Direct and Inverse Proportions — NCERT Solutions, AI Tutor & Practice

Since more pens cost more money, the number of pens and cost are in direct proportion.

In direct proportion, $\dfrac{x_1}{y_1} = \dfrac{x_2}{y_2}$.

Let the cost of 12 pens be $\text{₹}y$.

$\frac{5}{75} = \frac{12}{y}$

Cross-multiplying:

$5 \times y = 75 \times 12$ $y = \frac{75 \times 12}{5} = \frac{900}{5} = 180$

So the cost of 12 pens is ₹180.

(a) More workers finish the job in less time. As one increases, the other decreases and their product stays constant → Inverse proportion.

(b) More distance uses more fuel; their ratio stays constant → Direct proportion.

(c) Age and shoe size do not follow a fixed ratio or fixed product (shoe size stops changing in adulthood) → Neither.

Distance and petrol are in direct proportion.

$\frac{240}{16} = \frac{375}{y}$

Cross-multiplying:

$240 \times y = 16 \times 375$ $y = \frac{16 \times 375}{240} = \frac{6000}{240} = 25$

So 25 litres of petrol are needed.

More taps means less time, so this is inverse proportion.

In inverse proportion, $x_1 y_1 = x_2 y_2$.

Let the required time be $y$ minutes.

$6 \times 40 = 8 \times y$ $240 = 8y$ $y = \frac{240}{8} = 30$

So 8 taps will fill the tank in 30 minutes.

More soldiers means the food lasts fewer days → inverse proportion.

Total soldiers now = $120 + 30 = 150$.

Using $x_1 y_1 = x_2 y_2$:

$120 \times 30 = 150 \times y$ $3600 = 150y$ $y = \frac{3600}{150} = 24$

The food will now last 24 days.

Map distance and actual distance are in direct proportion.

$\frac{2}{50} = \frac{7}{y}$

Cross-multiplying:

$2 \times y = 50 \times 7$ $y = \frac{350}{2} = 175$

The actual distance is 175 km.

Fewer days available means more machines are needed → inverse proportion.

Using $x_1 y_1 = x_2 y_2$ where $x$ = machines, $y$ = days:

$42 \times 63 = y \times 54$ $2646 = 54y$ $y = \frac{2646}{54} = 49$

So 49 machines would be required.

The scale $1:150$ means every 1 unit of model = 150 units of real building → direct proportion.

Actual height $= 40 \times 150 = 6000$ cm.

Convert to metres:

$6000 \text{ cm} = \frac{6000}{100} = 60 \text{ m}$

The actual height of the building is 60 m.

In inverse proportion, $xy = k$ (constant).

$k = 15 \times 6 = 90$

When $x = 9$: $9 \times y = 90 \Rightarrow y = \frac{90}{9} = 10$

When $x = 30$: $30 \times y = 90 \Rightarrow y = \frac{90}{30} = 3$

So $y = 10$ when $x = 9$, and $y = 3$ when $x = 30$.

Convert time to minutes: $1$ hour $20$ minutes $= 80$ minutes.

Fewer pipes take more time → inverse proportion.

$6 \times 80 = 5 \times y$ $480 = 5y$ $y = \frac{480}{5} = 96 \text{ minutes}$

Convert back: $96$ minutes $= 1$ hour $36$ minutes.

So 5 pipes fill the tank in 1 hour 36 minutes.

More persons means fewer days → inverse proportion.

Using $x_1 y_1 = x_2 y_2$:

$3 \times 4 = 4 \times y$ $12 = 4y$ $y = \frac{12}{4} = 3$

So 4 persons should take 3 days to complete the job.

Number of books and weight are in direct proportion.

(a) Let weight of 40 books be $y$ kg. $\frac{72}{9} = \frac{40}{y}$ $72y = 9 \times 40 = 360$ $y = \frac{360}{72} = 5 \text{ kg}$

(b) Let the number of books weighing 6 kg be $x$. $\frac{72}{9} = \frac{x}{6}$ $9x = 72 \times 6 = 432$ $x = \frac{432}{9} = 48 \text{ books}$

So (a) 40 books weigh 5 kg, and (b) 48 books weigh 6 kg.