# Linear Equations

## Exercise 2.4 Part 2

Question 6: There is a narrow rectangular plot, reserved for a school, in Mahuli village. The length and breadth of the plot are in the ratio 11:4. At the rate Rs100 per metre it will cost the village panchayat Rs 75000 to fence the plot. What are the dimensions of the plot?

**Solution:** Cost of fence per meter = Rs 100

Total cost of fence the plot = Rs 75000

Given, ratio of length and breadth of the rectangular plot = 11:4

Let the length of the plot `=11x`

And the breadth of the plot `=4x`

∵ Rs. 100 is the cost to fence the plot of 1 meter

∴ Rs. 1 will be the cost to fence the plot of `(1)/(100)` meter

So, Rs. 75000 will be the cost of fencing `1/(100)xx75000=750` m

Hence, perimeter of plot = 750 m

We know that Perimeter = 2(length + breadth)

Or, `750m=2(11x+4x)`

Or, `750m=2(15x) = 30x`

After dividing both sides by 30, we get:

`(750)/(30)m=(30x)/(30)`

Or, `25m=x`

Or, `x=25m`

By substituting the value of x the length and breadth can be calculated as follows:

Length `=11x=11xx25=275m`

Breadth `=4x=4xx25=100m`

Thus, length = 275 m and Breadth = 100 m *Answer*

Question 7: Hasan buys two kinds of cloth materials for school uniforms, shirt material that costs him Rs 50 per metre and trouser material that costs him Rs 90 per metre. For every 3 meters of the shirt material he buys 2 metres of the trouser material. He sells the materials at 12% and 10% profit respectively. His total sale is Rs 36,600. How much trouser material did he buy?

**Solution:** Given, Rate of shirt material = Rs 50 per meter

Rate of trouser material = Rs 90 per meter

Profit on shirt material = 12%

Therefore, sale price of shirting material = cost price + 12% of cost price

`=50+50xx(12)/(100)=50+6` = Rs.56

Profit on trouser material = 10%

Since, profit on the cost price of trouser material = 10%

Therefore, sale price of trouser material = cost price of trouser material + 10% of cost price

`=90+90xx(10)/(100)=50+9` = Rs. 99

Total sale price = Rs 36600.00

Since Hasan buys 3m of shirt material for every 2 m of trouser material

So, let us assume that he buys `3x` m of shirting material and `2x` m of trousers material

Total sale price = Total SP of shirting material + Total SP of trouser material

Or, `36600=3x\xx\56+2x\xx\99`

Or, `36600=168x+198x`

Or, `36600=366x`

After dividing both sides by 366 we get:

`(36600)/(366)=(366x)/(366)`

Or, `x=100`

Since, purchase of trouser material `=2x`

So, after substituting the value of x, we get

Purchase of trouser material `=2xx100=200` m

Thus, Hasan buys 200 m of trouser material.

Question 8: Half of a herd of deer are grazing in the field and three fourths of the remaining are playing nearby. The rest 9 are drinking water from the pond. Find the number of deer in the herd.

**Solution:** Let the total number of deer = x

Number of deer grazing in the field `=x/2`

Number of deer playing nearby `=x/2xx3/4=(3x)/(8)`

Number of deer drinking water = 9

Now, total number of deer

`x=x/2+(3x)/(8)+9`

Or, `x=(4x+3x)/(8)+9=(7x)/(8)+9`

Now, after transposing `(7x)/(8)` to LHS we get:

`x-(7x)/(8)=9`

Or, `(8x-7x)/(8)=9`

Or, `x/8=9`

Now, after multiplying both sides by 8 we get:

`x/8xx8=9xx8`

Or, `x=72`

Question 9: A grandfather is ten times older than his granddaughter. He is also 54 years older than her. Find their present ages.

**Solution:** Let the age of granddaughter `=x`

As per question, age of grandfather `= 10x`

Moreover, as per question, age of grandfather `=x+54`

Therefore, `10x=x+54`

By transposing x to LHS we get:

`10x-x=54`

Or, `9x=54`

After dividing both sides by 9 we get:

`(9x)/(9)=(54)/(9)`

Or, `x=6`

Thus, age of granddaughter = 6 year

Age of grandfather `=6xx10=60` year

Question 10: Aman’s age is three times his son’s age. Ten years ago he was five times his son’s age. Find their present ages.

**Solution:** Let the age of Aman’s son `=x`

Therefore, age of Aman `=3x`

Ten years ago,

Present age of Aman – 10 year = (present age of his son – 10)5

Or, `3x-10=(x-10)5`

Or, `3x-10=5x-50`

By transposing 5x to LHS and -10 to RHS we get:

`3x-5x=-50+10`

Or, `-2x=-40`

After canceling the negative sign on both sides we get:

`2x=40`

After dividing both sides by 2 we get:

`(2x)/(2)=(40)/(2)`

Or, `x=20` year

Thus,present age of Aman's son=20 year

And present age of Aman`=20xx3=60` year