Proportion Notes

Proportional Changes: Occur when two related quantities change in such a way that one quantity is a fixed multiple or fraction of the other.
Proportionality Constant: The fixed number that gives one quantity as a multiple or fraction of the other in proportional changes. Also refered to k

Photos in Different Sizes: When enlarging or reducing a photo, the ratio of the sides should not change.
- Example: A photo with a width to height ratio of 4:3. The height is \frac{3}{4} of the width, and the width is \frac{4}{3} of the height.
- Example: A photo in 16:9 format. The height is \frac{9}{16} of the width, and the width is \frac{16}{9} \approx 1.77 of the height.
A-Size Paper: Width to height ratio is the same; height is proportional to the width.
- If width is x and height is kx, then the width and height of the next size are \frac{1}{2}kx and x.
- \frac{x}{\frac{1}{2}kx} = \frac{kx}{x}
- \frac{1}{ \frac{1}{2} } = k
- k^2 = 2
- k = \sqrt{2}
Circles: Circumference is 2\pi times the radius. If r is the radius and c is the circumference, then c = 2\pi r.
Object Traveling at Constant Speed: Distance traveled is proportional to the time of travel. For example, at 10 meters per second, the distance s is s = 10t, where t is time.

Changing the width of a picture proportionally changes the height.
- If h = \frac{3}{4}w, then the height changes proportionally with respect to width, and the proportionality constant is \frac{3}{4}.
- If w = \frac{4}{3}h, then the width changes proportionally with respect to height, and the proportionality constant is \frac{4}{3}.
For circles, circumference changes proportionally with respect to radius, and the proportionality constant is 2\pi. Radius changes proportionally with respect to circumference, and the proportionality constant is \frac{1}{2\pi}.
For an object moving at a constant speed of 10 m/s, distance changes proportionally with respect to time, and the proportionality constant is 10. Time changes proportionally with respect to distance, and the proportionality constant is \frac{1}{10}.

If the relation between two quantities x and y is y = kx, where k is a constant, then y changes proportionally with respect to x, and k is the proportionality constant.

The height of a point on a slanted line from the horizontal line changes proportionally with respect to its distance from the corner.

The extension of a spring is proportional to the weight suspended from it. This principle is used to calibrate spring scales.
If a weight of 1 kg extends the spring by 5 cm, then the extension for 100 grams would be 0.5 cm.

In proportional changes, if one quantity is scaled by a certain factor, the other quantity is scaled by the same factor.
Example: If the width of a photo is doubled, the height is also doubled.

In triangles with the same angles, sides are scaled by the same factor.
If the sides of a triangle are scaled by the same factor, the angles do not change.

In triangles with the same angles, the ratios between pairs of sides do not change even if the triangle is scaled.

The volume of water falling in a region is proportional to its area. The height of the rainwater collected is the same regardless of the size of the vessel. A rainfall of 1 millimeter means 1 liter of rainwater in each square meter.

The sum of the inner angles of polygons does not change proportionally with respect to the number of sides.
The sum of the inner angles of any polygon is given by s = 180(n-2), where s is the sum of the angles and n is the number of sides.
If we denote m = n - 2, then s = 180m. The sum of the inner angles is proportional to the number of sides reduced by 2.

The area A of a circle is not proportional to the radius r, but it is proportional to the square of the radius: A = \pi r^2.

The distance traveled by a falling object is not proportional to time, but it is proportional to the square of time.

For any regular polygon, the angle d made by two adjacent vertices at the center of the circle is related to the number of sides n by: d = \frac{360}{n}.

If two varying quantities are related in such a way that one quantity is a fixed multiple or fraction of the reciprocal of the other, the change is said to be inversely proportional.
Algebraically, if the relation between x and y is y = \frac{k}{x}, where k is a constant, then y changes inversely proportional to x, and k is the proportionality constant.

Changes given by the equation y = kx are sometimes called directly proportional to distinguish them from inversely proportional relationships.

If an object travels 100 meters, the relation between speed x (m/s) and time y (seconds) is: y = \frac{100}{x}. Therefore, y changes inversely proportional to x.

In objects made of the same material, weight changes proportionally with respect to volume.
- For iron, the proportionality constant (density) is 7.87 grams per cubic centimeter.
- For copper, the proportionality constant (density) is 8.96 grams per cubic centimeter.