Proportional Relationships
Two quantities are in a **proportional relationship** when they grow together at a constant rate. If one doubles, the other doubles. If one triples, the other triples. The ratio between them is always the same number — called the **constant of proportionality**.
Spotting the constant
Example: a recipe calls for 2 cups of flour per 3 cookies.\n- 4 cups flour → 6 cookies\n- 6 cups flour → 9 cookies\n- 10 cups flour → 15 cookies\n\nThe ratio flour:cookies is always 2:3. The constant of proportionality (cookies per cup of flour) is 3/2 = 1.5. Every cup of flour makes 1.5 cookies.\n\nThe equation is y = kx, where k is the constant. Here, cookies = 1.5 × flour.
A table shows: 2 books cost $10, 5 books cost $25, 8 books cost $40. What is the unit price (constant of proportionality)?
Is it proportional or not?
Not every "relationship" is proportional. Check two things:\n\n1. The ratio y/x is constant for every pair.\n2. When x = 0, y = 0 (the graph passes through the origin).\n\nExample that is NOT proportional: a phone plan charges $10 setup + $5 per GB. At 0 GB you still owe $10 — so the graph doesn't pass through the origin. It's a linear relationship, but not a proportional one. (This distinction matters — CCSS 7.RP.2 tests it explicitly.)
Which table shows a proportional relationship?
Graphs of proportional relationships
On a graph, a proportional relationship is a **straight line through the origin**. The slope of that line is the constant of proportionality.\n\n- Steeper line = bigger k = more y per unit of x\n- Flatter line = smaller k\n\nThat's why miles-per-hour graphs, price-per-pound graphs, and meters-per-second graphs all look the same shape: a straight line starting at zero.
Find the proportional relationships in your day
Write down 3 proportional relationships you used today. Examples: cost per song on a streaming service you don't have (rules out proportional), miles traveled at a steady speed, grams of sugar per serving of cereal, minutes per lap running. For each, identify the constant of proportionality and write the equation y = kx. If one of your examples isn't actually proportional (like the phone bill example above), say why.
Solving real problems (CCSS 7.RP.3)
Percent problems, tax problems, tip problems, markup and markdown, scale drawings — these are all proportional reasoning in disguise.\n\nExample: A $60 shirt is 25% off. Find the sale price.\n- Discount = 25% of 60 = 0.25 × 60 = $15\n- Sale price = 60 − 15 = $45\n\nOr: 8% sales tax on a $45 purchase?\n- Tax = 0.08 × 45 = $3.60\n- Total = $48.60\n\nThe pattern is always: set up the ratio, use k = percent/100, compute.
Build a tipping chart
Make a small table: for restaurant bills of $15, $30, $45, $60, and $75, compute 15% tip, 18% tip, and 20% tip. Do it in your head when you can — that's proportional reasoning under pressure. Bonus: notice that 20% is just "move the decimal point and double."
A map has a scale of 1 inch : 50 miles. How many miles is 3.5 inches on the map?
Proportional reasoning is one of the most useful real-world math skills. Every time you scale a recipe, calculate a tip, convert units, read a map, or compare prices per ounce at the grocery store — you're using it. Master it now and the rest of middle-school math gets a lot easier.
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