For a fraction of the form $(p / q)$ , the number in front of the division ( $p$ ) is the numerator. The number after the division ( $q$ ) is the denominator.

$\begin{aligned} \frac{p}{q} \quad \genfrac{}{}{0}{}{(\text{numerator})}{(\text{denominator})}\end{aligned}$ .

A positive fraction $(p/q)$ ( $p,\, q > 0$ ) is less than $1$ as long as the numerator $p$ is smaller than the denominator $q$ (that is, $p < q$ .) The reason is that as long as $p,\, q > 0$ :

$\begin{aligned} & \frac{p}{q} < 1 \\ \iff & (q)\, \left(\frac{p}{q}\right) < (q)\, (1) && (\text{given that $q > 0$}) \\ \iff & p < q\end{aligned}$ .

For example, in the fraction $(2/3)$ , $2$ is the numerator while $3$ is the denominator. Both the numerator and the denominator are positive, while the numerator $2\!$ is smaller than the denominator $3\!$ . Therefore, $(2/3)$ would be less than $1$ .

6 0

1 year ago

A circle centered at (-2,3) and contains point (5,9) which represents the equation

SIZIF [17.4K]

The standard form of the equation of a circle is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle, (x,y) is a point of the circle, and r is the length of the radius of the circle. When the equation of a circle is written, h,k, and r are numbers, while x and y are still variables. (x-2)^2 + (y-k)^2 = 16 is an example of a circle. The problem gives us two of the three things that a circle has, a point (5,9) and the center (-2,3). We need to find the radius in order to write the equation. We substitute -2 for h, 3 for k, 5 for x, and 9 for y to get (5 - (-2))^2 + (9 - 3)^2 = r^2 We simplify: 49 + 36 = r^2, r^2 = 85. We only need to know r^2 because the equation of a circle has r^2. We now have all the information to write the equation of a circle. (x + 2)^2 + (y - 3)^2 = 85.

8 0

3 years ago