Since x and y vary inversely as each other
xy = constant
10 x 6 = 15 xy
60 = 15y

y = 4

Using unitary method we can solve this as follows:
The cost of 2 notebooks = Rs. 24
The cost of 1 notebook = \(\frac { 24 }{ 2 } \) = Rs.12
Therefore, the cost of 9 notebooks = 9 x Rs.12
= Rs.108
Hence, Anbu has to pay Rs.108 for 9 notebooks.

Let x be the number of pencils required for 18 children. As the number of children increases, number of pencils also increases.
In the case of direct proportion we take \(\frac { { x }_{ 1 } }{ { y }_{ 1 } } =\frac { { x }_{ 2 } }{ { y }_{ 2 } } \)

\(\frac { 6 }{ 24 } =\frac { 18 }{ x } \)
\(\frac { 6\times x }{ 24 } \) = 18
x = \(\frac { 18\times 24 }{ 6 } \) = 72
Hence, 72 pencils are required for 18 children.

Let x be the number of notebooks, bought for Rs. 1863

This is direct proportion
\(\frac{x_{1}}{y_{1}} =\frac{x_{2}}{y_{2}} \)
\(\frac{105}{2415} =\frac{x}{1863} \)
x x 2415 = 1863 x 105
\(x=\frac{105 \times 1863}{2415} \Rightarrow x=81\)
81 notebooks can be bought for Rs. 1863.

6 cans of juice costs Rs. 210
1 can of juice costs \(=Rs. \frac{210}{6}=Rs. 35 \)
4 cans of juice costs = Rs. 35 x 4 = Rs. 140.

(i) Given that C = kd
The relation between C and d is direct proportion.
(ii) C = kd, when C = 30 and d = 6, k= ?
30 = k x 6
\(k=\frac{30}{6}\)
k = 5
(iii) C = kd when d = 10 and k = 5, c = ?
C = 10 x 5 = 50

This is an inverse proportion,
x₁ y₁ = x₂ y₂
6 x 120 = x \(\times\) 30
\(x=\frac{7 \times 180}{84} \Rightarrow x=24\)
24 gardeners are needed. Already 6 gardeners are available. Therefore, 18 more gardeners are needed.

No. of students in the hostel = 40
Now, the students are doubled.
Total no. of students = 80

This is an inverse proportion,
x₁ y₁ = x₂ y₂
40 x 30 = 80 \(\times\) x
\(x=\frac{40 \times 30}{80}\)
x = 15
If the students are doubled, then the stock will last in 15 days.

6 pumps filled the sump in hr 30 minutes = 90 minutes
Now, one pump in switched off.
Total no. of Pumps = 5

This is an inverse proportion, x₁y₁ = x₂y₂
6 x 90 = 5 \(\times\) x
\(x=\frac{6 \times 90}{5}\)
x = 108
The time taken to fill the pump = 108 minutes = 1 hr 48 minutes

30 men can reap a field in 15 days.
\(\therefore\) 1 men can reap the field in \(\cfrac { 15 }{ 30 } \) days = \(\cfrac { 1 }{ 2 } \)days
\(\therefore\) 20 men can reap the field in 10 days.

Since the given data are in direct proportion.
\(\frac{738}{6}=\frac{x}{9}\)

6x = 738 x 9
\(x=\frac{738 \times 9}{6}=1107\)
1107 letters can be sorted in 9 hours.

Rs. 840 paid for 21 students.
Rs. 1680 (double of its amount) paid for 42 students (double the students).

To weave 25 baskets time taken = 35days
\(\therefore\) To weave 1 basket time taken = \(\cfrac { 35 }{ 25 } days\)

Let x be the required number of days. The decrease in number of workers lead to the increase in number of days. (Therefore, both are in inverse proportion)
For inverse proportion x₁y₁ = x₂ y₂

Hence 60 x 7 = 42 \(\times\) x
42 \(\times\) x = 60 x 7
x = \(\frac { 60\times 7 }{ 42 } \)
x = 10
In 10 days 42 workers can spin a bale of cotton.