Algebra

System of Linear Equations: Level 2 Challenges

A donkey and a horse are carrying some bags of rice and some bags of beans.

The horse tells the donkey: "If you give me a bag of rice, then I would have twice as many bags of rice as you would, but if I give you one bag of rice instead, we will have the same number of bags of rice."

He continued, "Also, if you give me one bag of beans, I will have twice as many bags of beans but if I give you 2 bags of beans we will have the same number of bag of beans."

How many bags in total does the horse have?

Inspired by a problem a cousin gave me.

To keep track of time, an adventurer carries two cylindrical candles of equal heights but different radii. One takes 6 hours and the other takes 9 hours to burn out.

Now, the adventurer lights both candles and goes to sleep. If one candle is twice the height of the other when he wakes up, how much time has he slept?

Andy and Ben can paint a house in ten days; Andy and Chris can do it in twelve days; Ben and Chris can do it in twenty days. How many days will Chris take to do the work alone?

Two cars get closer by 9m every second while traveling in opposite directions (toward each other). They get closer by 1m every second while traveling in the same direction.

If the speeds of the cars are $X \text{ m/s}$ and $Y \text{ m/s}$, then find $X\times Y$.

In splitting up $1000, a group of three married couples agrees upon the following plan: The wives receive a total of$396, of which Mary gets $10 more than Diane, and Ellen gets$10 more than Mary.

Bill Brown gets twice as much as his wife, Henry Hobson gets the same as his wife, and John Jones gets 50 percent more than his wife.

What are the full names of the three wives?

Note: Assume that the wives have the same last names as their husbands.

Given that $x=y=0$ is not the only solution to the following system of linear equations, determine all the possible values of $k:$ \begin{aligned} x+2y &= kx \\ 2x+y &= ky . \end{aligned}

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