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Complex Number Puzzles with Words: Lesson for Kids

Instructor: Nick Rogers
In this lesson, you will take on more complex number puzzles. You will have to translate a problem into an algebraic expression (an equation involving variables) and solve for one or more of those variables.

Solving Mysteries with Numbers

Let's become number detectives! In order to solve the problems in this lesson, you must formalize your assumptions, determine the unknowns, and solve. To do so, you will write equations of two variables.

Adding and Multiplying with Words

Find two numbers that multiply to give 10 and add up to 7.

Let's approach this problem using algebra and symbols. How would you translate each of the requirements into a statement using variables?

We should give a name to each of the two numbers. Call the first one x, and the second one y. Then we know that they should multiply to give 10, so:

x*y = 10

We also know that they should add to 7:

x + y = 7

We have now written down two equations that tell us how x and y are related. This is enough information to determine what their values should be. In general, we will need one equation for each unknown. The first step is to solve for one of the variables in the second equation. For example:

y = 7 - x

Then we can substitute this value for y into the first equation:

x*(7-x) = 10

This equation can be expanded and simplified:

x^2 - 7x + 10 = 0

(x - 5)(x - 2) = 0

There are two values of x that will make this equation true, and you can read them off right away. x=5 or x=2. Remember that we previously showed that y = 7 - x. Therefore, if x=5, then y=2, and if x=2, then y=5. These are the two possible solutions to our problem.

Flying into the Wind

Let's try another popular type of puzzle involving speeds. The speed of a bird is either increased or slowed by the wind. The total distance a bird flies is given by multiplying the speed it flew at by the time it spent flying.

On a windy day, a peregrine falcon flew 40 miles in 20 minutes against the wind. When it flew with the wind, it traveled the same distance in only 8 minutes. How fast was the wind blowing, and how fast would the falcon fly if there was no wind at all?

First, capture these statements with variables - in math we use V to represent speed, and we can call the wind W. When the bird flew into the wind, it's speed was V-W, so we have:

40 = 20*(V - W) (i)

Then, when the bird flew with the wind, its speed was V+W, and its total time in flight was eight minutes:

40 = 8*(V + W) (ii)

Solve for V in equation (i) and substitute into equation (ii) as follows:

V = 2 + W

5 = V+W = 2+ 2W

3/2 = W

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