Explain the mathematical motivation for expanding to the rational number set.
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Answer:
Should be roughly 79.76% if I'm doing this right.
Step-by-step explanation:
Looking at the problem, you're given 3 things:
1) 6.8 million tonnes are recycled;
2) 6.8 million tonnes is 27% of the total household waste;
3) 20.1 million tonnes could be recycled.
Since they're asking for what percentage of <em>total</em> household waste could be recycled, you're looking at a multi-step cross multiplication problem.
It's not very difficult though, don't worry.
So I'm gonna start by telling you to just forget that the word "million" exists for the moment. You only need 6.8, 20.1, and 27%.
Setting up the first part of the problem requires you to convert 27% into a fraction. That's .
6.8 is 27% of some mysterious number that we need to figure out what percent of household waste could be recycled, so that'll be set up on the opposite side of the equal sign as the equivalent to 27.
So we're starting out with this:
Since you're trying to initially find the total household waste, we're setting x as the variable to solve for.
Cross-multiplying** means you multiply the numerator of one side of the equal sign to the denominator of the other side of the equal sign, and vice versa (denominator of one side to numerator of other). So multiply 6.8 to 100, and 27 to x.
Now that you have this, you have to isolate x. Divide both sides by 27 to do this.
I'm gonna round that for simplicity's sake, because 185 repeats.
So we have 25.2 million tonnes of total household waste. Great! Now we can find the <em>percentage </em>of household waste that could be recycled.
Since we have how much could be recycled and the total, we just need to solve for the percent this time. So this is what the set-up will look like:
x being the percent we're solving for.
Cross-multiply again:
Divide both sides by 25.2:
Then round for simplicity's sake:
And you get 79.76% of household waste could be recycled.
Hope this helped!
**Be careful with cross-multiplying when you get to higher level math though. Not every situation allows for cross-multiplying. In fact, you can only cross-multiply when there's an equal sign.