I’m struggling to spot when a situation is truly direct proportion. In exercises, cost vs kilos feels straightforward, but then they add a fixed delivery fee and I’m not sure if it still counts. Also, when units change (like cm to m), I get unsure what the constant k actually is and what units it should have. If a table gives pairs like (2 kg, $9) and (5 kg, $22.50), I try y/x, but if numbers are rounded the ratios aren’t exact, and then I’m not sure if it’s still “direct” or just noisy data.
What’s a clean checklist for deciding if it’s y = kx? Is “graph goes thru the origin” the main test, even with rounding? Any quick way to get k without messing up units? And in word problems, phrases like “varies directly with” vs “is proportional to” – are they the same thing? I’m probably overthinking, but I keep tripping on these lil gotchas. Any tips or simple examples to practise would help, thx.
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3 Responses
You’re not overthinking-“direct proportion” really is just one clean idea: y = kx with the same k for every x, and the graph is a straight line through the origin. Quick checklist I use: (1) if x = 0 then y must be 0 (no fixed fees, no intercepts), (2) the ratio y/x is (about) constant across pairs, and (3) units are consistent. If there’s a fixed delivery fee c, the model is y = c + kx: still a straight line, but not proportional because it doesn’t go through (0,0). About units: k always has “units of y per unit of x.” So if price varies directly with mass, k might be $/kg; switch x to grams and k becomes $/g (numerically 1/1000 of the $/kg value). Example: (2 kg, $9) gives k = 9/2 = 4.50 $/kg; in grams that’s 0.0045 $/g.
With rounding/noisy data, don’t demand perfection-compute k from a couple of pairs and see if they’re close; plot if you can and check whether a line through the origin fits “pretty well.” If it looks straight but misses the origin, suspect a fixed fee. “Varies directly with” and “is proportional to” mean the same thing; “inversely proportional” is the 1/x cousin. Nice practice pairs: price vs kilograms with no fee (direct), recipe ingredients vs number of servings (direct), circumference vs radius (C = 2πr, direct with k = 2π), taxi fare with a flag-fall (linear, not direct), and area vs side length (not direct-it’s kx²).
Quick checklist: convert units first, then see if y/x is (approximately) the same for all pairs and the line goes through the origin-any fixed fee means y = kx + b with b ≠ 0 so not direct; “varies directly with” and “is proportional to” mean the same. To get k, use k = y/x (units: “units of y per unit of x”), e.g., (2 kg,$9) and (5 kg,$22.50) both give k = $4.50/kg, with small deviations likely rounding-like paying only by weight for apples is direct, but a delivery or taxi base fare adds an intercept; nice refresher: https://www.khanacademy.org/math/algebra/one-variable-linear-equations/direct_and_inverse_variation/a/direct-variation.
Direct proportion is the “no sneaky entry fee” relationship: y = kx means when x is zero, y is zero, the ratio y/x is the same for every valid pair, and the graph is a straight line that hugs the origin. Phrases like “varies directly with” and “is proportional to” are the same thing. If there’s a fixed delivery fee b, the model is y = kx + b, which is still a straight line but no longer proportional because it misses the origin. Quick checks: does doubling x double y? does y/x stay (about) constant once units are consistent? and is the intercept near zero if you sketch it? With rounding, look for “close enough”: compute k from each point and see if they cluster. Your example (2 kg, $9) and (5 kg, $22.50) gives k = 9/2 = 4.5 and 22.5/5 = 4.5, so it’s nicely proportional at $4.50 per kg. If someone adds a $3 delivery fee, costs might be $12 and $25.5 for those same masses; now 12/2 = 6 and 25.5/5 = 5.1-ratios wobble, so it’s not direct proportion. In that case, you can spot the fee by solving k = (y2 − y1)/(x2 − x1) = (25.5 − 12)/(5 − 2) = 4.5, then b = y1 − kx1 = 12 − 4.5·2 = 3.
About units, k proudly wears the badge “units of y per units of x.” If y is dollars and x is kilograms, k is dollars per kilogram. If y is metres and x is centimetres in y = kx, then k = 0.01 m/cm (because 1 cm = 0.01 m). A handy habit: either convert everything first, or keep the units glued to your calculation so k’s units pop out correctly. For noisy tables, take k-tilde as the average of the ratios y/x, or use one clean point if the data looks consistent; then check whether y − k-tilde x stays close to zero (direct) or hovers near a constant b (fee alert!). A friendly walkthrough lives here: https://www.khanacademy.org/math/algebra/one-variable-linear-equations/alg1-eq-and-graphs/a/direct-variation.