Question 1037381
let a = the typing rate of josie
let b = the typing rate of jane
let x = the time it takes for each of them to finish typing.
this means that josie takes x minutes to type 100 words.
this means that jane takes x minutes to type 75 words.


the general formula is rate * time = quantity.


for josie, this equation becomes a * x = 100
for jane, this equation becomes b * x = 75


solve both equations for x and you get:


x = 100/a
x = 75/b


this means that 100/a = 75/b
cross multiply to get 100b = 75a
solve for b to get b = 75a/100 which simplifies to b = 3/4 * a


you are given that b = a - 8.
replace b with a-8 in this equation to get a-8 = 3/4 * a
add 8 to both sides of this equation and subtract 3/4 * a from both sides of this equation to get a - 3/4 * a = 8
simplify to get 1/4 * a = 8
multiply both sides of this equation by 4 to get a = 32.


since b = a-8, you get:


a = 32
b = 24


since b = jane's typing rate, your answer is that ....


jane's typing rate is 24 words per minute.


you can confirm the answer is correct since they both took the same time.


rate * time = quantity


for josie, this means 32 * time = 100.
for jane, this means 24 * time = 75


solve for time in each equation and you get:


for josie, time = 100 / 32 = 25 / 8.
for jane, time = 75 / 24 = 25 / 8.


the time is the same, confirming the solution is correct.


there was another way to solve this as well.


same setup.


ax = 100
bx = 75


subtract the second equation from the first to get ax - bx = 25
factor out the x to get (a-b)x = 25
solve for x to get x = 25/(a-b)


you are given that b = a-8.
replace b in the equation to get x = 25/(a - (a-8))
simplify to get x = 25/8.


now that you know the time, you can solve for a and b.


a * 25/8 = 100
a * 25 = 8*100
a = 8*100/25 = 8*4 = 32


b * 25/8 = 75
b * 25 = 8*75
b = 8*75/25 = 8*3 = 24


you get the same answer either way.