Can You Solve The Boys And Girls Painting Rooms Riddle?

Thanks to Oladele from Nigeria for suggesting this problem! Team A has 4 girls and 7 boys. Team A takes 5 days to paint 4 rooms. Team B has 7 girls and 10 boys. Team B takes 4 days to paint 5 rooms. Team C has 8 girls and 5 boys. How long will it take team C to paint 6 rooms?

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[MindYourDecisions]

MindYourDecisions is the only channel to feature math topics suggested by people from around the world. Sadly some textbooks, websites, and other channels think math is limited to specific topics from certain countries---and they are vocal about it. You can support this channel by subscribing and watching all of the videos (about 5-10% of the emails I get are for topics that already have videos!).

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[anthonyeaton]

Good stuff, Presh. I got this one, but - as usual - your solution was much more elegant (and efficient) than mine. Keep 'em coming!

[Tehom]

It shows the power of simultaneous equations.

g*4 + b*7 = 4/5, g*7 + b*10 = 5/4, so g=1/12, b=1/15, and (g*8 + b*5)*6 = 6, so 6 days.

[TC-eugy]

One boy paints 1/15 rooms a day, a girl 1/12 a day. Is that correct?

[Husqy]

I am only in 7th grade, and I seldom get anything right, yet i still watch your videos. I am just so FASCINATED by it. Keep up the good work!

[badrunna-im]

Since the painters are identical within a gender, I expressed team C as a linear combination of team A and B with their rates of 4/5 and 5/4 rooms per day respectively.
By the number of people, C=-5A+4B; substituting in their work rates, C paints 1 room per day.

[SuperMtheory]

These types of problems are sometimes challenging for algebra students. Your approach was clear and quite easy to understand. This video would make an excellent tutorial for an algebra student. Thank you!

[brightwolfz]

I set this up nearly the same, but a little differently. I did 4g+7b=.8 for the first one. Basically, The amount of work 4 girls do + 7 boys do os equal to the rate of .8 rooms per day. Follow this to eventually find the same problem. I think the huge hiccup is the initial want to do 4g+7b = 4R+5D which wont give you a correct answer

[Continental_Champ]

Here's how I solved it:

(G / B = amount of work done by one girl/boy in one day, x = variable for amount of work done)

First I made both Team A and B paint 20 rooms and got the following equations:
25 (4G + 7B) = 16 (7G + 10B)
100G + 175B = 112G + 160B
15B = 12G
5B = 4G
G = 5x
B = 4x

Then I used this information to calculate how much work needs to be done to paint one room:
5 (4G + 7B) = 4 rooms
5 (20x + 28x) = 4 rooms
240x = 4 rooms
60x = 1 room

Finally I calculated the amount of work Team C does:
8G + 5B = 60x = 1 day of work for Team C
6 rooms = 360x = 6 days of work for Team C.

[peterroder]

Should say every girl works at the same rate, not just a constant rate, and the same for boys.

[Tahgtahv]

While not solved in the video, it's easy to figure out for an arbitrary number of boys or girls. It takes 12 girls, or 15 boys to finish a room in a day. For team C, the girls paint 2/3 of the room, and the boys paint 1/3 of the room each day.

[AnonimityAssured]

Presumably, _g_ and _b_ are not actually percentages, but _fractions_ of a room painted per day by a girl and a boy respectively.

[Mahiu]

on a side note, it takes 13 people to paint 1 room, that room must be huge. LOL

Its been a very long time since i did algebra and i gotta admit, I went round in circles and didnt solve this one. great work as always

[NestorAbad]

It's an ellegant solution! I didn't realise that subtracting equations one gets the right piece to be multiplied by 6. I solved for g and b before.

[danmerget]

I solved for g=1/12 and b=1/15, rather than noticing that you could get 8g+5b without solving for g & b. The answer was the same either way. It wasn't very challenging, unfortunately.

[Continuum]

Another way to solve this problem:

Team A: 4 Girls & 7 Boys in 1 day paint 4/5 rooms
Team B: 7 Girls & 10 Boys in 1 day paint 5/4 rooms
Team C: 8 Girls & 5 Boys in 1 day paint 6/x rooms

Team A + Team C: 12 Girls & 12 Boys in 1 day paint ((4/5) + (6/x)) rooms

Team B - Team A: 3 Girls & 3 Boys in 1 day paint (5/4 - 4/5) = 9/20 rooms

4 x (Team B - Team A) = 12 Girls & 12 Boys in 1 day paint 36/20 = 9/5 rooms

Therefore, (4/5) + (6/x) = 9/5 → 6/x = 1 → x = 6 days

[SamvitAgarwal]

Using a system we could figure out the girls would paint 2/3 of a room in one day and the boys would paint 1/3, making a total of one room per day. Dividing by 6, we get 6 days.

[uiuiuiseraph]

Multiply both sides of the equations by the days:
20g+35b=4 and 28g+40b=5
SO: 28(0, 2-1, 75b)+40b=5
or: 5, 6 -9b = 5 which gives: b= 1/15 and g=1/12
Then put it in:
8*1/12 + 5* 1/15 = 6/n
n=6
It will take 6 days.

[MommyOfZoeAndLiam]

Omg, ip did the whole Algebraic solve for 2 variables. Got g in terms of b plugged in, got b as a number, plugged that in got g as a number. Then plugged into the he last equation. I did the math right, but Jesus I love how you showed a different perspective. That would have been so much nicer. Thanks!

[AyrtonTwigg]

I solved it by turning it into a linear system of equations, putting into matrix form, and then an rref. Because that’s what I’m learning in my math class in college.