grade 7 Direct & Inverse Proportion

Proportional & Inverse Proportional

Direct proportion means that when one quantity doubles or triples, the other does the same, written y = ax. Walking at a steady speed is a clear case: more time means proportionally more distance. Inverse proportion, y = a/x, works the other way, since a faster speed over a fixed distance takes less time, so the product xy stays constant. Change the constant a on each graph and the line through the origin and the hyperbola update.

Direct Proportion: y = ax

Constant a2

👀 What is Direct Proportion?

①Doubling x doubles y; tripling triples it

②Form y = ax (a ≠ 0)

③Graph: line through the origin

④a > 0 → upper right; a < 0 → lower right

Inverse Proportion: y = a/x

Constant a6

🔍 What is Inverse Proportion?

①Doubling x halves y

②Form y = a/x (a ≠ 0)

③Graph: hyperbola (approaches axes without touching)

④xy = a (product is constant)

Meaning of the Constant

Direct Constant

a = yx

y/x is constant and equals a

Inverse Constant

a = xy

xy is constant and equals a

💡 Finding the Constant

①Direct: one point (x, y) gives a = y/x

②Inverse: one point gives a = xy

③The sign of the constant determines the graph's position

Work It Out

Example 1

y is directly proportional to x, and y = 6 when x = 2. Write y as an expression in x.

Direct proportion is y = ax. Substitute x = 2, y = 6 to find a.

6 = a × 2 ⇒ a = 3

Put a = 3 into the expression.

y = 3x

▸ y = 3x

For y = ax, find a from a single (x, y) pair.

Example 2

y is inversely proportional to x, and y = 4 when x = 3. Find y when x = 6.

Inverse proportion is y = a/x. So a = xy = 3 × 4 = 12.

y = a/x, a = 3 × 4 = 12

Substitute x = 6.

y = 12/6 = 2

▸ y = 2

In inverse proportion the product xy is constant (= a).

Exam Key Points

Direct Proportion

y = ax → line through origin

Larger |a| → steeper line (closer to y-axis)

Inverse Proportion

y = ax

a > 0: Q1 & Q3 / a < 0: Q2 & Q4

Grade-7 school exam type

The graph of y = 12/x passes through the point (a, 3). What is a?

①2

②3

③4

④6

⑤12

▸ ③ 4

Substitute the point (a, 3) into the equation.

3 = 12/a

Solve for a.

a = 12/3 = 4

🎯 Exam Key Points

①Direct vs inverse: y/x constant vs xy constant

②Read the sign of the constant from the graph

③Direct: larger |a| → steeper

④Inverse: hyperbola asymptotic to axes

⑤Examples: speed and time for a fixed distance (inverse); distance and time at a fixed speed (direct)

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