Problems to work on together
Collaborative tasks usually involve two different characters doing the same work for different lengths of time. The challenge is to find out how long they will do a given job if they work together.
Sample problem
Imagine you are playing a game in which you are fighting a dragon. The Paladin can slice the dragon with his sword in 30 seconds, while the Barbarian can even smash it with his club in 20 seconds. The question is, how long would it take to kill the dragon if the Barbarian and Paladin fought the dragon together? How much "joint work" do they have to put in?
We solve the problem by first figuring out how much each of the fearless warriors hurt the dragon in one second. Since the Paladin kills the dragon in 30 seconds, he takes one thirtieth of the dragon's 1/30 lives in one second. If the dragon had 180 lives, then in order for the Paladin to kill it in thirty seconds, he would have to take 180 / 30 = 6 lives from it every second (that's one thirtieth of its lives). Yet 180 / 30 is the same as 180 · (1/30).
The Barbarian kills him in 20 seconds, so the dragon takes away 1/20 of his lives every second. So the same dragon that would have 180 lives would have 180 / 20 = 9 lives taken away every second.
So the Paladin takes 1/30 of the dragon's life every second, the Barbarian 1/20. Now we ask, how many seconds does it take for these men to kill him together? We can look at it one at a time - how many lives would they take off the dragon in two seconds?
$$ \frac{1}{30}+\frac{1}{20}+\frac{1}{30}+\frac{1}{20}=\frac{10}{60}=\frac16 $$
It would be one-sixth of his life. Let's add 1/30 + 1/20, because the warriors are fighting together, and let's add it up twice in total, because they're fighting for two seconds. We can see that we can generalize this to mean that if they fight for x seconds, they're taking away the dragon's
$$ x\cdot\frac{1}{30}+x\cdot\frac{1}{20} $$
lives. Now there's only one thing to do - the dragon dies when they take all its lives. All lives are represented by the number 1, because that fraction 1/30 represents one thirtieth of the dragon's lives - so the number 1 is the total number of lives of the dragon. So we just put the previous expression equal to one:
$$ x\cdot\frac{1}{30}+x\cdot\frac{1}{20} = 1 $$
and look for a solution. Multiplying the whole equation by 60, we get:
$$\begin{eqnarray} x\cdot\frac{60}{30}+x\cdot\frac{60}{20} &=& 60\\ x\cdot\frac{6}{3}+x\cdot\frac{6}{2} &=& 60\\ 2x+3x &=& 60\\ 5x &=& 60\qquad/\cdot\frac15\\ x &=& 12 \end{eqnarray}$$
x = 12That means the dragon will be killed in 12 seconds. That's not bad.
We can try to check by plugging in specific numbers. We had a dragon with 180 lives, and the result says that together they kill it in 12 seconds. Paladin takes 6 lives per second, Barbarian 9. That means that the Paladin takes a total of 12 · 6 = 72 lives off the dragon in 12 seconds, and the Barbarian takes 12 · 9 = 108 lives. They've taken a total of 72 + 108 = 180 lives off him, which is consistent with taking all his lives.
Check out the other solved examples to work together:
Examples
- Tony would really like to buy a new mobile phone. He made a deal with his mom, who told him that she would give him an allowance every week and that after 60 weeks he would have just enough money for the coveted cell phone. But Tony isn't today's boy, so he went to see his daddy too. He also promised him money and told him that he would even have enough for a new mobile phone after 30 weeks. How long in reality will it take Tony to get a new mobile phone if he takes weekly allowance from both his mommy and daddy?
Tony gets a weekly 1/60 of the price of the mobile phone from his mummy and a weekly 1/30 of the price of the mobile phone from his daddy. Let's set up the equation exactly the same as in the previous case:
$$ x\cdot\frac{1}{60}+x\cdot\frac{1}{30} = 1 $$
The expression x · 1/60 + x · 1/30 tells us how much of the mobile phone price Tony will have saved after x weeks. Since we are asking when he will have enough for the cell phone, we set this expression equal to one. If he wanted to buy three mobiles, the number three would be on the right-hand side.
We solve the equation. First we multiply it by 60:
$$\begin{eqnarray} x\cdot\frac{60}{60}+x\cdot\frac{60}{30} &=& 60\\ x\cdot\frac{1}{1}+x\cdot\frac{2}{1} &=& 60\\ x+2x &=& 60\\ 3x &=& 60\qquad /\cdot\frac13\\ x &=& 20 \end{eqnarray}$$
The result is that Tony will have a new cell phone in 20 days. We can check this again by substituting real values: if the mobile costs 9000, he would get 1/60 of this price, i.e. 150 crowns, from his mother every week. From daddy he would receive 1/30 of this price, which is 300 crowns. Well, he has quite generous parents. In 20 weeks, he would receive 20 · 150 = 3000 crowns from his mummy, whereas from his daddy he would receive 20 · 300 = 6000 crowns. All in all, he'd get 3000 + 6000 = 9000 crowns.
- There are three pipes leading to the pool. One fills the pool in 100 minutes, another in 75 minutes and the last in 50 minutes. How long will it take to fill the pool if all three pipes are filled?
Again, this example is the same as the previous examples, only instead of two actors working together, we have three actors. The equation will look like this:
$$ x\cdot\frac{1}{100}+x\cdot\frac{1}{75}+x\cdot\frac{1}{50} = 1 $$
We multiply the equation by 150 and continue to modify it further:
$$\begin{eqnarray} x\cdot\frac{150}{100}+x\cdot\frac{150}{75}+x\cdot\frac{150}{50} &=& 150\\ x\cdot\frac{3}{2}+x\cdot\frac{2}{1}+x\cdot\frac{3}{1} &=& 150\\ \frac32x+2x+3x &=& 150\\ \frac32x+5x &=& 150\qquad /\cdot 2\\ 3x + 10x &=& 300\\ 13x &=& 300\qquad/\cdot\frac{1}{13}\\ x &=& \frac{300}{13} \end{eqnarray}$$
We won't get a nicer result from this, the pool will fill in 300/13 minutes, which is approximately 23 minutes.
- The mowing company will mow the lawn in ten hours. The Dynamite Violet Tearing Company mows the lawn in six hours. How many hours would it take these firms to mow two such meadows if they worked together?
How much do both firms mow in x hours? That's a classic expression:
$$ x\cdot\frac{1}{6}+x\cdot\frac{1}{10} $$
But since we want them to mow two meadows, the right side of the equation will not be 1, but will be 2:
$$ x\cdot\frac{1}{6}+x\cdot\frac{1}{10} = 2 $$
Then the calculation proceeds in the same way. We multiply the whole equation by 30 and adjust:
$$\begin{eqnarray} x\cdot\frac{30}{6}+x\cdot\frac{30}{10} = 60\\ x\cdot\frac{5}{1}+x\cdot\frac{3}{1} = 60\\ 5x+3x&=& 60\\ 8x&=& 60\qquad/ \cdot \frac18\\ x &=& \frac{60}{8}\\ x &=& \frac{15}{2}\\ x &=& 7{,}5 \end{eqnarray}$$
The two meadows would be mowed in seven and a half hours.
- One woman would give birth to a child in nine months. In how many months will nine women give birth to a child?