Expenditure of paddy cultivation in 2 acres.

1. He spent more for wages Rs.10,000. Converting into percentage, We have
\(Wages=\cfrac { 10000 }{ 36000 } \times 100\%\) 
2. He spent Rs. 2000 for seeds. Converting into percentage,We have
\(\\ Seeds=\cfrac { 2000 }{ 36000 } \times 100\%=5.55\%\)

Centroid G divides each median in the ratio 2:1
ஃ XG = 2GP ⇒ XG = 2(2.5) = 5 cm and XP = XG + GP = 5 + 2.5 = 7.5 cm.
Similarly, GY = 2GQ ⇒ 6 = 2GQ ⇒ GQ = 3 cm and QY = 3 + 6 = 9 cm.
Now, ZR = 12 cm ⇒ GR + GZ = 12 cm ⇒ GR + 2GR = 12 cm ⇒ GR = 4 cm

(i) Since G is the centroid, XG : GL = 2 : 1 which gives XG : 2.5 = 2 : 1.
Therefore, we get 1 x (XG) = 2 x (2.5) ⇒ XG = 5 cm.
Hence, length XL = XG + GL = 5 + 2.5 = 7.5 cm.
(ii) If YG is of 2 parts then GM will be 1 part. (Why?)
This means YM has 3 parts.
3 parts is 9.3 cm long. So GM (made of 1 part) must be 9.3 ÷ 3 = 3.1 cm.

AM is median ⇒ M is the midpoint of BC.
Given that, BM = 3.5 cm, hence BC = twice the length BM = 2 x 3.5 cm = 7 cm.

Since they work on alternate days, let us consider a period of two days.
In the period of 2 days, work done by A and B = \(\cfrac { 1 }{ 12 } +\cfrac { 1 }{ 9 } =\cfrac { 7 }{ 36 } \) 
If we consider 5 such time periods for the fraction \(\cfrac { 7 }{ 36 } \) (we consider 5 periods because
7 goes 5 times completely in 36),
work done by A and B in 5\(\times\)2(=10)days = \(5\times \cfrac { 7 }{ 36 } =\cfrac { 35 }{ 36 } \) 
Therefore, the remaining work = \(1-\cfrac { 35 }{ 36 } =\cfrac { 1 }{ 36 } \) 
 This is done by A (why?) in \(\cfrac { 1 }{ 36 } \times 12=\cfrac { 1 }{ 3 } daysB\) 
 So, the total time taken = \(10days+\cfrac { 1 }{ 3 } days=10\cfrac { 1 }{ 3 } days\) 

If B does the work in 3 days, then A will do it in 1 day. That is, the difference is 2 days.
Here, given that the difference between A and B in completing the work is 24 days. Therefore,
A will take \(\cfrac { 24 }{ 2 } =12\) days and B will take 3 x 12 = 36 days to complete the work separately.
Hence, the time taken by A and B together to complete the work =\(\cfrac { ab }{ a+b } days\) 
= \(\cfrac { 12\times 36 }{ 12+36 } =\cfrac { 12\times 36 }{ 48 } =9days\)

Person days Method:
Here, let M and W denote a men and a women respectively.
Given that,\(6W=8M\Rightarrow 1W=\cfrac { 8 }{ 6 } M=\cfrac { 4 }{ 3 } M\) 
Now,\(7W+5M=7\times \cfrac { 4 }{ 3 } M+5M=\cfrac { 43M }{ 3 } \) 
If 8M can construct the room in 86 days, then \(\cfrac { 43M }{ 3 } \) can construct the same type of room in \(8M\times 86\div \cfrac { 43M }{ 3 } =8M\times 86\times \cfrac { 3 }{ 43M } =48days\)
Formula Method: 
Required time to construct the room = \(\cfrac { xyp }{ xb+ya } \)
= \(\cfrac { 6\times 8\times 86 }{ 6\times 5+8\times 7 } =\cfrac { 6\times 8\times 86 }{ 30+56 } =\cfrac { 6\times 8\times 86 }{ 86 } =48days\) 
(or)
Required time to construct the room = \(\cfrac { p }{ \frac { a }{ x } +\frac { b }{ y } } \) 
= \(\cfrac { 86 }{ \frac { 7 }{ 6 } +\frac { 5 }{ 8 } } =\cfrac { 86\times 48 }{ 86 } =48days\)

Multiplicative Factor Method:

Step 1:
Here, less men means more days. So, it is an inverse variation.
ஃ Th e multiplying factor is \(\cfrac { 48 }{ 28 } \) .
Step 2:
Also, more hours means less days. So, it is an inverse variation.
ஃ The multiplying factor is \(\cfrac { 7 }{ 8 } \) .
Step 3:
\(\therefore x=24\times \cfrac { 48 }{ 28 } \times \cfrac { 7 }{ 8 } =36days\) 
Formula Method:
Here, P₁= 48, D₁= 24, H₁ = 7 and W₁ = 1(why?)
P₂ = 28,D₂ = x,H₂ = 8 and W₂ = 1(why?)
Using the formula,\(\cfrac { { P }_{ 1 }\times { D }_{ 1 }\times { H }_{ 1 } }{ { W }_{ 1 } } =\cfrac { { P }_{ 2 }\times { D }_{ 2 }\times { H }_{ 2 } }{ { W }_{ 2 } } \) 
We have,\(\cfrac { 48\times 24\times 7 }{ 1 } =\cfrac { 28\times \times \times 8 }{ 1 } \) 
\(\Rightarrow x=\cfrac { 48\times 24\times 7 }{ 28\times 8 } =36days\)

Proportion Method:

Here, the unknown is the days (x). It is to be compared with the length and the women.
Step 1:
Here, more length means more days. So, it is a direct proportion.
∴ The proportion is 180 : 512 :: 12 : x  ➝1
Step 2:
Also, more women means less days. So, it is an inverse proportion
∴The proportion is 32 : 15 :: 12 : x➝2
Step 3:
Combining 1 and 2
\(\begin{matrix} 180:512 \\ 32:15 \end{matrix}\} ::12:x\) 
We know that the product of the extremes = the product of the means

So,\(180\times 32\times x=512\times 12\times 15\Rightarrow x\cfrac { 512\times 12\times 15 }{ 180\times 32 } =16days\) 
Here, the unknown is the days (x). It is to be compared with the length and the women
Step 1:
Here, more length means more days. So, it is a direct proportion.
ஃ The multiplying factor is \(\cfrac { 512 }{ 180 } \) (take the reciprocal).
Step 2:
Also, more women means less days. So, it is an inverse proportion.
ஃ The multiplying factor is \(\cfrac { 15 }{ 32 } \) (no change)
Step 3:
\(\therefore x=12\times \cfrac { 512 }{ 180 } \times \cfrac { 15 }{ 32 } =16\ days\) 
Formula Method:
Here, P₁ = 15,D₁ = 12 and W₁ = 180
P₂ = 32,D₂ = x and W₂ = 512
Using the formula,\(\cfrac { { P }_{ 1 }\times { D }_{ 1 } }{ { W }_{ 1 } } =\cfrac { { P }_{ 2 }\times { D }_{ 2 } }{ { W }_{ 2 } } \) 
We have,\(\cfrac { 15\times 12 }{ 180 } =\cfrac { 32\times x }{ 512 } \) 
\(\Rightarrow 1=\cfrac { 32\times x }{ 512 } \Rightarrow x=\cfrac { 512 }{ 32 } =16days\)
Remark: Students may answer in any of the three given methods dealt here.

By prime factorisation, we have 27000 = 2 x 2 x 2 x 3 x 3 x 3 x 5 x 5 x 5
\(\therefore \sqrt [ 3 ]{ 2700 } =2\times 3\times 5=30\)

By prime factorization, we have \(400=\underbrace { 2\times 2\times 2 } \times 2\times 5\times \)
There is only one triplet. To make further triplets, we will need two more 2’s and one more 5.
Therefore, 400 is not a perfect cube.

We find 250 = 5 x 5 x 5 x 2
= 5² x 5 x 2
Here, the prime factors 5 and 2 do not have pairs.
Therefore, we can either divide 250 by 10 (5 x 2) or multiply 250 by 10.
(i) If we multiply 250 by 10, we get 2500 = 5² × 5 × 2 × 5 × 2 and therefore the square root for 2500 would be 5 x 5 x 2 = 50.
(ii) If we divide 250 by 10 we get 25 and in that case we get \(\sqrt { 25 } =\sqrt { { 5 }^{ 2 } } =5\)

First, resolve the given number into prime factors. Group the identical factors in pairs and then take one from them to find the square root.

Now, 324 = 2 x 2 x 3 x 3 x 3 x 3
= 2² x 3² x 3 
= (2 x 3 x 3)²
\(\therefore \sqrt { 324 } =\sqrt { \left( 2\times 2\times 3 \right) ^{ 2 } } \) 
= 2 x 3 x 3
= 18

We have, 8² = 8 × 8 = 64, 15² = 15 × 15 = 225 and 17² = 17 × 17 = 289
We find, 8² + 15² = 64 + 225 = 289 = 17²
Thus, 8² + 15² = 172
∴ (8, 15, 17) is a Pythagorean triplet

Solution:
Here, there are 50 odd numbers from 1 to 99.
The sum of the fi rst n consecutive odd natural numbers = n²
The sum of the fi rst 50 consecutive odd natural numbers = 50²
= 50 × 50 = 2500