Solve the problem. Frank can type a report in 3 hours and James takes 4 hours. How long will it take the two of them typing together?
Solve the problem. Joe has a collection of nickels and dimes that is worth $6.00. If the number of dimes were doubled and the number of nickels were increased by 6, the value of the coins would be $9.90. How many dimes does he have?
Let Frank do a fraction x of the report and James the remaining fraction of
(1-x). The total time for them to finish is the time it takes for whoever finishes last so expressed in hours, it is:
f(x) = Max[3x, 4(1-x)]
We want to minimize this over x. It is not difficult to see that f(x) has its minimum where
3 x = 4 (1-x)
So, x = 4/7 and f(4/7) = 12/7 =
about 1 hour and 43 minutes.
let the number of nickels originally be x
let the number of dimes originally be y
then 5x + 10y = 600 or
x + 2y = 120
new condition:
5(x+6) + 10(2y) = 990
5x + 20y = 960
x + 4y = 192
subtract the two equations
2y = 72
y = 36
back in first
x + 72 = 120
x = 48
so 48 nickels ---> 2.40
and 36 dimes ----> 3.60
total = 6.00
54 nickels = 2.70
72 dimes - 7.20
total = 9.90
Frank --> 1/3 report/hr
James --> 1/4 report /hr
Frank + James --> 4/12 +3/12 = 7/12 report/hr
so together 12/7 hr or about and hour and 43 minutes
To solve this problem, we need to find the combined rate at which Frank and James can type.
First, let's determine their individual rates. Frank can type a report in 3 hours, so his rate is 1 report per 3 hours, or 1/3 reports per hour. Similarly, James can type a report in 4 hours, so his rate is 1 report per 4 hours, or 1/4 reports per hour.
To find their combined rate, we add their individual rates together. So, Frank's rate of 1/3 reports per hour plus James' rate of 1/4 reports per hour gives us a combined rate of 7/12 reports per hour.
To determine how long it will take for them to type the report together, we need to find the reciprocal of their combined rate. In other words, we need to invert the fraction 7/12 to get 12/7. This means it will take them 12/7 hours, or approximately 1.71 hours, to complete the report together.
Therefore, it will take Frank and James approximately 1.71 hours, or 1 hour and 42 minutes, to type the report together.