Proportional Relationships
Grade 6 · Math · Study Guide
Learn how to identify, represent, and solve proportional relationships.
Two quantities are proportional when they grow or shrink at the same rate. If you earn $12 per hour, working 2 hours earns $24 and 5 hours earns $60. The ratio of earnings to hours is always 12:1 — it never changes. That constant ratio is called the constant of proportionality.
You can check if a relationship is proportional by dividing the two quantities at several points. If every division gives the same result (the constant of proportionality, often called k), the relationship is proportional. For the earnings example: 24÷2 = 12, 60÷5 = 12. Same quotient — it's proportional!
Proportions are equations that say two ratios are equal: 3/4 = 6/8. To solve a proportion with a missing value, use cross-multiplication. If 3/4 = x/12, then 3×12 = 4×x, so 36 = 4x, meaning x = 9.
Proportional relationships show up everywhere — scaling recipes, reading maps, converting currencies. When you look at a map and the scale says '1 inch = 50 miles,' you're using a proportional relationship every time you measure a route.
| Hours | Earnings ($) | Ratio (Earnings/Hours) |
|---|---|---|
| 1 | 12 | 12 |
| 3 | 36 | 12 |
| 5 | 60 | 12 |
| 8 | 96 | 12 |
Key terms
- Proportion
- An equation stating that two ratios are equal, like 3/4 = 6/8
- Constant of Proportionality
- The constant ratio k in a proportional relationship, where y = kx
- Cross Multiplication
- A method to solve proportions: multiply diagonals and set them equal
- Direct Proportion
- When one quantity increases, the other increases at the same rate
Flashcards
- Proportion
- Two equal ratios — like 1/2 = 4/8
- Constant of proportionality (k)
- The number k in y = kx; the unit rate in a proportional relationship
- Cross multiplication
- If a/b = c/d, then a×d = b×c
- How to solve a proportion
- Cross multiply, then divide both sides to isolate the variable
- Proportional table check
- Divide y÷x at each row — if all quotients match, the relationship is proportional
- y = kx
- The equation of a proportional relationship; k is the constant of proportionality
Practice questions
If 5 apples cost $3.00, how much do 15 apples cost?
- $9.00
- $6.00
- $45.00
- $3.00
Show answer
Answer: $9.00
5 apples : $3.00 = 15 apples : ?. Multiply both by 3: $3.00 × 3 = $9.00.
In the proportion 2/3 = 8/12, cross multiplication gives 2×12 = 3×8.
Show answer
Answer: True
Cross multiply: 2×12=24 and 3×8=24. Both sides equal 24, so yes.
In the equation y = 7x, the constant of proportionality is ___.
Show answer
Answer: 7
In y=kx, k is always the coefficient of x — the number x is multiplied by.
Which pair of ratios forms a true proportion?
- 3/4 = 9/12
- 2/5 = 5/10
- 4/7 = 8/15
- 3/8 = 6/18
Show answer
Answer: 3/4 = 9/12
Cross multiply: 3×12=36 and 4×9=36. Equal products confirm a true proportion.
Solve for x: 4/6 = x/18
- 12
- 8
- 3
- 24
Show answer
Answer: 12
Cross multiply: 4×18 = 6×x → 72 = 6x → x = 12.
Fun facts
- Map makers use proportional relationships to scale the entire Earth onto a single sheet of paper.
- The human body is proportional — your arm span is roughly equal to your height in most people.
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