Question 1103455: Beth likes chocolate and goes to Rick and Morty's Chocolate Emporium every day. She walks quickly from her house to the shop so she is sure she gets her favorite chocolates, then walks 2 mph slower back home, savoring the candies and fresh air. Beth's house is 2 miles from the store and her entire walk takes 2 hours and 40 minutes. How fast does Beth walk on her way home?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the problem is not especially difficult, but the numbers used make it very clumsy to deal with, unless you round intermediate numbers.
unfortunately you had no instructions on rounding intermediate numbers or on the final answer.
i used unrounded intermediate numbers to get my answer.
unrounded numbers means that the rounding is done by the calculator and goes out at least 12 decimal places before rounding.
i stored the unrounded numbers in variables to make showing you the process easier by not muddling things up by showing all the details.
the basic formula to use is rate * time = distance.
going to the store, her rate was x miles per hour and the time she took was T hours and the distance was 2 miles.
the formula going to the store is therefore:
x * T = 2
coming back from the store, her rate was (x-2) miles per hour and the time she took was 2 hours and 40 minutes minus T and the distance was again 2 miles.
i converted 2 hours and 40 minutes to hours which is equal to 2.666666666..... hours which i stored in a variable called A.
the time it took to return is the total round trip time minus the time going which made the time coming back equal to A - T.
the formula coming back is therefore:
(x-2) * (A-T) = 2
i then expanded this formula by applying the distributive multiplication law to get:
x*A - x*T - 2*A + 2*T = 2
you have 2 equations that need to be solved simultaneously.
they are:
x * T = 2 and:
x*A - x*T - 2*A + 2*T = 2
solve for T in the first equation to get T = 2/x
replace T with 2/x in the second equation to get:
x*A - x*2/x - 2*A + 2*2/x = 2
simplify to get:
x*A - 2 - 2*A + 4/x = 2
multiply both sides of this equation by x to get:
x*A*x - 2*x - 2*A*x + (4/x)*x = 2*x
simplify to get:
A*x^2 - 2*x - 2*A*x + 4 = 2*x
subtract 2*x from both sides of this equation to get:
A*x^2 - 2*x - 2*A*x + 4 - 2*x = 0
combine like terms to get:
A*x^2 - 4*x - 2*A*x + 4 = 0
factor out the x from those 2 middle terms to get:
A*x^2 - (4+2*A)*x + 4 = 0
if you remember from above, i had previously set A equal to 2.666666666.....
replacing A with the number that it represents and rounding to 12 decimal places, i get:
2.666666666667 * x^2 - 9.333333333333 * x + 4 = 0
factoring this quadratic equation using a quadratic equation solver, i get:
x = 2.9999999999994 or x = 0.50000000000004
these look very suspiciously like x = 3 or x = .5
x can't be equal to .5 because then x-2 will be negative.
therefore, the only possible solution is x = 3.
when x = 3, the first equation becomes:
3 * T = 2
solve for T to get T = 2/3.
and the second equation becomes:
1 * (2 and 2/3 minus 2/3) = 2
this results in 1 * 2 = 2 which is true.
the solution is therefore good.
beth walks at 3 miles per hours going to the store and she walks at 1 mile per hour coming back.
converting to decimal rather than just using the fractions given might have made this problem a little more difficult to solve because of the rounding required, so i'll solve it using fractions below to see if it makes it easier.
going to the store, the formula is still x * T = 2
coming back from the store, the formula becomes (x-2) * (8/3 - T) = 2
the fraction of 8/3 is there because the round trip time is 2 hours and 40 minutes which is equal to 2 and 2/3 hours which is equal to 8/3 hours.
the two equations that need to be solved simultaneously are:
x * T = 2
(x-2) * (8/3 - T) = 2
solve for T in the first equation to get T = 2/x.
replace T in the second equation to get:
(x-2) * (8/3 - 2/x) = 2
simplify to get:
(8/3)*x - (2/x)*x - 2*(8/3) + 4/x = 2
simplify to get:
(8/3)*x - 2 - (16/3) + 4/x = 2
subtract 2 from both sides of this equation to get:
(8/3)*x - 4 - (16/3) + 4/x = 0
multiply both sides of this equation by x to get:
(8/3)*x*x - 4*x - (16/3)*x + 4 = 0
combine like terms and simplify to get:
(8/3)*x^2 - (4+16/3)*x + 4 = 0
simplify further to get:
(8/3)*x^2 - (28/3)*x + 4 = 0
factor using the quadratic formula to get:
x = .5 or x = 3
it's still messy but you avoid all those endlessly repeating decimals.
the solution to your problem is that she walks at a rate of 1 mile per hour coming home from the store.
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