13 men and 7 boys can finish a job in 7 days, while 6 boys and 13 women can finish the same job in 6 days. In how many days can 1 man, 1 boy, and 1 woman working together finish the same job (assume constant rates for each men, women, and boys)?
A) 42
B) 54
C) 21
D) 63
E) 24
Difficult Work/Rate Problem
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Tough and ugly problem, I'd love to know a faster way.
But algebraically here's how to do it:
Rate + Rate = Total Constant Rate
Remember that rate=job/day (time is always on the bottom)
m=rate of a man
b=rate of a boy
w=rate of a woman
So we know that
13(m)+7(b)=1/7
and
6(b) + 13(w)=1/6
Solve for m and w respectively so that we can calculate everything in terms of b.
m=(1/7-7b)/13
w=(1/6-6b)/13
b=b
We want to know what happens when we have one man, one woman and one boy
So
The combined constant rate = (1/7-7b)/13 + (1/6-6b)/13 + b
Simplify (all the b's cancel out) and get
(1/7 + 1/6) / 13
Find common denominators
This is where you want to keep everything in fractions til the end.
(1/7 + 1/6) / 13 = (1/13)*(1/7 + 1/6) =(1/13)*(6/(42) + 7/42)= (1/13)*(13/42) = 1/42 is the combined rate of one man, one boy, and one woman which verbally translates to
1 job per 42 days.
Answer A
But algebraically here's how to do it:
Rate + Rate = Total Constant Rate
Remember that rate=job/day (time is always on the bottom)
m=rate of a man
b=rate of a boy
w=rate of a woman
So we know that
13(m)+7(b)=1/7
and
6(b) + 13(w)=1/6
Solve for m and w respectively so that we can calculate everything in terms of b.
m=(1/7-7b)/13
w=(1/6-6b)/13
b=b
We want to know what happens when we have one man, one woman and one boy
So
The combined constant rate = (1/7-7b)/13 + (1/6-6b)/13 + b
Simplify (all the b's cancel out) and get
(1/7 + 1/6) / 13
Find common denominators
This is where you want to keep everything in fractions til the end.
(1/7 + 1/6) / 13 = (1/13)*(1/7 + 1/6) =(1/13)*(6/(42) + 7/42)= (1/13)*(13/42) = 1/42 is the combined rate of one man, one boy, and one woman which verbally translates to
1 job per 42 days.
Answer A
- selango
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13 Men and 7 Boys finish work in 7 days.
20 people finish work in 7 days.
20 people work for total of 140 days.
1 people work for 7 days.
13 Women and 6 Boys finish work in 6 days.
19 people finish work in 6 days.
19 people work for total of 114 days.
1 people work for 6 days.
Since the rate is same for all 3, 1 person(men or women or boy) fiinsh the job in 42 days.
20 people finish work in 7 days.
20 people work for total of 140 days.
1 people work for 7 days.
13 Women and 6 Boys finish work in 6 days.
19 people finish work in 6 days.
19 people work for total of 114 days.
1 people work for 6 days.
Since the rate is same for all 3, 1 person(men or women or boy) fiinsh the job in 42 days.
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13M+7B=1 job in 7 days.
6B+13W=1 job in 6 days.
13M+7B=(1/7)th job in 1 day
6B+13W=(1/6)th job in 1 day
________________________
13M+13B+13W=(1/7+1/6)th job in 1 day
-> 13(M+B+W)=(13/42)th job in 1 day
-> M+B+W=(1/42)th job in 1 day
Thus to complete (1/42)th job = 1 day
and to complete 1 job = ? days
-> 42 days is the answer
Hope this work helps!!!
6B+13W=1 job in 6 days.
13M+7B=(1/7)th job in 1 day
6B+13W=(1/6)th job in 1 day
________________________
13M+13B+13W=(1/7+1/6)th job in 1 day
-> 13(M+B+W)=(13/42)th job in 1 day
-> M+B+W=(1/42)th job in 1 day
Thus to complete (1/42)th job = 1 day
and to complete 1 job = ? days
-> 42 days is the answer
Hope this work helps!!!
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- Rahul@gurome
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One day work of 13M + 7B = 1/7
One day work of 6B + 13W = 1/6
One day work of (13M + 7B + 6B + 13W) or (13M + 13B + 13W) = 1/7 + 1/6 = 13/42
13M + 13B + 13W can finish the work in 42/13 days
1M + 1B + 1W can finish the work in (42/13)*13 = 42 days
The correct answer is (A).
One day work of 6B + 13W = 1/6
One day work of (13M + 7B + 6B + 13W) or (13M + 13B + 13W) = 1/7 + 1/6 = 13/42
13M + 13B + 13W can finish the work in 42/13 days
1M + 1B + 1W can finish the work in (42/13)*13 = 42 days
The correct answer is (A).
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When the job is undefined, we can plug in our own values for the job.mattocks wrote:13 men and 7 boys can finish a job in 7 days, while 6 boys and 13 women can finish the same job in 6 days. In how many days can 1 man, 1 boy, and 1 woman working together finish the same job (assume constant rates for each men, women, and boys)?
A) 42
B) 54
C) 21
D) 63
E) 24
13 men and 7 boys can finish a job in 7 days:
m = 1 unit/day and b = 1 unit/day.
13m + 7b = 13(1) + 7(1) = 20 units/day.
Over 7 days, 7 * 20 = 140 total units.
6 boys and 13 women can finish the same job in 6 days:
6b = 6 units/day.
Over 6 days, 6b = 6 * 6 = 36 units.
13w = 140 - 36 = 104 units.
13w = 104/6 = 52/3 units/day.
w = (52/3)/13 = 4/3 units/day.
So m + b + w = 1 + 1 + 4/3 = 10/3 units/day.
So 140 units would take 140/(10/3) = 42 days.
The correct answer is A.
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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
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As a tutor, I don't simply teach you how I would approach problems.
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For more information, please email me (Mitch Hunt) at [email protected].
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