Direct and Inverse Variations

Q1: Multiple Choice Type

i. The values of a and b from the table are:

x   3  5   b   8
y  18  a  90  48

Step 1: Check if x and y have a constant ratio (direct variation). \[ \frac{y}{x} = \frac{18}{3} = 6,\quad \frac{a}{5} = 6 \Rightarrow a = 30 \\ \frac{y}{x} = \frac{90}{b} = 6 \Rightarrow b = \frac{90}{6} = 15 \]Answer: b. a = 30 and b = 15

ii. In 5 hours, a machine makes 45 screws. In 12 hours, the same machine will make:

Step 1: Use direct proportion: More hours ⇒ More screws \[ \frac{\text{Screws}}{\text{Time}} = \frac{45}{5} = 9 \text{ screws/hour} \]Step 2: Multiply rate by 12 hours: \[ 9 \times 12 = 108 \text{ screws} \]Answer: d. 108 screws

iii. If the cost of 9 pens is ₹369, the cost of one dozen pens of the same make will be:

Step 1: Cost of 1 pen = ₹369 ÷ 9 = ₹41
Step 2: 1 dozen = 12 pens \[ 12 \times 41 = ₹492 \]Answer: d. ₹492

iv. If y varies directly to x and y = 80 when x = 400, find y when x = 25 is:

Step 1: Since y ∝ x, so \(\frac{y}{x} = \text{constant}\) \[ \frac{y}{x} = \frac{80}{400} = \frac{1}{5} \]Step 2: For x = 25: \[ y = \frac{1}{5} \times 25 = 5 \]Answer: a. 5

v. A man working 48 hours per week earns ₹2,400. If he works for 36 hours per week, he will earn:

Step 1: Earning per hour = ₹2400 ÷ 48 = ₹50
Step 2: For 36 hours: \[ 36 \times 50 = ₹1800 \]Answer: c. ₹1,800

Q2: In which of the following tables, x and y vary directly

i.

x:   3    5     8     11
y: 4.5   7.5   12    16.5

Step 1: If x and y vary directly, then y/x should be constant.
=> y/x = 4.5/3 = 1.5
=> y/x = 7.5/5 = 1.5
=> y/x = 12/8 = 1.5
=> y/x = 16.5/11 = 1.5
All ratios are same ⇒ x and y vary directly.
Answer: Yes, they vary directly.

ii.

x:  16    30    40    56
y:  32    60    80    84

Step 1: Check y/x ratio:
=> 32/16 = 2
=> 60/30 = 2
=> 80/40 = 2
=> 84/56 = 1.5
Last ratio differs ⇒ Not directly proportional.
Answer: No, they do not vary directly.

iii.

x:  27    45    54    75
y:  81   180   216   225

Step 1: Check y/x ratio:
=> 81/27 = 3
=> 180/45 = 4
=> 216/54 = 4
=> 225/75 = 3
Ratios are not the same ⇒ Not directly proportional.
Answer: No, they do not vary directly.

Q3: If x and y vary directly, find the value of x, y and z:

x:  3   x   y   10
y:  36  60  96  z

Step 1: Use the property of direct variation:
If x ∝ y ⇒ \(\frac{x_1}{y_1} = \frac{x_2}{y_2}\)
To find x: \[ \frac{3}{36} = \frac{x}{60} \Rightarrow x = \frac{3 \times 60}{36} = \frac{180}{36} = 5 \]To find y: \[ \frac{3}{36} = \frac{y}{96} \Rightarrow y = \frac{3 \times 96}{36} = \frac{288}{36} = 8 \]To find z: \[ \frac{3}{36} = \frac{10}{z} \Rightarrow z = \frac{10 \times 36}{3} = \frac{360}{3} = 120 \]Answer: x = 5, y = 8, z = 120

Q4: A truck consumes 28 litres of diesel for moving through a distance of 448 km. How much distance will it cover in 64 litres of diesel?

Step 1: Distance per litre = Total distance ÷ Diesel used
= 448 ÷ 28 = 16 km per litre
Step 2: For 64 litres of diesel:
= 64 × 16 = 1024 km
Answer: 1024 km

Q5: For 100 km, a taxi charges ₹1,800. How much will it charge for a journey of 120 km?

Step 1: Charge per km = ₹1800 ÷ 100 = ₹18 per km
Step 2: For 120 km:
= ₹18 × 120 = ₹2,160
Answer: ₹2,160

Q6: If 27 identical articles cost ₹1,890, how many articles can be bought for ₹1,750?

Step 1: Cost of 1 article = ₹1890 ÷ 27
= ₹70 per article
Step 2: Number of articles that can be bought for ₹1750:
= ₹1750 ÷ ₹70 = 25 articles
Answer: 25 articles

Q7: 7 kg of rice costs ₹1,120. How much rice can be bought for ₹3,680?

Step 1: Cost of 1 kg of rice = ₹1120 ÷ 7
= ₹160 per kg
Step 2: Quantity of rice for ₹3680:
= ₹3680 ÷ ₹160 = 23 kg
Answer: 23 kg

Q8: 6 notebooks cost ₹156. Find the cost of 54 such notebooks.

Step 1: Cost of 1 notebook = ₹156 ÷ 6
= ₹26
Step 2: Cost of 54 notebooks = 54 × ₹26
= ₹1,404
Answer: ₹1,404

Q9: 22 men can dig a 27 m long trench in one day. How many men should be employed for digging 135 m long trench of the same type in one day?

Step 1: This is a case of direct proportion:
More length ⇒ More men needed (to complete in the same 1 day).
Step 2: Let number of men required = x \[ \frac{22}{27} = \frac{x}{135} \]Step 3: Cross-multiply: \[ 22 \times 135 = 27 \times x \\ 2970 = 27x \\ x = \frac{2970}{27} = 110 \]Answer: 110 men

Q10: If the total weight of 11 identical articles is 77 kg, how many articles of the same type would weigh 224 kg?

Step 1: Weight of 1 article = 77 ÷ 11
= 7 kg
Step 2: Number of articles in 224 kg = 224 ÷ 7
= 32 articles
Answer: 32 articles

Q11: A train is moving with uniform speed of 120 km per hour.

i. How far will it travel in 36 minutes?

Step 1: Convert 36 minutes to hours:
36 minutes = 36 ÷ 60 = 0.6 hours
Step 2: Use formula: Distance = Speed × Time
= 120 × 0.6 = 72 km
Answer: 72 km

ii. In how much time will it cover 210 km?

Step 1: Use formula: Time = Distance ÷ Speed
= 210 ÷ 120 = 1.75 hours
Step 2: Convert decimal hours to hours and minutes:
0.75 hours = 0.75 × 60 = 45 minutes
So, 1.75 hours = 1 hour 45 minutes
Answer: 1 hour 45 minutes