3's the number!
3 times a certain number
Makes another number plus 3
But a third of that certain number
Is the other minus 3
Soon you'll get that certain number in a fraction
Reduce to lowest terms, please
Then give me the sum of the numerator and denominator
Multiplied by 3
- Don't let the title fool you.
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1 solution
When reading the problem, you should notice that he's setting up equations to solve. The 'certain' number is what is necessary to be solved for.
First, we set up the first equation, 3(certain number) = (another number) + 3 Let's simplify down and call certain number x, and another number, y. So, now we have 3x = y + 3.
The next bit of information is that a third of 'certain number' is 'other number' - 3. The key here is to recognize the repeated language of 'certain/an'other' number', and to realize you're supposed to make another equation using our original 'x' and 'y' variables.' From this we have the equation x/3 = y - 3.
Together, we have the 'system' of equations, 3x= y +3, and x/3= y - 3. There is a set of x and y variables that work in both equations; we need to find them.
Beyond here relies on some basic algebra. There are a few ways to solve systems of linear equations, you can look it up or look at how I did it. I solved it the way I did because I saw at first that I could multiple (x/3) by 9 to get 3x very easily, and I would be able to eliminate x from the set of equations and solve for y.
Once I found y=30/8, I created an x = (expression) equation by multiplying the original bottom equation by 3. I subsituted 30/8 for y in order to solve the equation for x. At some point I changed 9 into 72/8, in order to subtract the fractions more easily. x solves for 18/8. Both 18 and 8 divide by 2, so we reduce the fraction to 9/4, which can no longer be reduced. This gives us x=9/4. (I substituted both x and y values into both equations to check that they work; I erased the work for this, though.)
Remember from the beginning, that we assigned x for the 'certain number. And the 'certain number' (9/4) we got was indeed a fraction. 9 is the numerator, and 4 is the denominator for that fraction. We add them together to get 13. We multiply 13 by 3, and we get 39 for our answer.