Answer :
Joseph worked 8 hours last week and 12 hours this week, while Nicole worked 12 hours last week and 24 hours this week.
The first step is to set up equations to represent the given information. Let's denote the number of hours Joseph worked last week as "J" and the number of hours Nicole worked last week as "N".
According to the first sentence, the ratio of the number of hours Joseph worked to the number of hours Nicole worked last week was 2:3. This can be written as the equation J/N = 2/3.
In the second sentence, we're given that this week Joseph worked 4 hours more than last week, so his total hours worked this week is J + 4. We're also told that Nicole worked twice as many hours as last week, so her total hours worked this week is 2N.
According to the third sentence, the ratio of the hours Joseph worked to the number of hours Nicole worked this week is 1:2. This can be written as the equation (J + 4)/(2N) = 1/2.
Now we can solve the equations to find the values of J and N. Let's start by solving the first equation for J:
J/N = 2/3
Cross-multiplying gives us:
3J = 2N
Dividing both sides by 3:
J = (2N)/3
Now let's substitute this value of J into the second equation:
(J + 4)/(2N) = 1/2
Substituting (2N)/3 for J:
((2N)/3 + 4)/(2N) = 1/2
Simplifying the equation:
((2N + 12)/3)/(2N) = 1/2
To get rid of the fraction, we can multiply both sides by 2N:
2N + 12 = 3N
Subtracting 2N from both sides:
12 = N
Now we can substitute this value of N back into the first equation to find J:
J = (2N)/3 = (2*12)/3 = 24/3 = 8
So Joseph worked 8 hours last week, and Nicole worked 12 hours last week.
This week, Joseph worked 4 more hours, so he worked 8 + 4 = 12 hours.
Nicole worked twice as many hours as last week, so she worked 12 * 2 = 24 hours.
In conclusion, Joseph worked 8 hours last week and 12 hours this week, while Nicole worked 12 hours last week and 24 hours this week.
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