**Linear Equations In One Variable**

**Question 1.****Amina thinks of a number and subtracts from it. She multiplies the result by 8. The result now obtained is 3 times the same number she thought of. What is the number ?**

**Solution.**Let Amina thinks of a number x. It is given that

Hence, the number thought is 4.

**Question 2.****A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers ?**

**Solution.**

Let the numbers be x and 5x. it is given that

Hence, the required numbers are 7 and 35.

**Question 3.****Sum of the digits of a two-digit number is 9. When we interchange the digits, it is found that the resulting new number is greater than the original number by 27. What is the two-digit number ?**

**Solution.**

Let the unit’s digit be x.

∴ Unit’s digit = 6 and ten’s digit = 3

Hence, the number = 36.

**Question 4.****One of the two digits of a two digit number is three times the other digit. If you interchange the digits of this two-digit number and add the resulting number to the original number, you get 88. What is the original number ?**

**Solution.**

Let one’s digit be x. Then, ten’s digit 3x.

**Question 5.****Shobo’s mother’s present age is six times Shobo’s present age. Shobo’s age five years from now will be one third of his mother’s present age. What are their present ages ?**

**Solution.**

Let the present age of Shobo be x years. Then,

∴ Present age of Shobo = 5 years ,

∴ Present age of his mother = 30 years.

**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 of ₹ 100 per metre it will cost the village panchayat ₹ 75000 to fence the plot. What are the dimensions of the plot ?**

**Solution.**Let the length and breadth of the rectangular plot be 11x and 4x metres. Then,

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

**Solution.**Let material bought for trousers and shirts be 2x metres and 3x metres respectively.

Cost price of 2x metres of trouser material @ ₹ 90 per metre = ₹ (90 x 2x) = ₹ 180x

Cost price of 3x metres of shirt material @ ₹ 50 per metre = ₹ (50 x 3x) = ₹ 150x

**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 number of deer in the herd be x. Then,

**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 present age of grand daughter be x years. Then, the present age of grandfather is 10x years

According to the question.

10x – x = 54 ⇒ 9x = 54

⇒

∴ Granddaughter’s age = 6 years

and grandfather’s age = 10 x 6 = 60 years.

**Question 10.****Am.an’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 present age of son be x years. Then, the age of his father Aman is 3x.

**10 years ago :**Their ages will be (x – 10) years and (3x – 10) years respectively.