Grade 7

Direct and inverse proportion practice

Direct proportion is $y = ax$ and inverse proportion is $y = \frac{a}{x}$ . Almost every problem starts by finding the constant $a$ .

Basic, Standard: Grade 7 · Term 2 / Advanced: Grade 7 · Term 3

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Direct proportion

If $y = ax$ ( $a \neq 0$ ), then $y$ is directly proportional to $x$ with constant $a$ . To find $a$ , substitute known values: if $y = -12$ when $x = -3$ , then $a = 4$ and $y = 4x$ .

The graph is a straight line through the origin.

Inverse proportion

If $y = \frac{a}{x}$ , then $y$ is inversely proportional to $x$ , and $xy = a$ is constant. If $y = -3$ when $x = 5$ , then $a = -15$ and $y = -\frac{15}{x}$ .

The graph is a hyperbola: two smooth curves, symmetric about the origin.

Ranges and applications

To find the range of $y$ from a range of $x$ , substitute the endpoints and check whether $y$ increases or decreases. For $y = \frac{60}{x}$ with $3 \leq x \leq 20$ , $y$ decreases, so $3 \leq y \leq 20$ .

Applications

The weight of a wire is proportional to its length; for meshing gears, teeth × turns is constant, so turns are inversely proportional to teeth.

Q1BasicGr 7 · Term 2Direct and inverse proportion

$y$ is inversely proportional to $x$ , and $y = 7$ when $x = -5$ . Express $y$ in terms of $x$ .

Answer

y = -\frac{35}{x}

Solution

Since $a = xy$ ,
$a = (-5) \times 7 = -35 \quad\text{so}\quad y = -\frac{35}{x}$

Q2StandardGr 7 · Term 2Direct and inverse proportion

Write down the coordinates of points A and B in the figure.

Answer

A $(-2,\ -4)$
B $(0,\ 1)$

Solution

Point A has $x$ -coordinate $-2$ and $y$ -coordinate $-4$ .
$A(-2,\ -4)$
Point B has $x$ -coordinate $0$ and $y$ -coordinate $1$ .
$B(0,\ 1)$

Q3AdvancedGr 7 · Term 3Direct and inverse proportion

As shown, the graphs of $y = -2x$ and $y = \frac{a}{x}$ meet at two points A and B. Point A has $x$ -coordinate $2$ .

(1)Find the value of $a$ .
(2)Find the coordinates of point B.

Answer

(1) $a = -8$
(2) $(-2,\ 4)$

Solution

(1) A has $y = -2 \times 2 = -4$ . Since A $(2,\ -4)$ is also on the curve,
$a = 2 \times (-4) = -8$
(2) Both graphs are symmetric about the origin, so B is the reflection of A through the origin.
$B(-2,\ 4)$

Q1BasicGr 7 · Term 2Direct and inverse proportion

Which of the following shows $y$ directly proportional to $x$ ?

AThe amount of water $y$ L poured in $x$ minutes at 3 L per minute
BThe area $y$ cm $^2$ of a square with side $x$ cm
CThe time $y$ minutes to fill a 12 L tank at $x$ L per minute
DThe sum $y$ of the ages of a 10-year-old and a child aged $x$

Q2StandardGr 7 · Term 2Direct and inverse proportion

The point $(6,\ b)$ lies on the graph of $y = \frac{12}{x}$ . Find $b$ .

Q3AdvancedGr 7 · Term 3Direct and inverse proportion

8 m of wire weighs 240 g. How much does 9 m of the same wire weigh, in grams?

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