Proportional Relationships and Percent Problems: Grade 7 Study Guide
Many everyday relationships, from speed to recipes to sale prices, are proportional.
Two quantities are in a proportional relationship if their ratio is always the same. That constant ratio is the constant of proportionality, or unit rate, often written k in the equation y = kx. If 3 notebooks cost $4.50, then k = 4.50 ÷ 3 = $1.50 per notebook, and the cost of x notebooks is y = 1.5x.
On a graph, a proportional relationship is a straight line through the origin (0, 0). If the line does not pass through the origin, or the table has a different ratio in some rows, the relationship is not proportional.
Checking a table
Divide y by x in each row. If every quotient is the same, the relationship is proportional.
| Hours (x) | Miles (y) | y ÷ x |
|---|---|---|
| 1 | 55 | 55 |
| 2 | 110 | 55 |
| 3 | 165 | 55 |
| 4 | 220 | 55 |
Percent problems
A percent is a ratio out of 100. Percent change = (new − original) ÷ original × 100. A jacket goes from $80 to $60: change = −20 ÷ 80 = −25%, a 25% decrease. A markup raises a price, a discount lowers it, and sales tax or a tip is a percent of the price added on. A $40 meal with an 18% tip costs 40 + 7.20 = $47.20.
$80.
$80 − $60 = $20.
20 ÷ 80 = 0.25.
25% decrease.
Scale and unit rates
A map scale of 1 inch = 15 miles is a proportion: a distance of 4.5 inches on the map is 4.5 × 15 = 67.5 miles. Complex fractions can express unit rates: 1/2 mile in 1/4 hour is (1/2) ÷ (1/4) = 2 miles per hour. Always include units in the answer.
Worked examples and common mistakes
Example: A phone costs $250, then goes on sale for 20% off, with 8% sales tax. Sale price = $200. Tax = $16. Total = $216. Common mistakes: calculating percent change from the new number instead of the original, subtracting 20% twice for two successive 10% discounts (the discounts are applied one after the other, so the total is not 20%) and not checking whether a graph passes through the origin.
Check your understanding: model answers
1. k = 60, so y = 60x. 2. 10 ÷ 40 = 25% increase. 3. No: 10 ÷ 3 is not 3.
✨ Key Takeaways
- Proportional relationships have a constant ratio.
- Their graph is a line through the origin.
- Percent change uses the original amount.
- Always include units.
💬 Questions for Discussion & Reflection
- A car goes 180 miles in 3 hours. Write y = kx.
- A shirt goes from $40 to $50. What is the percent increase?
- Is the table (1, 3), (2, 6), (3, 10) proportional?
Print graph paper for graphs →