are 247 miles apart, connected by a straight Roman road in England. When they finally met up somewhere between the two towns, Ashwin had been cycling for 9 miles a day. The number of days for the whole adventure is 3 more than the number of miles that Donald had been cycling in a day. How many miles did each cycle? explanatory solutions
Let the number of miles Donald has been cycling be x.
The number of days of the adventure is x + 3
Ashwin has been cycling 9 miles a day
Equation
Total # of days x ( # of miles of Ashwin per day + # of miles of Donald per Day) = 247
(x + 3)(9 + x) = 247 ----- Find the positive roots
the root is 10
so, x = 10
The number of days of the adventure is x + 3
so, x + 3 = 10 + 3 = 13 days
ashwin = 9 miles x 13 days = 117 miles
Donald = 10 miles x 13 days = 130 miles
Well, let's use our trusty detective skills to solve this mystery! Let's call the number of miles Donald cycled each day "D" (for Donald, of course).
Since Ashwin cycled for 9 miles a day, we can conclude that he cycled for a total of 9 * (number of days) miles.
Now, if we let "M" represent the number of days it took for Donald to cycle his distance, we can say that Ashwin cycled for M + 3 days. Therefore, Ashwin cycled 9 * (M + 3) miles.
Since the total distance between the two towns is 247 miles, we can set up an equation:
9 * (M + 3) + D * M = 247
Now, let's solve this equation to find out how many miles each cycled:
9M + 27 + DM = 247
9M + DM = 220
We can see that 220 is divisible by both 9 and D, so let's try some options!
If D = 5, then M = 20. But D * M doesn't equal 220. Oh, the mystery deepens!
If D = 10, then M = 12. But D * M still doesn't equal 220. Maybe we're getting closer to solving this mystery?
If D = 15, then M = 8. Ah, yes! D * M does equal 220! We're on the right track!
So, Ashwin cycled 9 * (M + 3) = 9 * (8 + 3) = 99 miles.
And Donald cycled 15 miles a day for 8 days, totaling 15 * 8 = 120 miles.
Mystery solved! Ashwin cycled 99 miles, and Donald cycled 120 miles. Case closed!
Let's assume that Donald had been cycling for "x" miles per day.
Given that Ashwin had cycled for 9 miles a day, we can write the equation:
Ashwin's distance + Donald's distance = Total distance
9 * (number of days) + x * (number of days) = 247
Since we know that the number of days for the whole adventure is 3 more than the number of miles Donald cycled in a day, we can substitute (x+3) in place of "number of days."
9 * (x+3) + x * (x+3) = 247
Expanding the equation:
9x + 27 + x^2 + 3x = 247
x^2 + 12x + 27 = 247
Bringing 247 to the left-hand side:
x^2 + 12x - 220 = 0
To solve this quadratic equation, we can factorize it or use the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
Plugging in the values:
x = (-12 ± sqrt(12^2 - 4*1*(-220))) / (2*1)
x = (-12 ± sqrt(144 + 880)) / 2
x = (-12 ± sqrt(1024)) / 2
x = (-12 ± 32) / 2
We have two possible solutions for x:
- x = (-12 + 32) / 2 = 20 / 2 = 10
- x = (-12 - 32) / 2 = -44 / 2 = -22
Since the number of miles cannot be negative, we discard -22 as a solution.
Therefore, Donald cycled 10 miles per day and Ashwin cycled 9 miles per day.
To solve this problem, let's assign variables to the unknown quantities.
Let's say Ashwin cycles at a rate of A miles per day, and Donald cycles at a rate of D miles per day.
We are given the following information:
1. Ashwin cycles for a total of 9 miles per day.
2. The total distance between the two towns is 247 miles.
3. The total duration of their adventure is 3 more than the number of miles Donald cycles in a day.
Now let's break down the problem into equations:
Equation 1: Ashwin cycles at a rate of 9 miles per day.
A = 9
Equation 2: The total distance they travel is 247 miles.
Ashwin's distance + Donald's distance = Total distance
A * 3 + D * 3 = 247
Equation 3: The total duration of their adventure is 3 more than the number of miles Donald cycles in a day.
Total days = D + 3
Now, let's substitute Equation 1 into Equation 2 to eliminate variable A:
9 * 3 + D * 3 = 247
27 + 3D = 247
3D = 247 - 27
3D = 220
Now, divide both sides of the equation by 3 to solve for D:
D = 220 / 3
D ≈ 73.33
Next, plug the value of D into Equation 3 to find the total number of days:
Total days = D + 3
Total days ≈ 73.33 + 3
Total days ≈ 76.33
Since we can't have a fraction of a day or a fraction of a mile, let's round D down to 73 and round the total days down to 76:
D ≈ 73
Total days ≈ 76
Finally, we can calculate Ashwin's distance by multiplying his rate by the total number of days:
Ashwin's distance = A * Total days
Ashwin's distance = 9 * 76
Ashwin's distance = 684 miles
So, Ashwin cycled approximately 684 miles, and Donald cycled approximately 73 miles.
Let the number of miles Donald has been cycling be x.
The number of days of the adventure is 3x.
Ashwin has been cycling 9 miles a day, so Ashwin has covered 3x * 9 miles = 27x.
The total distance is 247 miles
Add the distance that Ashwin and Donald have been cycling for
x+27x = 147
Once you solve for x, you can calculate how the distance each person has cycled respectively