I tell you these facts about a mystery number, $c$: $\bullet$ $1.5 < c < 2$ $\bullet$ $c$ can be written as a fraction wit
h one digit for the numerator and one digit for the denominator. $\bullet$ Both $c$ and $1/c$ can be written as finite (non-repeating) decimals. What is this mystery number?
The simplest fraction for is . Write the upper bound as a fraction with the same denominator:
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Hence the range for would be:
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If the denominator of is also , then the range for its numerator (call it ) would be . Apparently, no whole number could fit into this interval. The reason is that the interval is open, and the difference between the bounds is less than .
To solve this problem, consider scaling up the denominator. To make sure that the numerator of the bounds are still whole numbers, multiply both the numerator and the denominator by a whole number (for example, 2.)
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At this point, the difference between the numerators is now . That allows a number ( in this case) to fit between the bounds. However, can't be written as finite decimals.
Try multiplying the numerator and the denominator by a different number.
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It is important to note that some expressions for can be simplified. For example, because of the common factor .
This answer is copied and pasted from an online site (aops) and this is what their explanation was, so please do not report for plagiarism!
Step-by-step explanation:
Because $c$ can be written as a finite decimal, we know it can be written as a fraction whose numerator is an integer and whose denominator is a power of $10$. Thus, after simplification, the denominator must still be a divisor of some power of $10$. That is, it must be factorable into $2$s and $5$s.
Since this denominator is a single digit, our choices are $1,$ $2,$ $4,$ $5,$ or $8$. We have the same options for the numerator, since we know $1/c$ also has a finite decimal. From here we could just test all the possibilities to see if they're between $1.5$ and $2,$ but with a little cleverness we can eliminate some of the remaining possibilities. If we don't use $5$ as the numerator or denominator, then $c$ is forced to be a power of $2$, so it can't be between $1.5$ and $2$. So, we must use $5$, and our only plausible choices are $5/2$ (which is $2.5$), $5/4$ (which is $1.25$), and $8/5$ (which is $1.6$). Of these, only $c=\boxed{8/5}$ works.
Slope-intercept form is y = mx + b where m is the slope and b is the y-intercept. Here, b is equal to 2 because the line crosses the y-axis at 2. THe slope is rise over run, or the change in y divided by the change in x. From the two points given, we get 3/3 which is equal to 1. 1x equals x, so this gives us "no slope." Putting it together gives us y = x + 2.
Have a nice day!
I hope this is what you are looking for, but if not - comment! I will edit and update my answer accordingly. (ノ^∇^)