2.3 Proportional relationships in quantitative contexts

Proportional Relationships in Quantitative Contexts

What Makes a Relationship Proportional

Two quantities are proportional when their ratio stays constant. In symbols, y = kx, where k is the constant of proportionality (also called the unit rate or constant of variation). Doubling x doubles y, and tripling x triples y. Crucially, a proportional relationship passes through the origin (0, 0): when x = 0, y = 0.

A non-proportional linear relationship has the form y = kx + b with b not equal to 0. It is still a straight line, but it does not pass through the origin, and the ratio y/x is not constant.

Finding the Constant of Proportionality

For any proportional pair, k = y / x.

Worked example: A worker earns $60 for 4 hours and $90 for 6 hours. Check the ratio: 60/4 = 15 and 90/6 = 15. The ratio is constant, so the relationship is proportional with k = 15 dollars per hour. The equation is y = 15x.

Worked example: A table shows x = 2 with y = 5, x = 4 with y = 10, and x = 6 with y = 15. The ratios are 5/2 = 2.5, 10/4 = 2.5, and 15/6 = 2.5. Proportional, k = 2.5, so y = 2.5x.

Testing a Table for Proportionality

Compute y/x for every row. If all the ratios are equal, it is proportional; if any differ, it is not.

Every ratio is 4, so y = 4x, which is proportional. Now compare a second table:

The ratios differ, so this is not proportional. Notice that y increases by 2 each time while x increases by 1, and the line would cross the y-axis at 3: the rule is y = 2x + 3, a non-proportional linear relationship.

Reading Graphs

On a graph, a proportional relationship is a straight line through the origin. The constant of proportionality is the steepness (rise over run) of that line, which equals the y-value when x = 1.

Worked example: A line passes through (0, 0) and (4, 10). The slope is 10 / 4 = 2.5, so k = 2.5 and y = 2.5x. To predict y when x = 10: y = 2.5 * 10 = 25.

If a line crosses the y-axis at any point other than the origin, say at (0, 3), the relationship is not proportional even though it is still linear.

Direct Variation

Saying "y varies directly with x" is another way to state the relationship is proportional: y = kx. You are usually given one pair to find k, then asked to predict another value.

Worked example: y varies directly with x, and y = 18 when x = 3. Find k: k = 18 / 3 = 6, so y = 6x. Then find y when x = 7: y = 6 * 7 = 42.

Worked example: The distance a car travels varies directly with time. It goes 150 miles in 3 hours. Find k, the speed: 150 / 3 = 50 mph, so d = 50t. In 5 hours it travels 50 * 5 = 250 miles.

Real-World Quantitative Reasoning

Proportional reasoning underlies scaling recipes, converting units, and comparing rates.

Worked example (unit conversion): If 1 kilogram is about 2.2 pounds, how many pounds is 15 kilograms? This is proportional with k = 2.2: pounds = 2.2 * 15 = 33 pounds.

Worked example (better buy): Brand A sells 3 pounds of rice for $6, so k = $2 per pound. Brand B sells 5 pounds for $8, so k = $1.60 per pound. Brand B has the smaller constant of proportionality, so it is the better value per pound.

Worked example (scaling): A blueprint is drawn so that 1 inch = 8 feet, a proportional relationship where feet = 8 * inches. A wall drawn 3.5 inches long is 8 * 3.5 = 28 feet in reality.

Worked example (rate table): A printer produces 90 pages in 2 minutes. Because printing is proportional, k = 90 / 2 = 45 pages per minute, so pages = 45 * minutes. In 5 minutes it prints 45 * 5 = 225 pages, and to print 360 pages it needs 360 / 45 = 8 minutes.

Proportional vs. Non-Proportional Checklist

• Does it pass through (0, 0)? Proportional requires yes.
• Is y/x the same for every pair? Proportional requires yes.
• Is it written as y = kx with no added constant? Then it is proportional.
• Is it written as y = kx + b with b not 0? Then it is non-proportional but still linear.

When a word problem includes a flat fee plus a per-unit charge, like a $25 base plus $15 per month, it is linear but not proportional, because the flat fee is the nonzero y-intercept. Recognizing that distinction is exactly the quantitative reasoning the TSIA2 tests.