Can you solve the following, without doing any Algebraic manipulation? Just by reading and thinking about what it says, can you figure out what x is? (The numbers a, x, and k are not zero.)
Given: 3a = k
And: ax = 4k
What must x be?
If you’re versed much at all in basic Algebra you will be tempted to substitute and solve. After all, this is a system of equations. But that will bypass the purpose and benefit of the exercise.
The intended benefit of this problem is that it promotes mathematical literacy, in particular, seeing relationships between terms. It’s not a complicated relationship but it is of utmost importance to this problem. Once you read and make sense of what the mathematical relationships are you can talk your way through the problem.
Once again, I believe the purpose of homework is learning. Sure, sometimes it is practice and familiarity, but those are the only times that answer-getting is important. Without understanding, having the right answer is often of little to no use. If it were, then copying the answers from the back of the book would be sufficient for learning, right?
If you’re ready to see the solution, you can watch the video or read the text after the video.
I understand that sometimes it’s appropriate to read, but not listen or watch a video. So here’s how this works.
Given: 3a = k
And: ax = 4k
The first statement says that the number k is three times bigger than the number a. We don’t know what a or k are but we know how they’re related and can think of lots of numbers that fit this relationship. One number that’s three times as large as the other.
The number k is three times as big as the number a.
Think of this relationship one more way, for a moment. The number k has two factors, 3 and a. Whether a is composite or prime is irrelevant really, it won’t change the fact that we could write k as the product of two numbers. I mention this, not because it helps solve this problem but because it might. Without knowing the path, sometimes it is a good idea to brain storm for a moment and list as many things you know about the information given, before seeking an answer. Sometimes, doing so, makes the answer apparent to you!
Let’s look at the second statement now.
Another number times a is four times as big as k. This is perhaps a bit distracting, but the key information is there. Remember, k is three times as big as a. Now we have something four times larger than k.
Let’s look at this a different way. The number 4k is not k at all, but instead, k and 4 are factors of new number.
If this new number is four times larger than k, and k is three times larger than a, how much larger is this new number than a?
You have three times as much money as me. Bobert has four times as much as you do. How much more money does Bob have than me?
For every dollar I have you have three. For every dollar you have, Bobert has four.
Still don’t see it? I know…picture good, word bad. Here you go.
If 3a = k, and ax = 4k, then x is 12 because