In the expression -1 / 3 , why is the remainder +2? How is this calculated?
We categorize numbers into terminal and non-terminal types. For non-terminal numbers, we have subcategories of repeating and non-repeating numbers. Are these subcategories exclusive to non-terminal numbers, or do terminal numbers also have similar classifications?
In the “Mixture Trick” lesson in Algebra, there was a question that asked, “By what factor is the amount of solution x larger than that of solution y?” We found the difference by subtracting the two amounts, and after calculating, we determined that x is 8/3 larger than y.
However, in a different lesson about distance, rate, and time, there was a question that asked, “By what percentage is Bill’s rate greater than Sarah’s rate?” In this case, we calculated the percentage by taking the difference between the two rates and dividing it by the smaller rate.
I’m confused about why we sometimes use just subtraction to find the difference, while other times we use a division method based on the smaller or larger quantity. Could you please clarify this for me?
Just add 3 to -1 to get 2/3, which leaves a remainder of 2.
Dk what you mean by “terminal and non-terminal types”. Every rational number is either repeating or terminating. Also, an irrational number doesn’t repeat nor terminate. I guess that’s practically what u wanted to know.
Better to show context first. Can you post the mixture question and the bill/sarah question.
Regarding the first question, do we always need to add the denominator to the numerator when the numerator is negative? you mean -1/3 + 3/3 = 2/3 ? but why do we do this? (I’d like to understand the logic behind this to help me remember it better.)
Secondly, I didn’t intend to categorize numbers as rational and irrational, but it actually helped me. I found it challenging to relate “terminal and non-terminal” to the “repeat or non-repeat” categories. Thanks!
Lastly, so does the percent decrease/increase method always work? I’m wondering if simply subtracting without dividing either the smaller or larger number can be misleading, as I explained earlier.
Adding multiples of 3 to a number doesn’t change its remainder when divided by 3.
Idk, maybe? It’s more of an algebra exercise, so once you’re comfortable with algebra in general, then “percent decrease/increase” isn’t really an independent topic.
For example, P + 0.1P is 10\% greater than P because you do (P + 0.1P) - P = 0.1P, which is 10\% of P. Alternatively, you could just factor it to have: P( 1 + 0.1) to get to the same conclusion of this quantity being 10\% greater than P.
The percent increase formula is just abstraction of this underlying algebra.
For the first question, does the rule only apply to the number 3? Are there any rules for the other numbers?
Regarding the last question, when we say something is 0.1 greater than P, yes we mean: P + 0.1P . But dividing can really change the outcome. For example, in the earlier question (11/3 y - y / y) we divided by 1, which didn’t affect anything. - But if we had to divide by something like 2y, the result would be different (11/3 y - y / 2y).
I’d also like to know when we should use percent increase or decrease in situations where it’s not directly mentioned, like asking about the percent increase from 2020 to 2022.
Nothing special about 3? If you want to find the remainder when dividing by a certain number, then adding multiples of that multiples of that number has no effect on the overall remainder.
For example, 47 has the same remainder as 52 when divided by 5 because 47 + 5 leaves the same remainder as just 47 alone.
The effect of dividing y is normalizing, but fundamentally it’s just algebra.
For example, when computing the percent increase of 2 from 4, you’d be doing:
2 \cdot k = 4 \implies k = 2, which is 200\% of 2 or rather a 100\% increase of 2.
Similarly for, y = k \cdot \frac{11}3 y \implies k = \frac{11}{3} or a (\frac {11}3 -1) \cdot 100 \% percent increase. Notice this is exactly what the formula is doing, so yeah it’s just algebra.
Yeah like mentioned earlier, it’s just algebra so use it whenever you deem appropriate (?)