Points to Remember- Direct and Inverse Proportions

Points to Remember

• When two quantities x and y change so that their ratio remains the same, they are in direct proportion.
• When two quantities x and y change so that one increases while the other decreases in such a way that their product remains constant, they are in inverse proportion.
• If x and y are in direct proportion then the ratio is constant: \( \dfrac{x}{y} = k \).
• If x and y are in inverse proportion then the product is constant: \( xy = k \).

We Know That

• Variable nature: A variable is a quantity whose value can change.
• Inter-related quantities: When one quantity depends on another so that a change in one causes a change in the other, this relation is called variation.
• Types of variation:There are two basic types of variation:
  (i) Direct variation (direct proportion)
  (ii) Inverse variation (inverse proportion)

Direct Proportion

If two quantities are related so that an increase in one causes a proportional increase in the other (and a decrease causes a proportional decrease), they are in direct proportion.

Two quantities x and y are in direct proportion if

\( \dfrac{x}{y} = k \)

or

\( x = ky \)

Important points

• In a direct proportion the ratio \( \dfrac{x}{y} \) remains constant for all corresponding values of x and y.
• The constant \( k = \dfrac{x}{y} \) is called the constant of proportionality or the constant of variation.
• For two pairs \( (x_1,y_1) \) and \( (x_2,y_2) \) in direct proportion, \( \dfrac{x_1}{y_1} = \dfrac{x_2}{y_2} \).

Inverse Proportion

If two quantities are related so that an increase in one causes a proportional decrease in the other (and vice versa), they are in inverse proportion.

Two quantities x and y are in inverse proportion if

\( x \propto \dfrac{1}{y} \)

or

\( x = \dfrac{k}{y} \)

which implies

\( xy = k \)

Important points

• In an inverse proportion the product \( xy \) remains constant for all corresponding pairs of values.
• The constant \( k = xy \) is the constant of variation for the inverse relation.
• For two pairs \( (x_1,y_1) \) and \( (x_2,y_2) \) in inverse proportion, \( x_1y_1 = x_2y_2 \).

Solved Examples

Q 1: Following are the car parking charges near an Airport up to
a. 2 hours Rs 60
b. 6 hours Rs 100
c. 12 hours Rs 140
d. 24 hours Rs 180
Check if the parking charges are in direct proportion to the parking time.

Solution: The charges and parking time are directly proportional if the ratio (charges ÷ time) is the same for all cases.

Compute the charge per hour in each case.

\( \text{Case (a): } \dfrac{60}{2} = 30 \)

\( \text{Case (b): } \dfrac{100}{6} \approx 16.666\ldots \)

\( \text{Case (c): } \dfrac{140}{12} \approx 11.666\ldots \)

\( \text{Case (d): } \dfrac{180}{24} = 7.5 \)

Since the charge per hour is not the same in all cases, the parking charges are not in direct proportion to the parking time.

Q 2: y is directly proportional to x, and y = 24 when x = 4. What is the value of y when x = 3?
a. 18
b. 20
c. 23
d. 43
Ans : a
Solution: Because y is directly proportional to x, we can write

\( y = kx \)

Use the given values to find k.

\( 24 = k \times 4 \)

\( k = 6 \)

Now substitute \( k \) and \( x = 3 \) into the relation.

\( y = 6 \times 3 \)

\( y = 18 \)

Q 3. The circumference (C cm) of a circle is directly proportional to its diameter (d cm). The circumference of a circle of diameter 3.5 cm is 11 cm. What is the circumference of a circle of diameter 4.2 cm?
a. 9.17 cm
b. 11.7 cm
c. 13.2 cm
d. 14 cm
Ans: c
Solution: Since circumference is directly proportional to diameter, write

\( C = kd \)

Use the given pair to find k.

\( 11 = k \times 3.5 \)

\( k = \dfrac{11}{3.5} = \dfrac{22}{7} \)

Now find the circumference for \( d = 4.2 \).

\( C = \dfrac{22}{7} \times 4.2 \)

\( C = \dfrac{22}{7} \times \dfrac{42}{10} \)

\( C = 22 \times \dfrac{6}{10} \)

\( C = \dfrac{132}{10} = 13.2 \text{ cm} \)

How to recognise and solve proportion problems 

• For direct proportion: check whether the ratios are equal. Use \( \dfrac{x_1}{y_1} = \dfrac{x_2}{y_2} \) or \( y = kx \).
• For inverse proportion: check whether the products are equal. Use \( x_1y_1 = x_2y_2 \) or \( x = \dfrac{k}{y} \).
• To find the constant of proportionality, substitute one known pair of values into the relation and solve for k.
• To find an unknown value, substitute k and the known value into the proportional relation and solve.